Asymptotic expansion for groupoids and Roe-type algebras

Xulong Lu; Qin Wang; Jiawen Zhang

doi:10.4171/jncg/666

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Asymptotic expansion for groupoids and Roe-type algebras

Xulong Lu
East China Normal University, Shanghai, P. R. China
- zbMATH
- ORCID
Qin Wang
East China Normal University, Shanghai, P. R. China
Jiawen Zhang
Fudan University, Shanghai, P. R. China

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Abstract

In this paper, we introduce a notion of expansion for groupoids, which recovers the classical notion of expander graphs by a family of pair groupoids and expanding actions in measure by transformation groupoids. We also consider an asymptotic version for expansion and establish structural theorems, showing that asymptotic expansion can be approximated by domains of expansions. On the other hand, we introduce dynamical propagation and quasi-locality for operators on groupoids and the associated Roe-type algebras. Our main results characterise when these algebras possess block-rank-one projections by means of asymptotic expansion, which generalises the crucial ingredients in previous works to provide counterexamples to the coarse Baum–Connes conjecture.

1. Introduction

Over the last few decades, the phenomenon of expansion has been discovered and extensively studied across various branches of mathematics. For instance, in graph theory, the expansion phenomenon leads to the notion of expander graphs, which plays an important role not only in pure and applied mathematics but also in theoretical computer science (see, e.g., [24

A. Lubotzky, Expander graphs in pure and applied mathematics. Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 1, 113–162 Zbl 1232.05194 MR 2869010

]). In dynamics, the expansion phenomenon leads to the notion of expanding actions in measure, which turns out to be equivalent to the classic notion of spectral gap for measure-preserving actions (see [32

F. Vigolo, Measure expanding actions, expanders and warped cones. Trans. Amer. Math. Soc. 371 (2019), no. 3, 1951–1979 Zbl 1402.05205 MR 3894040

]), and numerous examples have been discovered (e.g., [5

Y. Benoist and N. de Saxcé, A spectral gap theorem in simple Lie groups. Invent. Math. 205 (2016), no. 2, 337–361 Zbl 1357.22003 MR 3529116

, 6

J. Bourgain and A. Gamburd, On the spectral gap for finitely-generated subgroups of

SU (2)

. Invent. Math. 171 (2008), no. 1, 83–121 Zbl 1135.22010 MR 2358056

, 13

A. Gamburd, D. Jakobson, and P. Sarnak, Spectra of elements in the group ring of

SU (2)

. J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 51–85 Zbl 0916.22009 MR 1677685

]).

Recently, an asymptotic version of expansion was introduced in different areas of mathematics [20

K. Li, P. W. Nowak, J. Špakula, and J. Zhang, Quasi-local algebras and asymptotic expanders. Groups Geom. Dyn. 15 (2021), no. 2, 655–682 Zbl 1484.46057 MR 4303336

, 23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

], which is more stable under small perturbations and hence leads to important applications in operator algebras and higher index theory [17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, 21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, 22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

]. A crucial step therein is structural type theorems, showing that objects with asymptotic expansion can be approximated by those with expansion. In dynamics, asymptotic expansion also provides a new quantitative viewpoint on the classic notion of strong ergodicity, introduced in [8

A. Connes and B. Weiss, Property

T

and asymptotically invariant sequences. Israel J. Math. 37 (1980), no. 3, 209–210 Zbl 0479.28017 MR 0599455

, 29

K. Schmidt, Asymptotically invariant sequences and an action of

SL (2, Z)

on the

2 -

sphere. Israel J. Math. 37 (1980), no. 3, 193–208 Zbl 0485.28018 MR 0599454

, 30

K. Schmidt, Amenability, Kazhdan’s property

T,

strong ergodicity and invariant means for ergodic group-actions. Ergodic Theory Dynam. Systems 1 (1981), no. 2, 223–236 Zbl 0485.28019 MR 0661821

] in relation to the Ruziewicz problem, Kazhdan’s property (T) and amenability.

In this paper, we aim to generalise and unify the theory of asymptotic expansion from different areas (including graph theory and dynamical systems) in the language of groupoids. Groupoids provide a framework encompassing both groups and spaces. They arise naturally in a variety of research areas such as dynamical systems, topology and geometry, geometric group theory and operator algebras, building bridges between all these areas of mathematics (see, e.g., [7

R. Brown, From groups to groupoids: a brief survey. Bull. Lond. Math. Soc. 19 (1987), no. 2, 113–134 Zbl 0612.20032 MR 0872125

, 14

P. J. Higgins, Notes on categories and groupoids. Van Nostrand Reinhold Math. Stud. 32, Van Nostrand Reinhold, London-New York-Melbourne, 1971, 178 pp. Zbl 0226.20054 MR 0327946

]).

To achieve this, we introduce the notion of expansion and asymptotic expansion for groupoids, generalising both (asymptotic) expander graphs and (asymptotically) expanding actions in measure, and establish structural type theorems in the groupoid setting. Furthermore, we introduce two classes of operator algebras associated to the dynamics of groupoids, generalising the classical Roe and quasi-local algebras from higher index theory. Our main results show that the existence of certain projection operators in these operator algebras characterises asymptotic expansion of groupoids, which provide a unified approach to results in [17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, 21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, 22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

] and allow a boarder range of examples and applications.

We will now give a more detailed overview of this work.

1.1. Expansion and asymptotic expansion

To motivate our notion of asymptotic expansion, let us first recall the notion of asymptotically expanding actions in measure from [23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

].

Let

ρ : Γ ↷ X

be a measure-class-preserving action of a countable group

Γ

acting on a probability space

(X, μ)

and

ℓ_{Γ}

be a proper length function on

Γ

(i.e., for each

L > 0,

the ball

B_{L} ≔ {γ \in Γ ∣ ℓ_{Γ} (γ) \leq L}

is finite). The action is called asymptotically expanding in measure $μ$ if for any

α \in (0, \frac{1}{2}]

there exists

C_{α}, L_{α} > 0

such that for any measurable

A \subseteq X

with

α \leq μ (A) \leq \frac{1}{2},

we have

μ (B_{L_{α}} \cdot A) > (1 + C_{α}) μ (A) .

When the functions

α \mapsto L_{α}

and

α \mapsto C_{α}

can be taken to be constant functions, then the action is called expanding in measure $μ .$

To generalise the above to groupoids (see Section 2.1 for definitions), we consider the associated transformation groupoid

X ⋊ Γ

with source

s (x, γ) = γ^{- 1} x,

range

r (x, γ) = x

and inverse

{(x, γ)}^{- 1} = (γ^{- 1} x, γ^{- 1}) .

The length function

ℓ_{Γ}

naturally gives rise to a length function

ℓ

on

X ⋊ Γ

by

ℓ (x, γ) ≔ ℓ_{Γ} (γ)

for

x \in X

and

γ \in Γ .

This leads to the following setting of our paper: Let

𝒢

be a groupoid with a length function

ℓ

(see Definition 2.1) and the unit space

𝒢^{(0)}

be equipped with a measure

μ

on some

σ -

algebra

ℛ .

To abstract measure-class-preserving transformations, we introduce the following: A bisection

K \subseteq 𝒢

is called admissible if its source

s (K)

and range

r (K)

are measurable, the induced bijection

(1.1)

τ_{K} ≔ {r |}_{K} \circ {({s |}_{K})}^{- 1} : s (K) \to r (K)

is a measure-class-preserving measurable isomorphism and the length

ℓ (K) ≔ sup {ℓ (x) ∣ x \in K}

is finite. Moreover, a subset

K \subseteq 𝒢

is called decomposable if

K = ⋃_{i = 1}^{N} K_{i}

for

N \in ℕ

and admissible bisections

K_{i} .

Note that for transformation groupoids, any measurable subset (with respect to the product structure) with finite length is decomposable, while unfortunately, this does not hold in general.

Now we are in the position to introduce our notion of asymptotic expansion.

Definition A

(Definitions 3.1 and 3.2). Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a probability measure space. We say that

𝒢

is asymptotically expanding (in measure $μ$ ) if for any

α \in (0, \frac{1}{2}],

there exists a decomposable

K_{α} \subseteq 𝒢

with

ℓ (K_{α}) < \infty

such that for any

A \in ℛ

with

α \leq μ (A) \leq \frac{1}{2},

then

μ (r (K_{α} \cdot A) ∖ A) > C_{α} μ (A) .

If

K_{α} \equiv K,

then we say that

𝒢

is expanding (in measure $μ$ ).

It is clear that in the case of transformation groupoids, this recovers the notion of (asymptotically) expanding actions (see Section 6.1). On the other hand, when considering pair groupoids, a family version of Definition A recovers the notion of (asymptotic) expander graphs (see Section 6.2 for details).

We establish the following structure theorem for asymptotic expansion, showing that it can be approximated by domains of expansion.

Theorem B

(Theorems 3.9 and 3.18). Let $𝒢$ be a groupoid with a length function $ℓ$ and $(𝒢^{(0)}, ℛ, μ)$ be a probability measure space. Then the following are equivalent:

(1)

$𝒢$ is asymptotically expanding in measure.
(2)

$𝒢^{(0)}$ admits an exhaustion by domains of expansion with bounded ratio.
(3)

$𝒢^{(0)}$ admits an exhaustion by domains of Markov expansion with bounded ratio.

Roughly speaking, here the “domain of expansion” is a measurable subset

Y

of

𝒢^{(0)}

such that the reduction

𝒢_{Y}^{Y}

is expanding (see Definition 3.3), and “exhaustion” means a sequence of domains

Y_{n}

such that

μ (𝒢^{(0)} ∖ Y_{n}) \to 0 .

We also construct a Markov kernel for each domain (see Definition 3.14) and consider a Markovian version of expansion, which will play a key role later to produce certain projection operators. Details can be found in Sections 3.2 and 3.3.

Theorem B provides a unified approach for the structure results in [17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, 21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, 22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

] and also allows new examples (see Section 6). Even for asymptotic expander graphs and asymptotically expanding actions, our proof simplifies the original ones in a systematic way.

1.2. Dynamical propagation and quasi-locality

Now we introduce two classes of operator algebras associated to groupoids, which encode dynamical information and have their roots in higher index theory.

Recall that for a metric space, the Roe and the quasi-local algebra were introduced by Roe in his pioneering work on higher index theory [27

J. Roe, An index theorem on open manifolds. I. J. Differential Geom. 27 (1988), no. 1, 87–113 Zbl 0657.58041 MR 0918459

], where he discovered that higher indices of elliptic differential operators on open manifolds belong to

K -

theories of the Roe algebra (see also Engel’s work [10

A. Engel, Index theorems for uniformly elliptic operators. New York J. Math. 24 (2018), 543–587 Zbl 1401.58009 MR 3855638

, 11

A. Engel, Rough index theory on spaces of polynomial growth and contractibility. J. Noncommut. Geom. 13 (2019), no. 2, 617–666 Zbl 1436.58019 MR 3988758

]).

To compute their

K -

theories, a practical approach is to consult the coarse Baum–Connes conjecture [4

P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and

K -

theory of group

C^{*} -

algebras. In $C^{*} -$ algebras: 1943–1993 (San Antonio, TX, 1993), pp. 240–291, Contemp. Math. 167, American Mathematical Society, Providence, RI, 1994 Zbl 0830.46061 MR 1292018

], a central conjecture in higher index theory and closely related to other conjectures like the Novikov conjecture and the Gromov–Lawson conjecture. Unfortunately, counterexamples were discovered in [15

N. Higson, V. Lafforgue, and G. Skandalis, Counterexamples to the Baum–Connes conjecture. Geom. Funct. Anal. 12 (2002), no. 2, 330–354 Zbl 1014.46043 MR 1911663

, 17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, 22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

, 28

D. Sawicki, Warped cones violating the coarse Baum–Connes conjecture. Preprint, 2017

] using (asymptotic) expanders and (asymptotically) expanding actions. Due to their importance, Roe and quasi-local algebras have been extensively studied, and several variants have also been introduced (see [3

H. Bao, X. Chen, and J. Zhang, Strongly quasi-local algebras and their

K -

theories. J. Noncommut. Geom. 17 (2023), no. 1, 241–285 Zbl 1523.46055 MR 4565434

, 9

T. de Laat, F. Vigolo, and J. Winkel, Dynamical propagation and Roe algebras of warped space. J. Operator Theory 95 (2026), no. 1, 159–188 Zbl 08171289 MR 5040752

, 16

B. Jiang, J. Zhang, and J. Zhang, Quasi-locality for étale groupoids. Comm. Math. Phys. 403 (2023), no. 1, 329–379 Zbl 1534.46046 MR 4645718

, 22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

, 25

N. Ozawa, Embeddings of matrix algebras into uniform Roe algebras and quasi-local algebras. J. Eur. Math. Soc. (JEMS) 2025, 10.4171/JEMS/1672

, 31

J. Špakula and J. Zhang, Quasi-locality and Property A. J. Funct. Anal. 278 (2020), no. 1, article no. 108299, 25 pp. Zbl 1444.46016 MR 4027745

, 34

R. Willett and G. Yu, Higher index theory. Cambridge Stud. Adv. Math. 189, Cambridge University Press, Cambridge, 2020, 581 pp. Zbl 1471.19001 MR 4411373

, 35

G. Yu, The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139 (2000), no. 1, 201–240 Zbl 0956.19004 MR 1728880

]).

Inspired by these works, we introduce the following notions of dynamical propagation and quasi-locality for operators on groupoids. Note that in the context of transformation groupoids and pair groupoids, the following recovers the original (dynamical) Roe and quasi-local algebras (see Sections 6.1 and 6.2).

Definition C

(Definitions 4.1 and 4.2). Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a measure space. For an operator

T \in 𝔅 (L^{2} (𝒢^{(0)}, μ)),

we say:

(1)

$T$ has finite dynamical propagation if there exists a unital decomposable $K \subseteq 𝒢$ with $ℓ (K) < \infty$ such that for any $A, B \in ℛ$ with $μ (r (K \cdot A) \cap B) = 0,$ then $χ_{A} T χ_{B} = 0 .$
(2)

$T$ is dynamically quasi-local if for any $ε > 0,$ there exists a unital decomposable $K_{ε} \subseteq 𝒢$ with $ℓ (K_{ε}) < \infty$ such that for any $A, B \in ℛ$ with $μ (r (K_{ε} \cdot A) \cap B) = 0,$ then $∥ χ_{A} T χ_{B} ∥ < ε .$

The dynamical Roe algebra of $𝒢$ is the norm closure of all operators with finite dynamical propagation, denoted by

𝐂_{dyn}^{*} (𝒢) .

The dynamical quasi-local algebra of $𝒢$ is the set of all dynamically quasi-local operators, denoted by

𝐂_{dyn, q}^{*} (𝒢) .

Recall that in graph theory and dynamics, a key consequence of asymptotic expansion is that the Roe-type algebras possess block-rank-one projections (called the ghost projections), which are crucial to provide counterexamples to the coarse Baum–Connes conjecture (see [17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, 21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, 22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

]). In the context of groupoids, we study dynamical propagation and quasi-locality of block-rank-one projections and establish a more general result, which unifies and simplifies the previous ones.

More precisely, we first consider a rank-one projection

P \in 𝔅 (L^{2} (𝒢^{(0)}, μ))

and a unit vector

ξ \in L^{2} (𝒢^{(0)}, μ)

in the range of

P .

This induces a probability measure

ν

on

𝒢^{(0)}

given by

d ν (x) ≔ {| ξ (x) |}^{2} d μ (x)

for

x \in 𝒢^{(0)},

called the associated measure.

The following is our main result.

Theorem D

(Theorem 5.3). Suppose $𝒢$ is a groupoid with a length function $ℓ$ and $(𝒢^{(0)}, ℛ, μ)$ is a measure space. Let $P \in 𝔅 (L^{2} (𝒢^{(0)}, μ))$ be a rank-one projection, and $ν$ the associated probability measure on $𝒢^{(0)} .$ Then the following are equivalent:

(1)

$P \in 𝐂_{dyn}^{*} (𝒢) .$
(2)

$P \in 𝐂_{dyn, q}^{*} (𝒢) .$
(3)

$𝒢$ is asymptotically expanding in measure $ν .$

The proof of Theorem D is quite involved. Firstly, we reduce to averaging projections by changing measures. Applying Theorem B, we obtain an exhaustion by domain of Markov expansion, and on each domain, we obtain a rank-one projection in the dynamical Roe algebra by functional calculus. Finally, these projections converge in norm to the averaging projection, and we finish the proof.

After establishing Theorem D, we further consider a family version by chasing parameters. We obtain a family version of Theorem D (Theorem 5.3′), which characterises when the dynamical Roe and quasi-local algebras possess block-rank-one projections. This recovers the most technical results in [17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, 21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, 22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

].

Finally, we apply our results to several examples, including the known results on asymptotic expanders and asymptotically expanding actions. Moreover, we study new examples including groupoid actions on fibred spaces, the HLS groupoid and graph groupoids, and establish corresponding results therein.

1.3. Organisation

In Section 2, we recall background knowledge for groupoids and Markov kernels. In Section 3, we introduce the notion of (asymptotic) expansion for groupoids and establish the structure result (Theorem B). In Section 4, we introduce dynamical propagation and quasi-locality and relate asymptotic expansion to the quasi-locality of the averaging projection (Proposition 4.7). Then in Section 5, we accomplish the proof of Theorem D, making use of Theorem B. Finally, in Section 6, we explain in detail how our results recover the original ones for asymptotic expanders (by a family version of pair groupoids) and asymptotically expanding actions (by transformation groupoids) and provide new examples.

2. Preliminaries

In this section, we recall the notions of groupoids and Markov kernels.

2.1. Basic notions for groupoids

Recall that a groupoid

𝒢

is a small category which consists of a set

𝒢,

a subset

𝒢^{(0)}

called the unit space, two maps

s, r : 𝒢 \to 𝒢^{(0)}

called the source and the range maps, respectively, a composition law

(γ_{1}, γ_{2}) \mapsto γ_{1} γ_{2}

for

(γ_{1}, γ_{2}) \in 𝒢^{(2)} ≔ {(γ_{1}, γ_{2}) \in 𝒢 \times 𝒢 ∣ s (γ_{1}) = r (γ_{2})}

and an inverse map

γ \mapsto γ^{- 1} .

These operations satisfy a couple of axioms, including the associativity law and the fact that

𝒢^{(0)}

acts as units. For

A, B \subseteq 𝒢,

we denote

A^{- 1} ≔ {γ^{- 1} ∣ γ \in A}

and

A \cdot B ≔ {γ \in 𝒢 ∣ γ = γ_{1} γ_{2}, where γ_{1} \in A, γ_{2} \in B with s (γ_{1}) = r (γ_{2})} .

We say that

A

is symmetric if

A = A^{- 1}

and unital if

𝒢^{(0)} \subseteq A .

A subset

A \subseteq 𝒢

is called a bisection if the restrictions of

s, r

to

A

are injective, and recall from (1.1) that we have the induced bijection

τ_{A} ≔ {r |}_{A} \circ {({s |}_{A})}^{- 1} : s (A) \to r (A) .

Definition 2.1.

Let

𝒢

be a groupoid. A length function on

𝒢

is a map

ℓ : 𝒢 \to ℝ^{+}

such that

ℓ (𝒢^{(0)}) = {0},

ℓ (γ^{- 1}) = ℓ (γ)

for all

γ \in 𝒢

and

ℓ (γ_{1} γ_{2}) \leq ℓ (γ_{1}) + ℓ (γ_{2})

for all

γ_{1}, γ_{2} \in 𝒢

with

s (γ_{1}) = r (γ_{2}) .

Throughout the paper, let

𝒢

be a groupoid with length function

ℓ

and

𝒢^{(0)}

equipped with a (not necessarily finite) measure

μ

on some

σ -

algebra

ℛ .

Definition 2.2.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a measure space. A bisection

K \subseteq 𝒢

is called admissible if

s (K), r (K) \in ℛ,

τ_{K} : s (K) \to r (K)

is a measure-class-preserving measurable isomorphism and

ℓ (K) ≔ sup {ℓ (x) ∣ x \in K}

is finite.

We record the following, whose proof is straightforward and hence omitted.

Lemma 2.3.

If $K, K_{1}, K_{2} \subseteq 𝒢$ are admissible bisections, then $K^{- 1}$ and $K_{1} \cdot K_{2}$ are also admissible bisections.

Motivated by the case of group actions explained in Section 1, we introduce the following.

Definition 2.4.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a measure space. A subset

K \subseteq 𝒢

is called decomposable if

K = ⋃_{i = 1}^{N} K_{i}

for

N \in ℕ

and admissible bisections

K_{i} .

Given a decomposable

K \subseteq 𝒢,

a decomposition

K = ⋃_{i = 1}^{N} K_{i}

is called unital if

K_{i} = 𝒢^{(0)}

for some

i

and symmetric if there exists a bijection

σ : {1, 2, \dots, N} \to {1, 2, \dots, N}

such that

K_{i}^{- 1} = K_{σ (i)} .

We say that

K

is unital symmetric $N -$ decomposable if there exists a unital symmetric decomposition

K = ⋃_{i = 1}^{N} K_{i}

for admissible

K_{i} .

Applying Lemma 2.3, we obtain the following.

Lemma 2.5.

If $K$ is unital symmetric $N -$ decomposable with $ℓ (K) \leq L,$ then $K^{m}$ is unital symmetric $N^{m} -$ decomposable with $ℓ (K^{m}) \leq m L$ for any $m \in ℕ .$

The following observation is useful to define expansion for groupoids later.

Lemma 2.6.

For decomposable $K \subseteq 𝒢$ and measurable $A \subseteq 𝒢^{(0)},$ $r (K \cdot A)$ is measurable.

Proof.

By definition, we can write

K = ⋃_{i = 1}^{N} K_{i}

for admissible bisection

K_{i} .

Then

r (K \cdot A) = r (⋃_{i = 1}^{N} K_{i} \cdot A) = ⋃_{i = 1}^{N} τ_{K_{i}} (A \cap s (K_{i})),

which is measurable since each

τ_{K_{i}}

is a measurable isomorphism on

s (K_{i}) .

2.2. Markov kernels

Here, we recall a few elementary properties of reversible Markov kernels. We refer to the first chapters of [26

D. Revuz, Markov chains, 2nd edn., North-Holland Math. Libr. 11, North-Holland, Amsterdam, 1984, 374 pp. Zbl 0539.60073 MR 0758799

] for more details.

Definition 2.7.

Let

𝒮

be a

σ -

algebra on a set

X .

A Markov kernel on the measurable space

(X, 𝒮)

is a function

Π : X \times 𝒮 \to [0, 1]

such that:

(1)

For every $x \in X,$ the function $Π (x, -) : 𝒮 \to [0, 1]$ is a probability measure.
(2)

For every $A \in 𝒮,$ the function $Π (-, A) : X \to [0, 1]$ is $𝒮 -$ measurable.

The associated Markov operator

𝔓

is a linear operator on the space of bounded

𝒮 -

measurable functions, defined by

(2.1)

𝔓 f (x) ≔ \int_{X} f (y) Π (x, d y) .

Definition 2.8.

Given a measure

μ

on

(X, 𝒮)

and

A \in 𝒮,

the ( $μ -$ )size of the boundary of

A

(with respect to

Π

) is defined as

{| \partial_{Π} (A) |}_{μ} ≔ \int_{A} Π (x, X ∖ A) d μ (x) .

We are only interested in the special case of reversible Markov kernels.

Definition 2.9.

A Markov kernel

Π

is called reversible if there exists a measure

m

on

(X, 𝒮)

such that

\int_{X} f (x) 𝔓 g (x) d m (x) = \int_{X} 𝔓 f (x) g (x) d m (x)

for every pair of measurable bounded functions

f, g : X \to ℝ .

The measure

m

is called a reversing measure for

Π

(note that

m

need not be unique in general). In this case, we also say that

Π

is a reversible Markov kernel on $(X, m) .$

Given a reversing measure

m

on

X,

the Markov operator

𝔓

can be regarded as a bounded self-adjoint operator on

L^{2} (X, m)

with

∥ 𝔓 ∥ \leq 1 .

Define the Laplacian of

Π

as

Δ ≔ 1 - 𝔓,

which is positive self-adjoint with spectrum contained in

[0, 2] .

Let

Π

be a reversible Markov kernel on

(X, m),

where

m

is a finite measure. Then all constant functions on

X

belong to

L^{2} (X, m)

and are fixed by

𝔓 .

It follows that

∥ 𝔓 ∥ = 1

and

1

belongs to the spectrum of

𝔓 .

Denote the orthogonal complement of the constant functions in

L^{2} (X, m)

by

L_{0}^{2} (X, m),

that is,

L_{0}^{2} (X, m) ≔ {f \in L^{2} (X, m) | \int_{X} f (x) d m (x) = 0} .

Note that

L_{0}^{2} (X, m)

is

𝔓 -

invariant and that the spectrum of the restriction of

𝔓

on

L_{0}^{2} (X, m)

is contained in

[- 1, 1] .

We denote the supremum of this spectrum by

λ \in ℝ .

We make the following definition.

Definition 2.10.

A reversible Markov kernel on a finite measure space

(X, m)

is said to have a spectral gap if

λ < 1 .

On the other hand, we recall the notion of the Cheeger constant as follows.

Definition 2.11.

The Cheeger constant for a reversible Markov kernel

Π

on a finite measure space

(X, m)

is defined to be

κ ≔ inf {\frac{{| \partial_{Π} (A) |}_{m}}{m (A)} | A \in 𝒮, 0 < m (A) \leq \frac{1}{2} m (X)} .

Consequently, we have the following significant result relating the spectral gap to the Cheeger constant from [19

G. F. Lawler and A. D. Sokal, Bounds on the

L^{2}

spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 (1988), no. 2, 557–580 Zbl 0716.60073 MR 0930082

] (see also [22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

]).

Theorem 2.12

([19

G. F. Lawler and A. D. Sokal, Bounds on the

L^{2}

spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 (1988), no. 2, 557–580 Zbl 0716.60073 MR 0930082

, Theorem 2.1]). Let $Π$ be a reversible Markov kernel on $(X, m),$ where $m$ is finite. Then

\frac{κ^{2}}{2} \leq 1 - λ \leq 2 κ .

3. Asymptotic expansion in measure and structure theorems

In this section, we introduce the notion of expansion and asymptotic expansion for groupoids and then establish their structure theory. This generalises both (measured) asymptotic expanders in [17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, 20

K. Li, P. W. Nowak, J. Špakula, and J. Zhang, Quasi-local algebras and asymptotic expanders. Groups Geom. Dyn. 15 (2021), no. 2, 655–682 Zbl 1484.46057 MR 4303336

, 21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

] and asymptotic expansion in measure for group actions in [22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

, 23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

].

Again, we always assume that

𝒢

is a groupoid with a length function

ℓ

and

𝒢^{(0)}

is equipped with a measure

μ

on some

σ -

algebra

ℛ .

In this section, we require

μ

to be finite. For simplicity, further assume that

μ

is a probability measure. Note that all the results in this section are also available for finite measures by rescaling.

3.1. Expansion and asymptotic expansion

We first introduce the following.

Definition 3.1.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a probability measure space. We say that

𝒢

is expanding (in measure $μ$ ) if there exist

C, N, L > 0

and a unital symmetric¹

Since we are only interested in the existence of

K,

we can always make it unital and symmetric.

N -

decomposable

K \subseteq 𝒢

with

ℓ (K) \leq L

such that for any

A \in ℛ

with

0 < μ (A) \leq \frac{1}{2},

then

μ (r (K \cdot A) ∖ A) > C μ (A) .

In this case, we also say that

𝒢

is

(C, N, L) -

expanding.

We also consider the following asymptotic version of Definition 3.1.

Definition 3.2.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a probability measure space. We say that

𝒢

is asymptotically expanding (in measure $μ$ ) if for any

α \in (0, \frac{1}{2}],

there exist

C_{α}, N_{α}, L_{α} > 0

and a unital symmetric

N_{α} -

decomposable

K_{α} \subseteq 𝒢

with

ℓ (K_{α}) \leq L_{α}

such that for any

A \in ℛ

with

α \leq μ (A) \leq \frac{1}{2},

then

μ (r (K_{α} \cdot A) ∖ A) > C_{α} μ (A) .

The functions

α \mapsto C_{α},

α \mapsto N_{α}

and

α \mapsto L_{α}

are called expansion parameters of

𝒢

(which are not unique).

To establish the structure theorem, we also need the following notion.

Definition 3.3.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a probability measure space. A measurable subset

Y \subseteq 𝒢^{(0)}

is called a domain of asymptotic expansion²

This is equivalent to saying that the reduction

𝒢_{Y}^{Y} ≔ s^{- 1} (Y) \cap r^{- 1} (Y)

is asymptotically expanding.

if for any

α \in (0, \frac{1}{2}],

there exist

C_{α}, N_{α}, L_{α} > 0

and a unital symmetric

N_{α} -

decomposable

K_{α} \subseteq 𝒢

with

ℓ (K_{α}) \leq L_{α}

such that for any measurable

A \subseteq Y

with

α μ (Y) \leq μ (A) \leq \frac{1}{2} μ (Y),

we have

μ ((r (K_{α} \cdot A) ∖ A) \cap Y) > C_{α} μ (A) .

The functions

α \mapsto C_{α},

α \mapsto N_{α}

and

α \mapsto L_{α}

are called expansion parameters for

Y .

If

C_{α} \equiv C,

N_{α} \equiv N,

L_{α} \equiv L

and

K_{α} \equiv K,

then we also say that

Y

is a domain of $(C, N, L) -$ expansion (or simply a domain of $(N, L) -$ expansion or a domain of expansion).

Here, we collect several useful facts, generalising those in [21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, 23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

].

Lemma 3.4.

Let $Y \subseteq 𝒢^{(0)}$ be a domain of asymptotic expansion with expansion parameters $α \mapsto C_{α}, α \mapsto N_{α}, α \mapsto L_{α} .$ Then for any $α \in (0, \frac{1}{2}]$ and $β \in [\frac{1}{2}, 1),$ there exist $C, N, L > 0$ and a unital symmetric $N -$ decomposable $K \subseteq 𝒢$ with $ℓ (K) \leq L$ such that for any measurable $A \subseteq Y$ with $α μ (Y) \leq μ (A) \leq β μ (Y),$ we have

μ ((r (K \cdot A) ∖ A) \cap Y) > C μ (A) .

Here, $C, N, L$ only depend on $α, β$ and expansion parameters for $Y$ with $N = N_{α} + N_{\frac{1 - β}{2 β}},$ $L = \max {L_{α}, L_{\frac{1 - β}{2 β}}}$ and $C = \min {C_{α}, \frac{1 - β}{2 β} C_{\frac{1 - β}{2}}, \frac{1 - β}{2 β}} .$

The proof of Lemma 3.4 is similar to that of [23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

, Lemma 3.8] and hence omitted.

Lemma 3.5.

Assume $𝒢$ is asymptotically expanding in measure. Then for any $β \in (0, 1], α \in (0, \frac{1}{2}]$ and $C \in (0, 1),$ there exists a unital symmetric $N -$ decomposable $K \subseteq 𝒢$ with $ℓ (K) \leq L$ such that for any measurable $Y \subseteq 𝒢^{(0)}$ with $μ (Y) \geq β$ and measurable $A \subseteq Y$ with $α μ (Y) \leq μ (A) \leq \frac{1}{2} μ (Y),$ we have $μ ((r (K \cdot A) ∖ A) \cap Y) > C μ (A) .$ Here, $N, L$ only depend on $α, β, C$ and expansion parameters of $𝒢 .$

Proof.

Fix

α \in (0, \frac{1}{2}], β \in (0, 1],

C \in (0, 1)

and measurable

Y \subseteq 𝒢^{(0)}

with

μ (Y) \geq β .

Set

ε ≔ \frac{1 + C}{2} μ (Y)

and

γ ≔ μ (𝒢^{(0)} ∖ Y) + ε = 1 - \frac{1 - C}{2} μ (Y) .

It suffices to find a unital symmetric decomposable

K \subseteq 𝒢

such that for any measurable

A \subseteq Y

with

α μ (Y) \leq μ (A) \leq \frac{1}{2} μ (Y),

then

μ (r (K \cdot A)) > γ .

If this holds, then the proof would be completed by the following estimate:

μ (r (K \cdot A) \cap Y) > γ - μ (𝒢^{(0)} ∖ Y) = ε \geq \frac{2 ε}{μ (Y)} μ (A) = (1 + C) μ (A) .

Now we aim to find such a

K .

Since

𝒢

is asymptotically expanding, Lemma 3.4 provides

C^{'}, L^{'}, N^{'} > 0

and a unital symmetric

N^{'} -

decomposable

K^{'} \subseteq 𝒢

with

ℓ (K^{'}) \leq L^{'}

such that for any measurable

A^{'} \subseteq 𝒢^{(0)}

with

α β \leq μ (A^{'}) \leq 1 - \frac{1 - C}{2} β,

then

(3.1)

μ (r (K^{'} \cdot A^{'})) > (1 + C^{'}) μ (A^{'}) \geq (1 + C^{'}) α β .

Set

m

to be the minimal integer satisfying

{(1 + C^{'})}^{m} \geq \frac{1}{α β},

and take

K ≔ {(K^{'})}^{m} .

Then Lemma 2.5 shows that

K

is unital symmetric

{(N^{'})}^{m} -

decomposable and

ℓ (K) \leq m L^{'} .

Now for any measurable

A \subseteq Y

with

α μ (Y) \leq μ (A) \leq \frac{1}{2} μ (Y),

we need to show that

μ (r (K \cdot A)) > γ .

If not, then

μ (r ({(K^{'})}^{m} \cdot A)) \leq γ .

This shows that

α β \leq μ (r ({(K^{'})}^{i} \cdot A)) \leq γ \leq 1 - \frac{1 - C}{2} β for any i = 0, 1, \dots, m .

Applying inequality (3.1) inductively, we obtain

μ (r ({(K^{'})}^{m} \cdot A)) > {(1 + C^{'})}^{m} μ (A) \geq {(1 + C^{'})}^{m} α β \geq 1 > γ,

which leads to a contradiction.

3.2. The structure theorem

Here, we introduce the structure theorem for asymptotic expansion on groupoids. The main idea is to approximate by domains of expansions in the following sense.

Definition 3.6.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a probability measure space. We say that a sequence of measurable subsets

{Y_{n}}_{n \in ℕ}

in

𝒢^{(0)}

forms a (measured) exhaustion of $𝒢^{(0)}$ if

{lim}_{n \to \infty} μ (Y_{n}) = 1 .

For technical reasons, we also need to consider the following.

Definition 3.7.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a probability measure space. Given an admissible bisection

K \subseteq 𝒢,

we set

ℜ (K, x) ≔ \frac{{d {(τ_{K}^{- 1})}_{*} μ |}_{r (K)}}{{d μ |}_{s (K)}} (x)

the Radon–Nikodym derivative at

x \in s (K),

where

τ_{K}

comes from (1.1). Here,

{μ |}_{s (K)}

and

{μ |}_{r (K)}

are the restrictions of

μ,

and

{{(τ_{K}^{- 1})}_{*} μ |}_{K}

is the pushforward measure.

To simplify the statement of our main result, let us package the derivative information into the notion of a domain of expansion.

Definition 3.8.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a probability measure space. For measurable

Y \subseteq 𝒢^{(0)},

we say that

Y

is a domain of expansion with bounded ratio if there exist

C, N, L > 0

and a unital symmetric

N -

decomposable

K \subseteq 𝒢

with

ℓ (K) \leq L,

together with a unital symmetric decomposition

K = ⋃_{i = 1}^{N} K_{i},

satisfying the following:

(1)

For measurable $A \subseteq Y$ with $0 < μ (A) \leq \frac{1}{2} μ (Y),$ then we have

$μ ((r (K_{α} \cdot A) ∖ A) \cap Y) > C_{α} μ (A) .$
(2)

There exists $θ \geq 1$ such that $\frac{1}{θ} \leq ℜ (K_{i}, x) \leq θ$ for any $x \in Y$ and $i \in {1, 2, \dots, N}$ with $τ_{K_{i}} (x) \in Y .$ (As a priori to $τ_{K_{i}} (x) \in Y,$ we have $x \in s (K_{i})$ and hence $ℜ (K_{i}, x)$ makes sense.)

In this case, we say that

Y

is a domain of $(C, N, L) -$ expansion with ratio bounded by $θ .$

The following is our structure theorem.

Theorem 3.9.

Let $𝒢$ be a groupoid with a length function $ℓ$ and $(𝒢^{(0)}, ℛ, μ)$ be a probability measure space. Then the following are equivalent:

(1)

The groupoid $𝒢$ is asymptotically expanding in measure.
(2)

The unit space $𝒢^{(0)}$ admits an exhaustion by domains $Y_{n}$ of $(C_{n}, N_{n}, L_{n}) -$ expansion of ratio bounded³

The extra requirement on the Radon–Nikodym derivatives will play an important role in proving our main results in Section 5.
by $θ_{n}$ for $C_{n}, N_{n}, L_{n} > 0$ and $θ_{n} \geq 1 .$
(3)

The unit space $𝒢^{(0)}$ admits an exhaustion by domains of asymptotic expansion.

A key tool to prove Theorem 3.9 is to consider maximal Følner sets as follows.

Definition 3.10.

Given measurable

Y \subseteq 𝒢^{(0)},

decomposable

K \subseteq 𝒢

and

ε > 0,

we say that a measurable subset

F \subseteq Y

is $(ε, K) -$ Følner in $Y$ if

μ (F) \leq \frac{1}{2} μ (Y)

and

μ ((r (K \cdot F) ∖ F) \cap Y) \leq ε μ (F) .

Now fix measurable

Y \subseteq 𝒢^{(0)},

decomposable

K \subseteq 𝒢

and

ε > 0 .

Denote the set of all

(ε, K) -

Følner sets in

Y

by

ℱ_{ε, K} .

Consider the equivalence relation on

ℱ_{ε, K}

by setting

F \sim F^{'}

in

ℱ_{ε, K}

if and only if they differ by a null set. Define a partial order on

ℱ_{ε, K} / \sim

by setting

[F] ⊑ [F^{'}]

if

F \subseteq F^{'}

up to null sets.

The following lemmas are similar to [23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

, Lemmas 4.2 and 4.4], and hence we omit their proofs.

Lemma 3.11.

The partial ordered set $(ℱ_{ε, K} / \sim, ⊑)$ has maximal elements.

Lemma 3.12.

Given measurable $Y \subseteq 𝒢^{(0)},$ decomposable $K \subseteq 𝒢$ and $ε > 0,$ let $F_{ε, K} \subseteq Y$ be a maximal $(ε, K) -$ Følner set in $Y .$ Then for any measurable $A \subseteq Y ∖ F_{ε, K}$ with $0 < μ (A) \leq \frac{1}{2} μ (Y) - μ (F_{ε, K}),$ we have

μ (((r (K \cdot A) ∖ A) \cap Y) ∖ F_{ε, K}) > ε μ (A) .

Now we are ready to prove Theorem 3.9.

Proof of Theorem 3.9.

“(1)

\Rightarrow

(2)”: Fix

C \in (0, \frac{1}{2}),

and let

α_{n} ≔ \frac{C}{(4 + 2 C) (n + 1)}

for each

n \in ℕ .

Since

𝒢

is asymptotically expanding, Lemma 3.5 provides a unital symmetric

N_{n} -

decomposable

K_{n}

with

ℓ (K_{n}) \leq L_{n}

satisfying the condition therein for

β = \frac{1}{2},

α = α_{n}

and

C .

Take a unital symmetric decomposition

K_{n} = ⋃_{i = 1}^{N_{n}} K_{n, i}

for admissible

K_{n, i}

and set

θ_{n} ≔ N_{n} \cdot (n + 1) .

Denote

Z_{n} ≔ {τ_{K_{n, i}} (x) | x \in s (K_{n, i}) and i = 1, \dots, N_{n} such that ℜ (K_{n, i}, x) < \frac{1}{θ_{n}}} .

Then

μ (Z_{n}) < \frac{N_{n}}{θ_{n}} \cdot μ (𝒢^{(0)}) = \frac{1}{n + 1},

which implies that

μ (𝒢^{(0)} ∖ Z_{n}) \geq \frac{n}{n + 1} \geq \frac{1}{2} .

Let

X_{n} ≔ 𝒢^{(0)} ∖ Z_{n},

and take

F_{n}

to be a maximal

(C, K_{n}) -

Følner sets in

X_{n},

ensured by Lemma 3.11. Then we have

μ (F_{n}) < α_{n} \cdot μ (X_{n}) .

Setting

Y_{n} ≔ X_{n} ∖ F_{n},

then

μ (Y_{n}) > (1 - α_{n}) μ (X_{n}) \geq (1 - α_{n}) \cdot \frac{n}{n + 1} .

Now we claim that

Y_{n}

is a domain of

(\frac{C}{2}, N_{n}, L_{n}) -

expansion. In fact, we take an arbitrary measurable

A \subseteq Y_{n}

with

0 < μ (A) \leq \frac{1}{2} μ (Y_{n})

and divide into two cases.

If

μ (A) \leq \frac{1}{2} μ (X_{n}) - μ (F_{n}),

then

μ ((r (K_{n} \cdot A) ∖ A) \cap Y_{n}) > C μ (A)

by Lemma 3.12.

If

μ (A) > \frac{1}{2} μ (X_{n}) - μ (F_{n}),

then

\frac{1}{2} μ (X_{n}) \geq μ (A) > (\frac{1}{2} - α_{n}) μ (X_{n}) \geq α_{n} μ (X_{n}) .

It follows from the requirement on

K_{n}

that

μ ((r (K_{n} \cdot A) ∖ A) \cap X_{n}) > C μ (A) .

Since

μ ((r (K_{n} \cdot A) ∖ A) \cap Y_{n}) \geq μ ((r (K_{n} \cdot A) ∖ A) \cap X_{n}) - μ (F_{n}) > C μ (A) - μ (F_{n})

and

μ (A) > (\frac{1}{2} - α_{n}) μ (X_{n}) \geq (\frac{1}{2} - \frac{C}{4 + 2 C}) μ (X_{n}) = \frac{1}{C + 2} μ (X_{n}),

then we have

C μ (A) - μ (F_{n}) > C μ (A) - α_{n} μ (X_{n}) > C μ (A) - α_{n} (C + 2) μ (A) \geq \frac{C}{2} μ (A) .

In conclusion, we showed that

Y_{n}

is a domain of

(\frac{C}{2}, N_{n}, L_{n}) -

expansion with ratio bounded by

θ_{n}

and

μ (Y_{n}) > (1 - α_{n}) \cdot \frac{n}{n + 1} .

“(2)

\Rightarrow

(3)” is trivial.

“(3)

\Rightarrow

(1)”: Take an exhaustion of

𝒢^{(0)}

by domains

Y_{n}

of asymptotic expansion. Assume that

𝒢

were not asymptotically expanding. Then there exists

α_{0} \in (0, \frac{1}{2})

such that for any

C > 0

and any unital symmetric decomposable

K \subseteq 𝒢,

there exists

A_{C, K} \in ℛ

with

α_{0} \leq μ (A_{C, K}) \leq \frac{1}{2}

such that

μ (r (K \cdot A_{C, K}) ∖ A_{C, K}) \leq C μ (A_{C, K}) .

Take

n \in ℕ

such that

μ (Y_{n}) \geq 1 - \frac{α_{0}}{2} .

Then for any measurable

A \subseteq 𝒢^{(0)}

with

α_{0} \leq μ (A) \leq \frac{1}{2},

direct calculations show that

\frac{α_{0}}{2} μ (Y_{n}) \leq μ (A \cap Y_{n}) \leq \frac{1}{2 - α_{0}} μ (Y_{n}) .

Then Lemma 3.4 provides

ε > 0

and unital symmetric decomposable

K \subseteq 𝒢

such that

μ ((r (K \cdot (A \cap Y_{n})) ∖ (A \cap Y_{n})) \cap Y_{n}) > ε μ (A \cap Y_{n}) \geq ε (μ (A) - \frac{α_{0}}{2}) \geq \frac{ε}{2} μ (A),

which implies that

μ (r (K \cdot A) ∖ A) \geq μ ((r (K \cdot (A \cap Y_{n})) ∖ A) \cap Y_{n}) > \frac{ε}{2} μ (A) .

This leads to a contradiction.

Remark 3.13.

From the proof above, if

𝒢

is asymptotically expanding, then in condition (2) we can take

C_{n} \equiv \frac{C}{2}

for any chosen

C \in (0, \frac{1}{2}),

domains

Y_{n}

satisfying

μ (Y_{n}) > (1 - \frac{C}{(4 + 2 C) (n + 1)}) \cdot \frac{n}{n + 1},

N_{n}, L_{n}

and

θ_{n}

only depend on the expansion parameters.

3.3. Markov kernels on groupoids

Here, we construct reversible Markov kernels on groupoids and study the relation between their Cheeger constants and the expansion of groupoids.

Firstly, we construct a Markov kernel for decomposable subsets. Let us fix a unital symmetric decomposable subset

K

together with a unital symmetric decomposition

K = ⋃_{i = 1}^{N} K_{i} .

Definition 3.14.

For measurable

Y \subseteq 𝒢^{(0)}

and

x \in Y,

denote

(3.2)

K_{Y, x} := {i ∣ x \in s (K_{i}) and τ_{K_{i}} (x) \in Y} and σ_{Y, K} (x) = \sum_{i \in K_{Y, x}} ℜ {(K_{i}, x)}^{\frac{1}{2}} .

The normalised local Markov kernel associated to

Y

and

K

is the Markov kernel on

Y

defined as follows:

(3.3)

Π_{Y, K} (x, -) = \frac{1}{σ_{Y, K} (x)} \sum_{i \in K_{Y, x}} ℜ {(K_{i}, x)}^{\frac{1}{2}} δ_{τ_{K_{i}} (x)} (-) for x \in Y .

Here,

δ_{y}

is the Dirac delta measure on

y .

Since the decomposition is unital, it is clear that

σ_{Y, K} > 0

on

Y

and (3.3) makes sense. It is also routine to check that (3.3) is indeed a Markov kernel on

Y .

To see that

Π_{Y, K}

is reversible, we consider the measure on

Y

defined by

(3.4)

d {\tilde{μ}}_{Y, K} ≔ σ_{Y, K} \cdot d ({μ |}_{Y}) .

Note that both

Π_{Y, K}

and

{\tilde{μ}}_{Y, K}

depend on the decomposition of

K .

We collect several useful properties of

Π_{Y, K}

in the following. The proof is similar to that of [22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

, Proposition 3.10], and hence we omit the details.

Proposition 3.15.

In the setting above, we have:

(1)

The measure ${\tilde{μ}}_{Y, K}$ is reversing for the Markov kernel $Π_{Y, K} .$
(2)

For measurable $A \subseteq Y,$ we have $μ (A) \leq {\tilde{μ}}_{Y, K} (A) \leq N \sqrt{μ (A) μ (Y)} .$ Hence ${\tilde{μ}}_{Y, K}$ is equivalent to the restriction ${μ |}_{Y} .$

Definition 3.16.

For a measurable

Y \subseteq 𝒢^{(0)}

and

C, N, L > 0,

we call

Y

a domain of Markov $(C, N, L) -$ expansion (or simply, domain of Markov expansion) if there exists a unital symmetric

N -

decomposable

K

together with a unital symmetric decomposition

K = ⋃_{i = 1}^{N} K_{i}

such that

ℓ (K) \leq L

and the associated normalised local Markov kernel

Π_{Y, K}

on

(Y, {\tilde{μ}}_{Y, K})

has the Cheeger constant greater than

C .

Moreover, if the decomposition for

K

above has ratio bounded by

θ \geq 1,

then we say that

Y

is a domain of Markov $(C, N, L) -$ expansion with ratio bounded by $θ .$

The following lemma relates the domain of Markov expansion to the ordinary expansion. The proof is similar to that of [22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

, Lemma 3.14] and hence omitted.

Lemma 3.17.

For a measurable $Y \subseteq 𝒢^{(0)},$ we have:

(1)

If $Y$ is a domain of $(C, N, L) -$ expansion with ratio bounded by $θ,$ then $Y$ is a domain of $(\frac{C}{N θ}, N, L) -$ Markov expansion.
(2)

If $Y$ is a domain of $(κ, N, L) -$ Markov expansion with ratio bounded by $θ,$ then $Y$ is a domain of $(\frac{κ}{N \sqrt{θ} + κ}, N, L) -$ expansion.

Combining Theorem 3.9 with Lemma 3.17, we obtain the following Markovian version of the structure theorem.

Theorem 3.18.

Let $𝒢$ be a groupoid with a length function $ℓ$ and $(𝒢^{(0)}, ℛ, μ)$ be a probability measure space. Then the following are equivalent:

(1)

The groupoid $𝒢$ is asymptotically expanding in measure.
(2)

The unit space $𝒢^{(0)}$ admits an exhaustion by domains $Y_{n}$ of $(C_{n}, N_{n}, L_{n}) -$ Markov expansion with ratio bounded by $θ_{n}$ for $C_{n}, N_{n}, L_{n} > 0$ and $θ_{n} \geq 1 .$

Remark 3.19.

Combining Remark 3.13 with the dependence of parameters established in Lemma 3.17, we know that if

𝒢

is asymptotically expanding, then in Theorem 3.18 (2) we can take domains

Y_{n}

to satisfy

μ (Y_{n}) > t_{n}

for some universal

t_{n}

independent of

𝒢,

while

C_{n}, N_{n}, L_{n}

and

θ_{n}

only depend on expansion parameters of

𝒢 .

4. Dynamical propagation and quasi-locality

In this section, we introduce the notion of dynamical propagation and dynamical quasi-locality for operators on groupoids. Again we always assume that

𝒢

is a groupoid with a length function

ℓ

and

𝒢^{(0)}

is equipped with a measure

μ

on some

σ -

algebra

ℛ .

To include more examples, here we do not require

μ

to be finite.

4.1. Basic notions

We start with the following key notions.

Definition 4.1.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a measure space. An operator

T \in 𝔅 (L^{2} (𝒢^{(0)}, μ))

is said to have finite dynamical propagation if there is a unital symmetric

N -

decomposable subset

K \subseteq 𝒢

with

ℓ (K) \leq L

for some

N, L > 0

such that for any

A, B \in ℛ

with

μ (r (K \cdot A) \cap B) = 0,

then

χ_{A} T χ_{B} = 0 .

In this case, we also say that

T

has $(N, L) -$ dynamical propagation.

Denote the set of all operators with finite dynamical propagation by

ℂ_{dyn} (𝒢)

and its norm completion by

𝐂_{dyn}^{*} (𝒢),

called the dynamical Roe algebra of $𝒢 .$

Similarly, we consider its quasi-local counterpart.

Definition 4.2.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a measure space. An operator

T \in 𝔅 (L^{2} (𝒢^{(0)}, μ))

is called dynamically quasi-local if for any

ε > 0,

there is a unital symmetric

N_{ε} -

decomposable subset

K_{ε} \subseteq 𝒢

with

ℓ (K_{ε}) \leq L_{ε}

for some

N_{ε}, L_{ε} > 0

such that for any

A, B \in ℛ

with

μ (r (K_{ε} \cdot A) \cap B) = 0,

then

∥ χ_{A} T χ_{B} ∥ < ε .

The maps

ε \mapsto N_{ε}, ε \mapsto L_{ε}

are called quasi-local parameters for

T .

Let us denote the set of all dynamically quasi-local operators in

𝔅 (L^{2} (𝒢^{(0)}, μ))

by

𝐂_{dyn, q}^{*} (𝒢),

called the dynamical quasi-local algebra of $𝒢 .$

The following shows that

𝐂_{dyn}^{*} (𝒢)

and

𝐂_{dyn, q}^{*} (𝒢)

are indeed

C^{*} -

algebras. The proof is straightforward from definitions and hence omitted.

Lemma 4.3.

Given $𝒢,$ $ℓ$ and $μ$ as above, we have:

(1)

The set $ℂ_{dyn} (𝒢)$ is a $* -$ algebra, and hence $𝐂_{dyn}^{*} (𝒢)$ is a $C^{*} -$ algebra.
(2)

The set $𝐂_{dyn, q}^{*} (𝒢)$ is a $C^{*} -$ algebra.

For a sequence of groupoids, we combine them into a single groupoid and translate the notion of dynamical propagation and quasi-locality thereon in a uniform version. More precisely, for each

n \in ℕ,

let

𝒢_{n}

be a groupoid with a length function

ℓ_{n}

and

(𝒢_{n}^{(0)}, ℛ_{n}, μ_{n})

be a measure space. Form a groupoid

𝒢

to be their disjoint union

⨆_{n \in ℕ} 𝒢_{n}

and equip

𝒢^{(0)} = ⨆_{n \in ℕ} 𝒢_{n}^{(0)}

with a measure

μ

on some

σ -

algebra

ℛ

generated by

⨆_{n \in ℕ} ℛ_{n},

determined by

{μ |}_{𝒢_{n}^{(0)}} ≔ μ_{n}

for each

n .

Also take a length function

ℓ

on

𝒢

to be the disjoint union of

ℓ_{n} .

Then we have

L^{2} (𝒢^{(0)}, μ) = ⨁_{n \in ℕ} L^{2} (𝒢_{n}^{(0)}, μ_{n}) .

The following shows that operators in

𝐂_{dyn, q}^{*} (𝒢)

are always diagonal and can be described in a uniform version.

Lemma 4.4.

Given $T \in 𝔅 (L^{2} (𝒢^{(0)}, μ)),$ we have:

(1)

$T \in ℂ_{dyn} (𝒢)$ if and only if there exists $T_{n} \in 𝔅 (L^{2} (𝒢_{n}^{(0)}, μ_{n}))$ for each $n \in ℕ$ with ${sup}_{n \in ℕ} ∥ T_{n} ∥ < \infty$ such that $T = (SOT) - \sum_{n \in ℕ} T_{n}$ and there exist $N, L > 0$ satisfying that for any $n \in ℕ,$ there exists a unital symmetric $N -$ decomposable $K_{n} \subseteq 𝒢_{n}$ with $ℓ_{n} (K_{n}) \leq L$ such that for any $A, B \in ℛ_{n}$ with $μ (r (K_{n} \cdot A) \cap B) = 0,$ we have $χ_{A} T_{n} χ_{B} = 0 .$
(2)

$T \in 𝐂_{dyn, q}^{*} (𝒢)$ if and only if there exists $T_{n} \in 𝔅 (L^{2} (𝒢_{n}^{(0)}, μ_{n}))$ for each $n \in ℕ$ with ${sup}_{n \in ℕ} ∥ T_{n} ∥ < \infty$ such that $T = (SOT) - \sum_{n \in ℕ} T_{n}$ and for any $ε > 0,$ there exist $N, L > 0$ satisfying that for any $n \in ℕ,$ there exists a unital symmetric $N -$ decomposable $K_{n} \subseteq 𝒢_{n}$ with $ℓ_{n} (K_{n}) \leq L$ such that for any $A, B \in ℛ_{n}$ with $μ (r (K_{n} \cdot A) \cap B) = 0,$ we have $∥ χ_{A} T_{n} χ_{B} ∥ < ε .$

Proof.

(1): We divide the proof into two directions.

Necessity.

Firstly, we show that

T

is diagonal. Note that for any

n \neq m

and

A \in ℛ_{n},

B \in ℛ_{m},

we have

(K \cdot A) \cap B = \emptyset

for any decomposable

K \subseteq 𝒢 .

Then by definition,

χ_{A} T χ_{B} = 0 .

Hence

T = (SOT) - \sum_{n \in ℕ} T_{n}

for

T_{n} = χ_{𝒢_{n}^{(0)}} T χ_{𝒢_{n}^{(0)}} .

Furthermore, there exists a unital symmetric decomposable

K \subseteq 𝒢

such that for any

A, B \in ℛ

with

μ (r (K \cdot A) \cap B) = 0,

then

χ_{A} T χ_{B} = 0 .

For each

n \in ℕ,

set

K_{n} ≔ K \cap 𝒢_{n} .

It is easy to check that

K_{n}

satisfies the requirement.

Sufficiency.

Assume

K_{n}

satisfies the requirement, and write

K_{n} = ⋃_{i = 1}^{N} K_{n}^{(i)}

for admissible

K_{n}^{(i)} \subseteq 𝒢_{n}

with

ℓ_{n} (K_{n}^{(i)}) \leq L .

Define

K ≔ ⨆_{n \in ℕ} K_{n} \subseteq 𝒢

and for

i = 1, \dots, N,

define

K^{(i)} ≔ ⨆_{n \in ℕ} K_{n}^{(i)} .

Since

K_{n}^{(i)}

is admissible, it is easy to see that

K^{(i)} \subseteq 𝒢

is admissible for each

i .

Hence

K

is

N -

decomposable and

ℓ (K) \leq L .

Now for any

A, B \in ℛ

with

μ (r (K \cdot A) \cap B) = 0,

set

A_{n} ≔ A \cap 𝒢_{n}^{(0)}

and

B_{n} ≔ B \cap 𝒢_{n}^{(0)},

and then we have

μ_{n} (r (K_{n} \cdot A_{n}) \cap B_{n}) = 0

for each

n \in ℕ .

By assumption, we have

χ_{A_{n}} T_{n} χ_{B_{n}} = 0 .

Finally, note that

χ_{A} T χ_{B} = \sum_{n \in ℕ} χ_{A_{n}} T_{n} χ_{B_{n}} = 0,

which concludes (1). Since (2) is similar to (1), the details are omitted.

4.2. Quasi-locality of the averaging projection

Now we focus on a special projection operator in

𝔅 (L^{2} (𝒢^{(0)}, μ))

when

μ

is a probability measure, whose dynamical quasi-locality is closely related to the asymptotic expansion of the groupoid.

Definition 4.5.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a measure space. For any measurable

Y \subseteq 𝒢^{(0)}

with

0 < μ (Y) < \infty,

denote by

P_{Y} \in 𝔅 (L^{2} (𝒢^{(0)}, μ))

the averaging projection on $Y$ , which is the orthogonal projection onto the one-dimensional subspace in

L^{2} (𝒢^{(0)}, μ)

spanned by

χ_{Y} .

In other words,

P_{Y} (f) ≔ \frac{⟨ f, χ_{Y} ⟩}{μ (Y)} χ_{Y} for f \in L^{2} (𝒢^{(0)}, μ) .

If

μ (𝒢^{(0)}) < \infty,

we simply write

P_{𝒢}

for

P_{𝒢^{(0)}} .

Similar to [20

K. Li, P. W. Nowak, J. Špakula, and J. Zhang, Quasi-local algebras and asymptotic expanders. Groups Geom. Dyn. 15 (2021), no. 2, 655–682 Zbl 1484.46057 MR 4303336

, Lemma 3.8], we have the following.

Lemma 4.6.

Assume that $μ (𝒢^{(0)}) = 1 .$ Then for any measurable $A, B \subseteq 𝒢^{(0)},$ we have

∥ χ_{A} P_{𝒢} χ_{B} ∥ = \sqrt{μ (A) μ (B)} .

The following relates the quasi-locality of

P_{𝒢}

to asymptotic expansion of

𝒢 .

Proposition 4.7.

Let $𝒢$ be a groupoid with a length function $ℓ$ and $(𝒢^{(0)}, ℛ, μ)$ be a probability measure space. Then the averaging projection $P_{𝒢}$ is dynamically quasi-local if and only if $𝒢$ is asymptotically expanding in measure $μ .$

Proof.

We divide the proof into two directions.

Necessity.

Assume

ε \mapsto N_{ε},

ε \mapsto L_{ε}

are quasi-local parameters for

P_{𝒢} .

Given

α \in (0, \frac{1}{2}],

set

N ≔ N_{\sqrt{α} / 2}

and

L ≔ L_{\sqrt{α} / 2} .

Then there exists a unital symmetric

N -

decomposable

K

with

ℓ (K) \leq L

such that for any

A, B \in ℛ

with

μ (r (K \cdot A) \cap B) = 0,

we have

∥ χ_{A} T χ_{B} ∥ < \frac{\sqrt{α}}{2} .

Then for any

A \in ℛ

with

α \leq μ (A) \leq \frac{1}{2},

Lemma 4.6 shows

\sqrt{μ (A) \cdot μ (𝒢^{(0)} ∖ r (K \cdot A))} = ∥ χ_{A} P_{𝒢} χ_{𝒢^{(0)} ∖ r (K \cdot A)} ∥ < \frac{\sqrt{α}}{2} .

Hence we obtain

μ (A) \cdot μ (𝒢^{(0)} ∖ r (K \cdot A)) < \frac{α}{4},

which implies that

μ (𝒢^{(0)} ∖ r (K \cdot A)) < \frac{1}{4} .

Therefore, we obtain

μ (r (K \cdot A)) > \frac{3}{4} \geq (1 + \frac{1}{2}) \cdot μ (A) .

This concludes that

𝒢

is asymptotically expanding in measure.

Sufficiency.

Assuming

α \mapsto C_{α},

α \mapsto N_{α}

and

α \mapsto L_{α}

are expansion parameters for

𝒢,

we take

K_{α}

as in Definition 3.2. Given

0 < ε \leq \frac{1}{2},

take

n \in ℕ

to be the smallest number such that

{(1 + C_{ε})}^{n} \cdot ε > \frac{1}{2} .

Set

K ≔ K_{ε}^{2 n},

which is unital symmetric

N_{ε}^{2 n} -

decomposable with length at most

2 n L_{ε}

by Lemma 2.5.

For measurable

A, B \subseteq 𝒢^{(0)}

with

μ (r (K \cdot A) \cap B) = 0,

we have

μ (r (K_{ε}^{n} \cdot A) \cap r (K_{ε}^{n} \cdot B)) = 0 .

Hence we can assume

μ (r (K_{ε}^{n} \cdot A)) \leq \frac{1}{2} .

If

μ (A) < ε,

it follows from Lemma 4.6 that

∥ χ_{A} P_{𝒢} χ_{B} ∥ = \sqrt{μ (A) μ (B)} < \sqrt{ε} .

If

μ (A) \geq ε,

then using asymptotic expansion inductively, we have

\frac{1}{2} \geq μ (r (K_{ε}^{n} \cdot A)) \geq (1 + C_{ε}) μ (r (K_{ε}^{n - 1} \cdot A)) \geq \dots \geq {(1 + C_{ε})}^{n} μ (A) \geq {(1 + C_{ε})}^{n} \cdot ε .

This leads to a contradiction to the choice of

n .

Remark 4.8.

From the proof above, it is clear that if

P_{𝒢}

is dynamically quasi-local, then we can choose expansion parameters for

𝒢

only depending on quasi-local parameters for

P_{𝒢},

and vice versa.

5. Main results

Now we are ready to prove the following fundamental case of the main result.

Theorem 5.1.

Let $𝒢$ be a groupoid with a length function $ℓ$ and $(𝒢^{(0)}, ℛ, μ)$ be a probability measure space. Then the following are equivalent:

(1)

The averaging projection $P_{𝒢} \in 𝐂_{dyn}^{*} (𝒢) .$
(2)

The averaging projection $P_{𝒢} \in 𝐂_{dyn, q}^{*} (𝒢) .$
(3)

The groupoid $𝒢$ is asymptotically expanding in measure.

Proof.

“(1)

\Rightarrow

(2)” is trivial, and “(2)

\Leftrightarrow

(3)” comes from Proposition 4.7. Hence we only focus on “(3)

\Rightarrow

(1)”.

Fix

C \in (0, \frac{1}{2}) .

By Theorem 3.18 and Remarks 3.13 and 3.19, there exists a sequence of measurable subsets

Y_{n} \subseteq 𝒢^{(0)}

with

μ (Y_{n}) > (1 - \frac{C}{(4 + 2 C) (n + 1)}) \cdot \frac{n}{n + 1}

such that each

Y_{n}

is a domain of

(C_{n}, N_{n}, L_{n}) -

Markov expansion with ratio bounded by

θ_{n}

for

C_{n}, N_{n}, L_{n} > 0

and

θ_{n} \geq 1

only depending on expansion parameters of

𝒢 .

From Definition 3.16, we can choose a unital symmetric

N_{n} -

decomposable

K_{n}

together with a unital symmetric decomposition

K_{n} = ⋃_{i = 1}^{N_{n}} K_{n}^{(i)}

such that

ℓ (K_{n}) \leq L_{n},

and the associated normalised local Markov kernel

Π_{n} ≔ Π_{Y_{n}, K_{n}}

from (3.3) on

(Y_{n}, {\tilde{μ}}_{n})

has the Cheeger constant greater than

C_{n},

where

{\tilde{μ}}_{n} ≔ {\tilde{μ}}_{Y_{n}, K_{n}}

is the reversing measure defined in (3.4). We have a function

σ_{n} ≔ σ_{Y_{n}, K_{n}}

from (3.2) such that

σ_{n} \geq 1

on

Y_{n} .

Denote the associated Markov operator by

𝔓_{n}

from (2.1) with spectral gap

λ_{n} .

For each

n \in ℕ,

consider the embedding

I_{n} : L^{2} (Y_{n}, {\tilde{μ}}_{n}) \to L^{2} (𝒢^{(0)}, μ)

simply by extending each function in

L^{2} (Y_{n}, {\tilde{μ}}_{n})

by zero on

𝒢^{(0)} ∖ Y_{n} .

Thanks to Proposition 3.15, this is well defined and

∥ I_{n} ∥ \leq 1 .

Direct calculations show that

I_{n}^{*} (g) = {\frac{1}{σ_{n}} \cdot g |}_{Y_{n}}

for

g \in L^{2} (𝒢^{(0)}, μ) .

Denote the adjoint map

{Ad}_{n} : 𝔅 (L^{2} (Y_{n}, {\tilde{μ}}_{n})) \to 𝔅 (L^{2} (𝒢^{(0)}, μ)), T \mapsto I_{n} \circ T \circ I_{n}^{*} for T \in 𝔅 (L^{2} (Y_{n}, {\tilde{μ}}_{n})) .

Recall from Definition 4.5 that we have the averaging projection

P_{n} ≔ P_{Y_{n}}

in

𝔅 (L^{2} (𝒢^{(0)}, μ)) .

Denote the orthogonal projection

{\tilde{P}}_{n} \in 𝔅 (L^{2} (Y_{n}, {\tilde{μ}}_{n}))

onto constant functions on

Y_{n}

in

L^{2} (Y_{n}, {\tilde{μ}}_{n}) .

Direct calculations show that

(5.1)

{Ad}_{n} ({\tilde{P}}_{n}) = \frac{μ (Y_{n})}{{\tilde{μ}}_{n} (Y_{n})} \cdot P_{n} .

On the other hand, Theorem 2.12 shows that

\frac{C_{n}^{2}}{2} \leq 1 - λ_{n} \leq 2 C_{n} .

Hence

\frac{1}{2} χ_{Y_{n}} + \frac{1}{2} 𝔓_{n}

has spectrum contained in

[- \frac{3}{4}, 1 - \frac{C_{n}^{2}}{4}] \cup {1} .

Therefore, for any

m \in ℕ,

we have

∥ {(\frac{1}{2} χ_{Y_{n}} + \frac{1}{2} 𝔓_{n})}^{m} - {\tilde{P}}_{n} ∥ \leq {(1 - \frac{C_{n}^{2}}{4})}^{m} .

Applying

{Ad}_{n}

and using (5.1), we obtain

∥ \frac{{\tilde{μ}}_{n} (Y_{n})}{μ (Y_{n})} \cdot {Ad}_{n} [{(\frac{1}{2} χ_{Y_{n}} + \frac{1}{2} 𝔓_{n})}^{m}] - P_{n} ∥ \leq \frac{{\tilde{μ}}_{n} (Y_{n})}{μ (Y_{n})} \cdot {(1 - \frac{C_{n}^{2}}{4})}^{m} .

Note that

\frac{{\tilde{μ}}_{n} (Y_{n})}{μ (Y_{n})} \leq {∥ σ_{n} ∥}_{\infty} \leq {∥ \sum_{i = 1}^{N_{n}} ℜ {(K_{n}^{(i)}, x)}^{\frac{1}{2}} ∥}_{\infty} \leq N_{n} \sqrt{θ_{n}} .

Combining the above together, we obtain

(5.2)

∥ \frac{{\tilde{μ}}_{n} (Y_{n})}{μ (Y_{n})} \cdot {Ad}_{n} [{(\frac{1}{2} χ_{Y_{n}} + \frac{1}{2} 𝔓_{n})}^{m}] - P_{n} ∥ \leq N_{n} \sqrt{θ_{n}} \cdot {(1 - \frac{C_{n}^{2}}{4})}^{m} .

Hence given

ε > 0,

we can choose

m_{n} \in ℕ

such that

(5.3)

N_{n} \sqrt{θ_{n}} \cdot {(1 - \frac{C_{n}^{2}}{4})}^{m_{n}} < \frac{ε}{2} .

Moreover, direct calculations show that for each

n \in ℕ,

we have

∥ P_{n} - P_{𝒢} ∥ \leq \sqrt{μ (𝒢^{(0)} ∖ Y_{n})} = \sqrt{\frac{1}{n + 1} + \frac{n C}{(4 + 2 C) {(n + 1)}^{2}}} .

For

ε

above, we can further choose

\tilde{N} \in ℕ

such that for any

n > \tilde{N},

we have

(5.4)

\sqrt{\frac{1}{n + 1} + \frac{n C}{(4 + 2 C) {(n + 1)}^{2}}} < \frac{ε}{2} .

Combining with (5.2), (5.3) and (5.4), we obtain that for any

n > \tilde{N},

we have

(5.5)

\begin{matrix} ∥ \frac{{\tilde{μ}}_{n} (Y_{n})}{μ (Y_{n})} \cdot {Ad}_{n} [{(\frac{1}{2} χ_{Y_{n}} + \frac{1}{2} 𝔓_{n})}^{m_{n}}] - P_{𝒢} ∥ \\ < ∥ \frac{{\tilde{μ}}_{n} (Y_{n})}{μ (Y_{n})} \cdot {Ad}_{n} [{(\frac{1}{2} χ_{Y_{n}} + \frac{1}{2} 𝔓_{n})}^{m_{n}}] - P_{n} ∥ + ∥ P_{n} - P_{𝒢} ∥ \\ < N_{n} \sqrt{θ_{n}} \cdot {(1 - \frac{C_{n}^{2}}{4})}^{m_{n}} + \sqrt{\frac{1}{n + 1} + \frac{n C}{(4 + 2 C) {(n + 1)}^{2}}} < \frac{ε}{2} + \frac{ε}{2} = ε . \end{matrix}

Finally, for

n \in ℕ

and measurable

A, B \subseteq Y_{n}

with

μ ((K_{n} \cdot A) \cap B) = 0,

we have

{\tilde{μ}}_{n} ((K_{n} \cdot A) \cap B) = 0

by Proposition 3.15 (2). For

x \in Y_{n}

and

ξ \in L^{2} (Y_{n}, {\tilde{μ}}_{n}),

we have

\begin{matrix} (χ_{A} 𝔓_{n} χ_{B} ξ) (x) & = χ_{A} (x) \cdot \int_{B} ξ (y) Π_{n} (x, dy) \\ = χ_{A} (x) \cdot \frac{1}{σ_{n} (x)} \sum_{i : τ_{K_{n}^{(i)}} (x) \in B} ℜ {(K_{n}^{(i)}, x)}^{\frac{1}{2}} ξ (τ_{K_{n}^{(i)}} (x)) . \end{matrix}

It follows that

χ_{A} 𝔓_{n} χ_{B} = 0 .

Note that the operator

I_{n}

does not change the propagation, and hence the following operator

\frac{{\tilde{μ}}_{n} (Y_{n})}{μ (Y_{n})} \cdot {Ad}_{n} [{(\frac{1}{2} χ_{Y_{n}} + \frac{1}{2} 𝔓_{n})}^{m_{n}}] in 𝔅 (L^{2} (𝒢^{(0)}, μ))

has

(N_{n}^{m_{n}}, m_{n} \cdot L_{n}) -

propagation. Combining with (5.5), we conclude the proof.

Remark 5.2.

From the proof above together with Remarks 3.13 and 3.19, we know that if

𝒢

is asymptotically expanding in measure, then for any

ε > 0,

we can choose

T_{ε} \in ℂ_{dyn} (𝒢)

with

(N_{ε}, L_{ε}) -

dynamical propagation such that

∥ T_{ε} - P_{𝒢} ∥ < ε

and the functions

ε \mapsto N_{ε}

and

ε \mapsto L_{ε}

only depend on expansion parameters of

𝒢 .

Conversely, it follows from Remark 4.8 that parameters of

𝒢

only depend on functions

ε \mapsto N_{ε}

and

ε \mapsto L_{ε}

satisfying the conditions above.

In the following, we consider general rank-one projections on

L^{2} (𝒢^{(0)}, μ)

for general

μ .

Let

P \in 𝔅 (L^{2} (𝒢^{(0)}, μ))

be a rank-one projection and

ξ \in L^{2} (𝒢^{(0)}, μ)

be a unit vector in the range of

P .

Then

P (η) = ⟨ η, ξ ⟩ ξ

for any

η \in L^{2} (𝒢^{(0)}, μ) .

This induces a probability measure

ν

on

(𝒢^{(0)}, ℛ)

defined by

d ν (x) ≔ {| ξ (x) |}^{2} d μ (x) for x \in 𝒢^{(0)} .

It is clear that the measure

ν

only depends on

P,

called the associated measure to $P .$

Then we have the following.

Theorem 5.3.

Let $𝒢$ be a groupoid with a length function $ℓ$ and $(𝒢^{(0)}, ℛ, μ)$ be a (not necessarily finite) measure space. Let $P \in 𝔅 (L^{2} (𝒢^{(0)}, μ))$ be a rank-one projection and $ν$ the associated probability measure on $𝒢^{(0)} .$ Then the following are equivalent:

(1)

$P \in 𝐂_{dyn}^{*} (𝒢) .$
(2)

$P \in 𝐂_{dyn, q}^{*} (𝒢) .$
(3)

$𝒢$ is asymptotically expanding in measure $ν .$

Our idea is to apply Theorem 5.1 to the probability measure

ν .

Firstly, note that

ν

might not be equivalent to

μ,

and hence consider

Z ≔ {x \in 𝒢^{(0)} ∣ ξ (x) = 0} .

Here,

Z

is well defined up to

μ -

null sets. We set

Y ≔ 𝒢^{(0)} ∖ Z

and then the restrictions

{μ |}_{Y}

and

{ν |}_{Y}

are equivalent. Now we consider the reduction

𝒢_{Y}^{Y} = s^{- 1} (Y) \cap r^{- 1} (Y),

equipped with the restriction

ℓ_{Y}

of the length function

ℓ .

Denote

Q : L^{2} (Y, {μ |}_{Y}) \to L^{2} (𝒢^{(0)}, μ)

the embedding by extending functions in

L^{2} (Y, {μ |}_{Y})

to

0

on

Z .

Then we have the following, whose proof is similar to [21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, Lemmas 4.5 and 4.6] and hence omitted.

Lemma 5.4.

With the notation above, we have:

(1)

$P \in 𝐂_{dyn, q}^{*} (𝒢)$ if and only if $Q^{*} P Q \in 𝐂_{dyn, q}^{*} (𝒢_{Y}^{Y}) .$
(2)

$P \in 𝐂_{dyn}^{*} (𝒢)$ if and only if $Q^{*} P Q \in 𝐂_{dyn}^{*} (𝒢_{Y}^{Y}) .$

Here, we equip $𝒢_{Y}^{Y}$ with $ℓ_{Y}$ and $Y$ with ${μ |}_{Y}$ to define $𝐂_{dyn}^{*} (𝒢_{Y}^{Y})$ and $𝐂_{dyn, q}^{*} (𝒢_{Y}^{Y}) .$

Note that for

T \in 𝐂_{dyn, q}^{*} (𝒢),

Q^{*} T Q

and

T

have the same quasi-local parameters, and the same holds for

T \in ℂ_{dyn} (𝒢) .

Hence to prove Theorem 5.3, it suffices to consider that

μ

and

ν

are equivalent, that is,

μ (Z) = 0 .

Therefore, in the following, we assume

Z = \emptyset .

To tell the difference, denote

𝐂_{dyn}^{*} (𝒢; μ)

and

𝐂_{dyn, q}^{*} (𝒢; μ)

the dynamical Roe and quasi-local algebras defined using

μ,

while

𝐂_{dyn}^{*} (𝒢; ν)

and

𝐂_{dyn, q}^{*} (𝒢; ν)

for those using

ν .

We have the following.

Lemma 5.5.

With the notation above and assuming $Z = \emptyset,$ we have:

(1)

$P \in 𝐂_{dyn, q}^{*} (𝒢; μ)$ if and only if $P_{𝒢} \in 𝐂_{dyn, q}^{*} (𝒢; ν) .$
(2)

$P \in 𝐂_{dyn}^{*} (𝒢; μ)$ if and only if $P_{𝒢} \in 𝐂_{dyn}^{*} (𝒢; ν) .$

Here, $P_{𝒢}$ is the averaging projection in $𝔅^{2} (L^{2} (𝒢^{(0)}, ν))$ from Definition 4.5.

Proof.

We only prove (1), since (2) is similar. Denote

U : L^{2} (𝒢^{(0)}, ν) \to L^{2} (𝒢^{(0)}, μ), f \mapsto f \cdot ξ for f \in L^{2} (𝒢^{(0)}, ν) .

By the construction and the assumption

Z = \emptyset

(which implies that

ξ

is non-zero everywhere), it is clear that

U

is unitary with

U^{*} η = \frac{1}{ξ} \cdot η

for

η \in L^{2} (𝒢^{(0)}, μ) .

Denote

{Ad}_{U} : 𝔅 (L^{2} (𝒢^{(0)}, ν)) \to 𝔅 (L^{2} (𝒢^{(0)}, μ)), T \mapsto U T U^{*} for T \in 𝔅 (L^{2} (𝒢^{(0)}, ν)) .

We claim that

{Ad}_{U} (P_{𝒢}) = P .

In fact, given

η \in L^{2} (𝒢^{(0)}, μ),

we have

\begin{matrix} {Ad}_{U} (P_{𝒢}) η & = U P_{𝒢} (\frac{1}{ξ} \cdot η) = U ({⟨ \frac{1}{ξ} \cdot η, χ_{𝒢^{(0)}} ⟩}_{L^{2} (𝒢^{(0)}, ν)} χ_{𝒢^{(0)}}) \\ = {⟨ \frac{1}{ξ} \cdot η, χ_{𝒢^{(0)}} ⟩}_{L^{2} (𝒢^{(0)}, ν)} ξ \\ = (\int_{𝒢^{(0)}} \frac{1}{ξ} (x) η (x) d ν (x)) \cdot ξ = (\int_{𝒢^{(0)}} η (x) ξ (x) d μ (x)) \cdot ξ \\ = {⟨ η, ξ ⟩}_{L^{2} (𝒢^{(0)}, μ)} ξ = P η . \end{matrix}

Hence we prove the claim. Therefore, for any measurable

A, B \subseteq 𝒢^{(0)},

we have

{Ad}_{U} (χ_{A} P_{𝒢} χ_{B}) = {Ad}_{U} (χ_{A}) \circ {Ad}_{U} (P_{𝒢}) \circ {Ad}_{U} (χ_{B}) = χ_{A} P χ_{B} .

Finally, since

μ

and

ν

are equivalent, we have

K \subseteq 𝒢

is decomposable with respect to

μ

if and only if it is decomposable with respect to

ν .

Moreover, for any measurable

A, B \subseteq 𝒢^{(0)}

and unital symmetric decomposable

K \subseteq 𝒢,

we have

μ (r (K \cdot A) \cap B) = 0

if and only if

ν (r (K \cdot A) \cap B) = 0 .

So we finish the proof.

Combining Theorem 5.1 with Lemmas 5.4 and 5.5, we obtain Theorem 5.3.

Remark 5.6.

From the proof above together with Remark 5.2, we know that if the rank-one projection

P

belongs to

𝐂_{dyn, q}^{*} (𝒢),

then for any

ε > 0,

we can choose

T_{ε} \in ℂ_{dyn} (𝒢)

with

(N_{ε}, L_{ε}) -

dynamical propagation such that

∥ T_{ε} - P ∥ < ε

and the functions

ε \mapsto N_{ε}

and

ε \mapsto L_{ε}

only depend on quasi-local parameters of

P .

Conversely, quasi-local parameters of

P

only depend on functions

ε \mapsto N_{ε}

and

ε \mapsto L_{ε}

satisfying the conditions above.

Finally, we introduce two variants of Theorem 5.3. The first one deals with a family version, following the discussion in Section 4.1. Combining Lemma 4.4 and Theorem 5.3 together with Remark 5.6, we obtain the following.

Theorem 5.3′.

For each $n \in ℕ,$ let $𝒢_{n}$ be a groupoid with a length function $ℓ_{n}$ and $(𝒢_{n}^{(0)}, ℛ_{n}, μ_{n})$ be a measure space. Form the groupoid $𝒢 = ⨆_{n \in ℕ} 𝒢_{n}$ together with a length function $ℓ$ and a measure $μ$ on $𝒢^{(0)}$ as in Section 4.1. Moreover, let $P_{n} \in 𝔅 (L^{2} (𝒢_{n}^{(0)}, μ_{n}))$ be a rank-one orthogonal projection for each $n \in ℕ$ and consider their direct sum

P ≔ (SOT) - \sum_{n \in ℕ} P_{n} \in 𝔅 (L^{2} (𝒢^{(0)}, μ)) .

Then the following are equivalent:

(1)

$P \in 𝐂_{dyn}^{*} (𝒢) .$
(2)

$P \in 𝐂_{dyn, q}^{*} (𝒢) .$
(3)

$𝒢_{n}$ is asymptotically expanding in the associated measure $ν_{n}$ to $P_{n}$ uniformly in the sense that they have the same expansion parameters.

The second focuses on a specific family of admissible and decomposable subsets allowed to build the notion of (asymptotic) expansion, dynamical propagation and quasi-locality. We introduce the following refined version of Definition 2.4.

Definition 5.7.

Let

𝒢

be a groupoid with a length function

ℓ

and

(𝒢^{(0)}, ℛ, μ)

be a measure space. Let

𝒦

be a family of admissible bisections which is closed under taking compositions and inverses and

𝒢^{(0)} \in 𝒦 .

A subset

K \subseteq 𝒢

is called $𝒦 -$ decomposable if

K = ⋃_{i = 1}^{N} K_{i}

for

N \in ℕ

and admissible bisection

K_{i} \in 𝒦 .

Given such

𝒦,

we replace the word “decomposable” by “

𝒦 -

decomposable” in Definitions 3.1, 3.2, 4.1 and 4.2 to define the notion of

𝒦 -

expansion (in measure),

𝒦 -

asymptotic expansion (in measure),

𝒦 -

dynamical propagation and

𝒦 -

dynamical quasi-local, together with

C^{*} -

algebras

𝐂_{dyn}^{*} (𝒢, 𝒦)

and

𝐂_{dyn, q}^{*} (𝒢, 𝒦) .

Remark 5.8.

Note that if

𝒦

is cofinal in the sense that any admissible bisection is contained in the union of finitely many elements in

𝒦,

then these notions coincide with the original ones of possibly different parameters.

Applying exactly the same proofs, we obtain the following.

Theorem 5.3″.

Let $𝒢$ be a groupoid with a length function $ℓ,$ $(𝒢^{(0)}, ℛ, μ)$ be a measure space and $𝒦$ be a family of admissible bisections which is closed under taking compositions and inverses and $𝒢^{(0)} \in 𝒦 .$ Let $P \in 𝔅 (L^{2} (𝒢^{(0)}, μ))$ be a rank-one projection and $ν$ the associated probability measure on $𝒢^{(0)} .$ Then the following are equivalent:

(1)

$P \in 𝐂_{dyn}^{*} (𝒢, 𝒦) .$
(2)

$P \in 𝐂_{dyn, q}^{*} (𝒢, 𝒦) .$
(3)

$𝒢$ is $𝒦 -$ asymptotically expanding in measure $ν .$

6. Examples

In this section, we apply our theory to several classes of groupoids.

6.1. Transformation groupoids

Our first example comes from group actions, which is one of our main motivations for this work.

Let

ρ : Γ ↷ X

be a countable group

Γ

acting on a set

X

and

ℓ_{Γ}

be a proper word length function on

Γ

in the sense that for any

L > 0,

the closed ball denoted by

B_{L} ≔ {γ \in Γ ∣ ℓ_{Γ} (γ) \leq L}

is finite. Consider the transformation groupoid

X ⋊ Γ

as in Section 1 with the length function

ℓ

by

ℓ (x, γ) ≔ ℓ_{Γ} (γ)

for

x \in X

and

γ \in Γ .

Moreover, let

μ

be a measure on

(X, ℛ)

for some

σ -

algebra

ℛ

and assume that the action

ρ

is measure class preserving. Then for

γ \in Γ,

the map

τ_{X \times {γ}} : X \to X

coincides with

ρ (γ)

(simply denoted by

γ

) and

X \times {γ}

is admissible.

Concerning decomposable subsets in

X ⋊ Γ,

we have the following. The proof is straightforward and hence omitted.

Lemma 6.1.

For any $L > 0,$ the subset $X \times B_{L}$ is unital symmetric $| B_{L} | -$ decomposable with $ℓ (X \times B_{L}) \leq L,$ and it admits a unital symmetric decomposition $X \times B_{L} = ⋃_{γ \in B_{L}} X \times {γ},$ where each $X \times {γ}$ is admissible. Here, we use $| \cdot |$ to denote the cardinality. Conversely, for any decomposable $K \subseteq X ⋊ Γ,$ we have $K \subseteq X \times B_{ℓ (K)} .$

Hence it suffices to consider decomposable subsets of the form

X \times B_{L},

which is determined by its length. Therefore, the notion of (asymptotic) expansion in measure can be translated as follows.

Proposition 6.2.

In the above setting, the transformation groupoid $X ⋊ Γ$ is asymptotically expanding in measure in the sense of Definition 3.2 if and only if the action is asymptotically expanding in measure. A similar result holds for the notion of expansion and the domain of (asymptotic) expansion.

Consequently, Theorem 3.9 and Theorem 3.18 recover [23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

, Theorem 4.6] and [22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

, Theorem 3.20], respectively.

On the other hand, applying Lemma 6.1 again, the notion of dynamical propagation and quasi-locality can be translated as follows.

Proposition 6.3.

In the above setting, an operator $T \in L^{2} (X, μ)$ is dynamically quasi-local in the sense of Definition 4.2 if and only if $T$ is $ρ -$ quasi-local in the sense of [22
K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250
, Definition 4.3], that is, for any $ε > 0,$ there exists $L_{ε} > 0$ such that for any measurable $A, B \subseteq X$ with $μ ((B_{L_{ε}} \cdot A) \cap B) = 0,$ then $∥ χ_{A} T χ_{B} ∥ < ε .$ A similar result holds for the notion of finite dynamical propagation.

Consequently, Theorem 5.1 recovers [22

K. Li, F. Vigolo, and J. Zhang, A Markovian and Roe-algebraic approach to asymptotic expansion in measure. Banach J. Math. Anal. 17 (2023), no. 4, article no. 74, 54 pp. Zbl 1530.37007 MR 4636250

, Proposition 4.6 and Theorem 4.16] for the averaging projection when

μ

is finite.

Corollary 6.4.

Let $ρ : Γ ↷ X$ be a countable group $Γ$ acting on a probability measure space $(X, μ)$ and assume that $ρ$ is measure class preserving. For the averaging projection $P_{X} \in 𝔅 (L^{2} (X, μ)),$ the following are equivalent:

(1)

$P$ is $ρ -$ quasi-local.
(2)

$P$ is a norm limit of operators in $B (L^{2} (X, μ))$ with finite $ρ -$ propagations.
(3)

The action is asymptotically expanding in measure $μ .$

Furthermore, we have the following generalised version from Theorem 5.3, which deals with an arbitrary rank-one projection.

Corollary 6.5.

Let $ρ : Γ ↷ X$ be a countable group $Γ$ acting on a (not necessarily finite) measure space $(X, μ)$ and assume that $ρ$ is measure class preserving. Then for any rank-one projection $P \in 𝔅 (L^{2} (X, μ)),$ we have that $P$ is $ρ -$ quasi-local if and only if $P$ is a norm limit of operators in $B (L^{2} (X, μ))$ with finite $ρ -$ propagations.

6.2. Pair groupoids: A family version

Now we consider pair groupoids and aim to recover the results for uniform Roe and quasi-local algebras of metric spaces.

Let

(X, d)

be a discrete metric space,

x \in X

and

R > 0 .

Denote

B (x, R)

the closed ball with radius

R

and centre

x .

We say that

(X, d)

has bounded geometry if

{sup}_{x \in X} | B (x, R) |

is finite for each

R > 0 .

A sequence of metric spaces

{(X_{n}, d_{n})}_{n \in ℕ}

has uniformly bounded geometry if for all

R > 0,

{sup}_{n \in ℕ} {sup}_{x \in X_{n}} | B (x, R) |

is finite.

Definition 6.6.

Let

{(X_{n}, d_{n})}_{n \in ℕ}

be a sequence of finite metric spaces. A coarse disjoint union of

{(X_{n}, d_{n})}_{n \in ℕ}

is a metric space

(X, d),

where

X = ⨆_{n \in ℕ} X_{n}

is the disjoint union of

{X_{n}}

as a set and

d

is a metric on

X

such that:

The restriction of $d$ on each $X_{n}$ coincides with $d_{n} .$
$d (X_{n}, X_{m}) \to \infty$ as $n + m \to \infty$ and $n \neq m .$

Definition 6.7.

Let

(X, d)

be a discrete metric space with bounded geometry. For an operator

T \in 𝔅 (ℓ^{2} (X)),

we say that:

(1)

$T$ has finite propagation if there exists $R > 0$ such that for any $A, B \subseteq X$ with $d (A, B) > R,$ we have that $χ_{A} T χ_{B} = 0 .$
(2)

$T$ is quasi-local if for any $ε > 0,$ there exists $R > 0$ such that for any $A, B \subseteq X$ with $d (A, B) > R,$ we have $∥ χ_{A} T χ_{B} ∥ < ε .$

The set of all finite propagation operators in

𝔅 (ℓ^{2} (X))

forms a

* -

algebra, called the algebraic uniform Roe algebra and denoted by

ℂ_{u} [X] .

The uniform Roe algebra of $X$ is defined to be the operator norm closure of

ℂ_{u} [X]

in

𝔅 (ℓ^{2} (X)),

which is a

C^{*} -

algebra and denoted by

𝐂_{u}^{*} (X) .

The set of all quasi-local operators in

𝔅 (ℓ^{2} (X))

forms a

C^{*} -

algebra, called the uniform quasi-local algebra of $X$ and denoted by

𝐂_{uq}^{*} (X) .

In the following, we focus on a coarse disjoint union

(X, d)

of a sequence of finite metric spaces

{(X_{n}, d_{n})}_{n \in ℕ}

with uniformly bounded geometry.

For each

n \in ℕ,

we consider the pair groupoid

X_{n} \times X_{n}

with

(x, y) \cdot (y, z) = (x, z),

{(x, y)}^{- 1} = (y, x)

for

x, y, z \in X_{n} .

Its unit space can be identified with

X_{n}

with source and range maps given by

s (x, y) = y

and

r (x, y) = x

for

x, y \in X_{n} .

Moreover, let

μ_{n}

be a finite measure on

(X_{n}, ℛ_{n})

for

ℛ_{n} = 𝒫 (X_{n})

and consider the length function

ℓ_{n}

on

𝒢_{n}

defined by

ℓ_{n} (x, y) ≔ d_{n} (x, y)

for

x, y \in X_{n} .

For

L > 0,

we denote

E_{L}^{(n)} ≔ {(x, y) \in X_{n} \times X_{n} ∣ ℓ_{n} (x, y) \leq L} .

It is clear that any bisection in

𝒢_{n}

is admissible, and moreover, we have the following. The proof follows from a well-known fact in coarse geometry (see, e.g., [34

R. Willett and G. Yu, Higher index theory. Cambridge Stud. Adv. Math. 189, Cambridge University Press, Cambridge, 2020, 581 pp. Zbl 1471.19001 MR 4411373

, Lemma 12.2.3]) and hence omitted.

Lemma 6.8.

For any $L > 0,$ there exists $N_{L} \in ℕ$ such that for any $n \in ℕ,$ the set $E_{L}^{(n)}$ is unital symmetric and $N_{L} -$ decomposable. Conversely, for any $n \in ℕ$ and decomposable $K \subseteq 𝒢_{n},$ we have $K \subseteq E_{ℓ (K)}^{(n)} .$

Hence it suffices to consider decomposable subsets of the form

E_{L}^{(n)} .

For

A \subseteq X_{n},

r (E_{L}^{(n)} \cdot A) ∖ A = \partial_{L} A,

where

\partial_{L} A ≔ {x \in X_{n} ∖ A ∣ d_{n} (x, A) \leq L} .

Therefore, we recover⁴

Note that although we define the notion of (asymptotic) expansion in the setting of probability measure in Definitions 3.1 and 3.2, these can be naturally extended to all finite measure spaces as mentioned at the beginning of Section 3.

the notion of measured asymptotic expanders introduced in [23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

].

Proposition 6.9.

In the above setting, the following are equivalent:

(1)

There exist functions $\underline{C}, \underline{N}, \underline{L} : (0, \frac{1}{2}] \to (0, \infty)$ such that for all $n \in ℕ,$ $𝒢_{n}$ is asymptotically expanding with parameters $\underline{C}, \underline{N}, \underline{L} .$
(2)

${(X_{n}, d_{n}, μ_{n})}_{n \in ℕ}$ forms a sequence of measured asymptotic expanders in the sense of [23
K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232
, Definition 6.1].

Consequently, Theorem 3.18 and Remark 3.19 recover [21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, Theorem 4.15].

Now we consider finite propagation and quasi-local operators. Choose

μ_{n}

to be the counting measure on

X_{n} .

From the discussions in Section 4.1,

𝒢_{n}

gives rise to a single groupoid

𝒢 = ⨆_{n \in ℕ} 𝒢_{n}

together with a length function

ℓ

and a measure

μ

on

𝒢^{(0)} = X .

Combining Lemmas 4.4 and 6.8, it is easy to see the following.

Corollary 6.10.

Given $T \in 𝔅 (ℓ^{2} (X)),$ we have $T \in 𝐂_{dyn, q}^{*} (𝒢)$ if and only if there exists $T_{n} \in 𝔅 (ℓ^{2} (X_{n}))$ for each $n \in ℕ$ with ${sup}_{n \in ℕ} ∥ T_{n} ∥ < \infty$ such that $T = (SOT) - \sum_{n \in ℕ} T_{n}$ and for any $ε > 0,$ there exists $L > 0$ satisfying that for any $n \in ℕ$ and $A_{n}, B_{n} \subseteq X_{n}$ with $d_{n} (A_{n}, B_{n}) > L,$ we have $∥ χ_{A_{n}} T_{n} χ_{B_{n}} ∥ < ε .$ A similar result holds for $𝐂_{dyn}^{*} (𝒢) .$

Combining Lemma 4.4 and Proposition 6.9 with [3

H. Bao, X. Chen, and J. Zhang, Strongly quasi-local algebras and their

K -

theories. J. Noncommut. Geom. 17 (2023), no. 1, 241–285 Zbl 1523.46055 MR 4565434

, Lemmas 3.13 and 3.14], we obtain the following.

Proposition 6.11.

With the same notation as above, we have:

(1)

$𝐂_{dyn}^{*} (𝒢) = 𝐂_{u}^{*} (X) \cap Π_{n \in ℕ} 𝔅 (ℓ^{2} (X_{n})) .$
(2)

$𝐂_{dyn}^{*} (𝒢) + 𝔎 (ℓ^{2} (X)) = 𝐂_{u}^{*} (X) .$
(3)

$𝐂_{dyn, q}^{*} (𝒢) = 𝐂_{uq}^{*} (X) \cap Π_{n \in ℕ} 𝔅 (ℓ^{2} (X_{n})) .$
(4)

$𝐂_{dyn, q}^{*} (𝒢) + 𝔎 (ℓ^{2} (X)) = 𝐂_{uq}^{*} (X) .$

Finally, combining with Theorem 5.3′ and Proposition 6.9, we recover both [17

A. Khukhro, K. Li, F. Vigolo, and J. Zhang, On the structure of asymptotic expanders. Adv. Math. 393 (2021), article no. 108073, 35 pp. Zbl 1487.46051 MR 4340225

, Theorem C] and [21

K. Li, J. Špakula, and J. Zhang, Measured asymptotic expanders and rigidity for Roe algebras. Int. Math. Res. Not. IMRN 2023 (2023), no. 17, 15102–15154 Zbl 1534.51008 MR 4637459

, Theorem B].

Corollary 6.12.

Let ${(X_{n}, d_{n})}_{n \in ℕ}$ be a sequence of finite metric spaces with uniformly bounded geometry and $(X, d)$ be their coarse disjoint union. For each $n \in ℕ,$ let $P_{n} \in 𝔅 (ℓ^{2} (X_{n}))$ be a rank-one projection and $ν_{n}$ the associated probability measure on $X_{n} .$ For $P = (SOT) - \sum_{n \in ℕ} P_{n} \in 𝔅 (ℓ^{2} (X)),$ the following are equivalent:

(1)

$P \in 𝐂_{u}^{*} (X) .$
(2)

$P \in 𝐂_{uq}^{*} (X) .$
(3)

${(X_{n}, d_{n}, ν_{n})}_{n \in ℕ}$ forms a sequence of measured asymptotic expanders.

6.3. Semi-direct product groupoids

Here, we consider groupoid actions on fibre spaces, generalising those in Section 6.1. Our idea is to package the information of groupoid actions into semi-direct product groupoids.

Let us start with some preliminaries for groupoid actions. A fibre space over a set

X

is a pair

(Y, p),

where

Y

is a set and

p : Y \to X

is a surjective map. For two fibre spaces

(Y_{1}, p_{1})

and

(Y_{2}, p_{2})

over

X,

we form their fibred product to be

Y_{1} {}_{_{p_{1}}}*_{_{p_{2}}} Y_{2} ≔ {(y_{1}, y_{2}) \in Y_{1} \times Y_{2} ∣ p_{1} (y_{1}) = p_{2} (y_{2})} .

Definition 6.13.

Let

𝒢

be a groupoid. A left $𝒢 -$ space is a fibre space

(Y, p)

over

𝒢^{(0)},

equipped with a map

(γ, y) \mapsto γ y

from

𝒢 {}_{_{s}}*_{_{p}} Y

to

Y

(called a $𝒢 -$ action on $(Y, p)$ ) which satisfies the following:

$p (γ y) = r (γ)$ for $(γ, y) \in 𝒢 {}_{_{s}}*_{_{p}} Y,$ and $p (y) y = y .$
$γ_{2} (γ_{1} y) = (γ_{2} γ_{1}) y$ for $(γ_{1}, y) \in 𝒢 {}_{_{s}}*_{_{p}} Y$ and $s (γ_{2}) = r (γ_{1}) .$

Given a

𝒢 -

space

(Y, p),

the associated semi-direct product groupoid

Y ⋊ 𝒢

is defined (as a set) to be

Y {}_{_{p}}*_{_{r}} 𝒢 .

For

(y, γ) \in Y ⋊ 𝒢,

define its range to be

(y, r (γ))

and its source to be

(γ^{- 1} y, s (γ)) .

The product and inverse are given by

(y, γ) (γ^{- 1} y, γ^{'}) = (y, γ γ^{'}) and {(y, γ)}^{- 1} = (γ^{- 1} y, γ^{- 1}) .

Clearly,

(y, p (y)) \mapsto y

is a bijection from the unit space of

Y ⋊ 𝒢

onto

Y .

Given a bisection

K \subseteq 𝒢,

we define the following map:

(6.1)

β_{K} : p^{- 1} (s (K)) \to p^{- 1} (r (K)), y \mapsto γ_{y} \cdot y for y \in p^{- 1} (s (K)),

where

γ_{y}

is the unique point in

K

such that

s (γ_{y}) = p (y) .

The following is easy.

Lemma 6.14.

For a groupoid $𝒢$ acting on a fibre space $(Y, p)$ and a bisection $K \subseteq 𝒢,$ then $Y {}_{_{p}}*_{_{r}} K$ is a bisection in $Y ⋊ 𝒢 .$ Moreover, we have $τ_{Y {}_{_{p}}*_{_{r}} K} = β_{K} .$

Analogous to Definitions 2.2 and 2.4, we introduce the following.

Definition 6.15.

Let

𝒢

be a groupoid with a length function

ℓ

and

(Y, p)

be a

𝒢 -

space equipped with a probability measure structure

(Y, ℛ, μ) .

A bisection $K \subseteq 𝒢$ is called dynamically admissible if $p^{- 1} (s (K)), p^{- 1} (r (K))$ are measurable, the bijection $β_{K}$ from (6.1) is a measure-class-preserving measurable isomorphism and $ℓ (K)$ is finite.
A subset $K \subseteq 𝒢$ is called dynamically decomposable if $K = ⋃_{i = 1}^{N} K_{i}$ for $N \in ℕ$ and dynamically admissible bisection $K_{i} \subseteq 𝒢 .$

To translate the notions above in terms of

Y ⋊ 𝒢,

we first notice that a length function

ℓ

on

𝒢

gives rise to a length function

\hat{ℓ}

on

Y ⋊ 𝒢

by

(6.2)

\hat{ℓ} (y, γ) ≔ ℓ (γ) for (y, γ) \in Y ⋊ 𝒢 .

Then Lemma 6.14 implies the following.

Lemma 6.16.

A bisection $K \subseteq 𝒢$ is dynamically admissible if and only if $Y {}_{_{p}}*_{_{r}} K$ is admissible in $Y ⋊ 𝒢 .$

Hence we consider the following specific family of admissible bisections in

Y ⋊ 𝒢

:

𝒦_{𝒢, Y} = {Y {}_{_{p}}*_{_{r}} K \subseteq Y ⋊ 𝒢 ∣ K \subseteq 𝒢 dynamically admissible} .

It is clear that

𝒦_{𝒢, Y}

is closed under taking composition and inverse and contains

Y .

Moreover, Lemma 6.16 directly implies the following.

Corollary 6.17.

Let $𝒢$ be a groupoid with a length function $ℓ$ and $(Y, p)$ be a $𝒢 -$ space equipped with a probability measure structure $(Y, ℛ, μ) .$ Equip $Y ⋊ 𝒢$ with the length function $\hat{ℓ}$ from (6.2). Then for $K \subseteq 𝒢,$ $K$ is dynamically decomposable if and only if $Y {}_{_{p}}*_{_{r}} K \subseteq Y ⋊ 𝒢$ is $𝒦_{𝒢, Y} -$ decomposable (see the end of Section 5).

Similar to Lemma 2.6, it is clear that for dynamically decomposable

K \subseteq 𝒢

and measurable

A \subseteq Y,

K \cdot A

is also measurable. Moreover, direct calculations show

K \cdot A = r ((Y {}_{_{p}}*_{_{r}} K) \cdot A) .

Hence the

𝒦_{𝒢, Y} -

asymptotic expansion of

𝒢 ⋊ Y

is equivalent to the following.

Definition 6.18.

Let

𝒢

be a groupoid with a length function

ℓ

and

(Y, p)

be a

𝒢 -

space equipped with a probability measure structure

(Y, ℛ, μ) .

We say that the

𝒢 -

action on

(Y, p)

is asymptotically expanding if for any

α \in (0, \frac{1}{2}],

there exist

C_{α}, N_{α}, L_{α} > 0

and a unital symmetric dynamically

N_{α} -

decomposable subset

K_{α} \subseteq 𝒢

with

ℓ (K_{α}) \leq L_{α}

such that for any

A \in ℛ

with

α \leq μ (A) \leq \frac{1}{2},

then

μ ((K_{α} \cdot A) ∖ A) > C_{α} μ (A) .

Combining Theorem 5.3′′ with the discussions above, we obtain the following.

Corollary 6.19.

Let $𝒢$ be a groupoid with a length function $ℓ$ and $(Y, p)$ be a $𝒢 -$ space equipped with a probability measure structure $(Y, ℛ, μ) .$ Equip $Y ⋊ 𝒢$ with the length function $\hat{ℓ}$ from (6.2). Let $P$ be a rank-one projection in $𝔅 (L^{2} (Y, μ))$ and $ν$ the associated measure to $P$ constructed in Section 5, then the following are equivalent:

(1)

$P \in 𝐂_{dyn}^{*} (Y ⋊ 𝒢, 𝒦_{𝒢, Y}) .$
(2)

$P \in 𝐂_{dyn, q}^{*} (Y ⋊ 𝒢, 𝒦_{𝒢, Y}) .$
(3)

$Y ⋊ 𝒢$ is $𝒦_{𝒢, Y} -$ asymptotically expanding in measure $ν .$
(4)

The $𝒢 -$ action on $(Y, p)$ is asymptotically expanding in measure $ν .$

Remark 6.20.

In the case of a group

G

acting on a space

Y,

Lemma 6.1 shows that when the action is measure class preserving, then the family

𝒦_{G, Y}

is cofinal in the sense of Remark 5.8. Hence combining Remark 5.8 with Proposition 6.3, Corollary 6.19 generalises Corollary 6.5.

6.4. The HLS groupoid and its variant

Throughout this subsection, let

Γ

be a finitely generated group and

{N_{i}}_{i \in ℕ}

be a family of nested, finite index normal subgroups of

Γ

with trivial intersection. For each

i \in ℕ,

denote the quotient map

π_{i} : Γ \to Γ / N_{i}

and

π_{\infty} : Γ \to Γ

the identity map.

The associated HLS groupoid (after Higson, Lafforgue and Skandalis from [15

N. Higson, V. Lafforgue, and G. Skandalis, Counterexamples to the Baum–Connes conjecture. Geom. Funct. Anal. 12 (2002), no. 2, 330–354 Zbl 1014.46043 MR 1911663

C^{*} -

algebras are the same. Münster J. Math. 8 (2015), no. 1, 241–252 Zbl 1369.46064 MR 3549528

])

𝒢

is defined to be

𝒢 ≔ ⨆_{i \in ℕ \cup {\infty}} {i} \times X_{i}, where X_{i} = {\begin{matrix} Γ / N_{i} & if i \in ℕ; \\ Γ & if i = \infty . \end{matrix}

Since

𝒢

is a bundle of groups, it cannot be asymptotically expanding for any given measure and length function. Regarding

𝒢

as a family of groups, it is uniformly asymptotically expanding since each unit space consists of a single point.

Now we consider a variant of the HLS groupoid from [2

V. Alekseev and M. Finn-Sell, Non-amenable principal groupoids with weak containment. Int. Math. Res. Not. IMRN 2018 (2018), no. 8, 2332–2340 Zbl 1406.22016 MR 3801485

]. To recall their construction, we use the same notation as above and denote

\hat{Γ} ≔ \lim_{\leftarrow} Γ / N_{i}

the profinite completion of

Γ

with respect to the family

{Γ / N_{i}}_{i \in ℕ},

that is, the inverse limit of

{Γ / N_{i}}_{i \in ℕ} .

For each

i \in ℕ,

let

𝒢_{i}^{AFS}

be the transformation groupoid

X_{i} ⋊ (Γ / N_{i})

with the action by left multiplication. For

i = \infty,

let

𝒢_{\infty}^{AFS}

be the transformation groupoid

\hat{Γ} ⋊ Γ

with the natural free action. Then the groupoid constructed in [2

V. Alekseev and M. Finn-Sell, Non-amenable principal groupoids with weak containment. Int. Math. Res. Not. IMRN 2018 (2018), no. 8, 2332–2340 Zbl 1406.22016 MR 3801485

] is defined to be the disjoint union (thanks to [2

V. Alekseev and M. Finn-Sell, Non-amenable principal groupoids with weak containment. Int. Math. Res. Not. IMRN 2018 (2018), no. 8, 2332–2340 Zbl 1406.22016 MR 3801485

, Lemma 2.1])

𝒢^{AFS} ≔ ⨆_{i \in ℕ \cup {\infty}} 𝒢_{i}^{AFS} .

For

i \in ℕ,

we take the normalised counting measure

μ_{i}

on

Γ / N_{i} .

For

i = \infty,

we take the induced probability

Γ -

invariant measure

μ_{\infty}

on

\hat{Γ} .

On the other hand, fix a length function

ℓ

on

Γ,

which induces a quotient length function

ℓ_{i}

on

Γ / N_{i}

for each

i \in ℕ .

Following Section 6.1, we obtain a length function on

𝒢_{i}^{AFS}

for each

i \in ℕ \cup {\infty} .

By the discussions in Section 4.1, these can be combined to provide a measure and a length function for

𝒢^{AFS} .

We have the following characterisation.

Proposition 6.21.

With the same notation as above, the following are equivalent:

(1)

$𝒢_{i}^{AFS}$ is asymptotically expanding in measure uniformly for $i \in ℕ \cup {\infty}$ (i.e., having the same expansion parameters).
(2)

The natural action of $Γ$ on $\hat{Γ}$ is asymptotically expanding in measure.
(3)

The natural action of $Γ$ on $\hat{Γ}$ is expanding in measure.
(4)

The sequence ${Γ / N_{i}}_{i \in ℕ}$ forms a sequence of asymptotic expander graphs.
(5)

The sequence ${Γ / N_{i}}_{i \in ℕ}$ forms a sequence of expander graphs.
(6)

$𝒢_{i}^{AFS}$ is expanding in measure uniformly for $i \in ℕ \cup {\infty} .$

Proof.

“(1)

\Rightarrow

(2)”: Condition (1) implies that

𝒢_{\infty}^{AFS}

is asymptotically expanding in measure. Then by Proposition 6.2, this implies (2).

“(2)

\Leftrightarrow

(3)” is due to [1

M. Abért and G. Elek, Dynamical properties of profinite actions. Ergodic Theory Dynam. Systems 32 (2012), no. 6, 1805–1835 Zbl 1297.37004 MR 2995875

, Theorem 4] and [23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

, Proposition 3.5].

“(3)

\Leftrightarrow

(5)” is fairly well known (see, e.g., [1

M. Abért and G. Elek, Dynamical properties of profinite actions. Ergodic Theory Dynam. Systems 32 (2012), no. 6, 1805–1835 Zbl 1297.37004 MR 2995875

, Lemma 2.2]).

“(2)

\Leftrightarrow

(4)” is due to [23

K. Li, F. Vigolo, and J. Zhang, Asymptotic expansion in measure and strong ergodicity. J. Topol. Anal. 15 (2023), no. 2, 361–399 Zbl 1522.37002 MR 4585232

, Theorem 6.16, Corollary 6.17].

“(5)

\Rightarrow

(6)”: For each

i \in ℕ,

note that

𝒢_{i}^{AFS}

is isomorphic to the pair groupoid

(Γ / N_{i}) \times (Γ / N_{i}),

where the length function corresponds to the one given by

(x, y) \mapsto d_{i} (x, y)

for the left-invariant metric

d_{i}

on

Γ / N_{i}

induced by

ℓ_{i} .

According to Proposition 6.9, (4) is equivalent to that

𝒢_{i}^{AFS}

is expanding uniformly for

i \in ℕ .

This concludes (5) since we already showed “(4)

\Rightarrow

(3)”.

“(6)

\Rightarrow

(1)” is trivial.

Finally, combining with Theorem 5.3′, we obtain the following.

Corollary 6.22.

For the groupoid $𝒢^{AFS},$ take $P_{i} \in 𝔅 (L^{2} (X_{i}, μ_{i}))$ to be the averaging projection for $i \in ℕ \cup {\infty} .$ For $P = (SOT) - \sum_{i \in ℕ \cup {\infty}} P_{i},$ the following are equivalent:

(1)

$P \in 𝐂_{u}^{*} (𝒢^{AFS}) .$
(2)

$P \in 𝐂_{uq}^{*} (𝒢^{AFS}) .$
(3)

The natural action of $Γ$ on $\hat{Γ}$ is asymptotically expanding in measure.
(4)

The sequence ${Γ / N_{i}}_{i \in ℕ}$ forms a sequence of expander graphs.

6.5. Graph groupoids

We first recall background knowledge from [18

A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs, groupoids, and Cuntz–Krieger algebras. J. Funct. Anal. 144 (1997), no. 2, 505–541 Zbl 0929.46055 MR 1432596

].

Throughout this subsection, let

G = (V, E, r, s)

be a directed uniformly locally finite connected graph, where

V

is the vertex set,

E

is the edge set and

r, s : E \to V

describe the range and source of edges. Here,

G

is called uniformly locally finite if

{sup}_{v \in V} | s^{- 1} (v) |

and

{sup}_{v \in V} | r^{- 1} (v) |

are finite. Given

v \in V,

denote

s - \deg (v) ≔ | s^{- 1} (v) | .

A finite path in

G

is a sequence

α = (α_{1}, \dots, α_{k})

of edges in

E

with

s (α_{j + 1}) = r (α_{j})

for

1 \leq j \leq k - 1 .

Write

s (α) = s (α_{1})

and

r (α) = r (α_{k}) .

The length of

α

is

| α | ≔ k .

Denote by

v

the path of length

0

with

s (v) = r (v) = v .

Denote

F (G)

(resp.

P (G)

) the set of all finite (resp. infinite) paths in

G

and

F (G, v)

(resp.

P (G, v)

) those starting at

v \in V .

For

α, μ \in F (G)

satisfying

r (α) = s (μ),

we define a path

α μ \in F (G)

of length

| α | + | μ |

by

α μ = (α_{1}, \dots, α_{| α |}, μ_{1}, \dots, μ_{| μ |}) .

We can similarly define

α x \in P (G)

for

α \in F (G)

and

x \in P (G)

satisfying

s (x) = r (α) .

Equip

P (G)

with the induced topology from the product topology on the infinite product

Π E,

which is locally compact,

σ -

compact, totally disconnected and Hausdorff. Moreover, it has a basis consisting of the following subsets:

Z (α) = {x \in P (G) ∣ x_{1} = α_{1}, \dots, x_{| α |} = α_{| α |}}, where α \in F (G) .

By [18

A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs, groupoids, and Cuntz–Krieger algebras. J. Funct. Anal. 144 (1997), no. 2, 505–541 Zbl 0929.46055 MR 1432596

, Lemma 2.1],

Z (α) \cap Z (β) \neq \emptyset

if and only if either

α

is a prefix of

β

(i.e.,

β = α β^{'}

for some

β^{'} \in F (G)

) or

β

is a prefix of

α .

To construct the groupoid associated to the graph

G,

we consider the following equivalence relation on

P (G)

: two paths

x, y \in P (G)

are shift equivalent with lag

k \in ℤ

(written

x \sim_{k} y

) if there exists

N \in ℕ

such that

x_{i} = y_{i + k}

for all

i \geq N .

Definition 6.23.

For a directed uniformly locally finite connected graph

G = (V, E, r, s),

the associated graph groupoid

𝒢

is defined to be (as a set)

𝒢 = {(x, k, y) \in P (G) \times ℤ \times P (G) ∣ x \sim_{k} y}

with

(x, k, y) \cdot (y, l, z) ≔ (x, k + l, z)

and

{(x, k, y)}^{- 1} ≔ (y, - k, x) .

The range and source maps are given by

r (x, k, y) = x

and

s (x, k, y) = y,

respectively, with

𝒢^{(0)} = P (G) .

For

α, β \in F (G)

with

r (α) = r (β),

denote

Z (α, β) ≔ {(x, k, y) ∣ x \in Z (α), y \in Z (β), k = | β | - | α |, x_{i} = y_{i + k} for i \geq | α |},

which is a bisection in

𝒢

with

r (Z (α, β)) = Z (α)

and

s (Z (α, β)) = Z (β) .

The sets

Z (α, β)

form a basis for a locally compact Hausdorff topology on

𝒢,

which makes it a second countable étale groupoid where each

Z (α, β)

is compact open. The induced topology on

𝒢^{(0)} = P (G)

coincides with the product topology above.

To apply our results to the graph groupoid

𝒢,

we first endow

𝒢

with the following length function

ℓ .

For

(x, k, y) \in 𝒢,

choose the smallest

N \in ℕ

such that

x_{i} = y_{i + k}

for all

i \geq N,

denoted by

N (x, k, y),

and set

(6.3)

ℓ (x, k, y) = 2 N (x, k, y) + k .

Note that

N (x, k, y) + k \geq 0

for any

(x, k, y) \in 𝒢 .

Intuitively,

ℓ (x, k, y)

is the total length of the shortest starting segments in

x

and

y,

deleting which

x

and

y

are the same with lag

k .

It is routine to check that

ℓ

is indeed a length function on

𝒢

taking values in

ℕ .

Moreover, given

n \in ℕ,

the ball in

𝒢

defined by

B_{n} ≔ {(x, k, y) \in 𝒢 ∣ ℓ (x, k, y) \leq n}

can be reconstructed as follows:

(6.4)

B_{n} = ⋃ {Z (α, β) ∣ α, β \in F (G) with r (α) = r (β) and | α | + | β | \leq n} .

Moreover, we have

(6.5)

B_{n} = {(B_{1})}^{n} for any n \in ℕ .

These can be verified by direct calculations, and we omit the details.

Next, we will define a Borel measure

μ

on

𝒢^{(0)} = P (G) .

Given

v \in V

and

b_{v} > 0,

we define a measure

μ_{v}

on

P (G, v)

by first setting

(6.6)

μ_{v} (Z (α)) ≔ b_{v} \cdot \prod_{i = 1}^{k} \frac{1}{s - \deg (s (α_{i}))} for α = (α_{1}, α_{2}, \dots, α_{k}) \in F (G, v) .

Since the sets

Z (α)

for

α \in F (G, v)

form a basis for the topology on

P (G, v),

then the above gives rise to a Borel measure

μ_{v}

on

P (G, v) .

Combine

μ_{v}

into a single probability Borel measure

μ

on

P (G) .

Since

G

is connected and uniformly locally finite,

V

is countable. For each

v \in V,

choose

b_{v} > 0

such that

\sum_{v \in V} b_{v} = 1 .

Then the measures

μ_{v}

defined above give rise to a Borel measure

μ

on

P (G) .

In the sequel, we fix such a measure

μ .

By the construction of

ℓ

and

μ

above, it is clear that for any

α, β \in F (G)

with

r (α) = r (β),

the subset

Z (α, β)

is an admissible bisection. Moreover, we have the following.

Lemma 6.24.

Each ball $B_{n} = {(x, k, y) \in 𝒢 ∣ ℓ (x, k, y) \leq n}$ is decomposable.

Proof.

Fix

n \in ℕ .

We aim to decompose the ball

B_{n}

into finitely many admissible bisections. To achieve this, we construct another graph

\hat{G} = (\hat{V}, \hat{E})

with

\hat{V} ≔ {(α, β) ∣ α, β \in F (G) with r (α) = r (β) and | α | + | β | \leq n},

and two vertices

(α_{1}, β_{1})

and

(α_{2}, β_{2})

in

\hat{V}

are connected by an edge if and only if at least one of the following four cases holds: (i)

α_{1}

is a prefix of

α_{2};

(ii)

α_{2}

is a prefix of

α_{1};

(iii)

β_{1}

is a prefix of

β_{2};

(iv)

β_{2}

is a prefix of

β_{1} .

Since

G

is uniformly locally finite and connected, then

\hat{G}

is uniformly locally finite as well. It is a well-known fact (see, e.g., [31

J. Špakula and J. Zhang, Quasi-locality and Property A. J. Funct. Anal. 278 (2020), no. 1, article no. 108299, 25 pp. Zbl 1444.46016 MR 4027745

, Lemma 4.2]) that

\hat{V}

can be decomposed into

\hat{V} = ⨆_{p = 1}^{N} {\hat{V}}_{p}

such that for each

p,

any two distinct vertices in

{\hat{V}}_{p}

are not connected by an edge.

For each

p = 1, \dots, N,

set

K_{p} ≔ ⨆_{(α, β) \in {\hat{V}}_{p}} Z (α, β) .

By the construction of

\hat{E},

it is routine to check that

K_{p}

is an admissible bisection. Finally, (6.4) implies

B_{n} = ⋃_{p = 1}^{N} K_{p},

and hence we conclude the proof.

Lemma 6.24 shows that the set

{B_{n} ∣ n \in ℕ}

is cofinal in the set of all decomposable subsets of

𝒢

in the sense of Remark 5.8. Hence we can merely use these balls to define asymptotic expansion, dynamical propagation and quasi-locality for graph groupoids by the discussion at the end of Section 5.

Thanks to Lemma 6.24 and (6.5), we have the following.

Lemma 6.25.

The graph groupoid $𝒢$ is asymptotically expanding in measure if and only if for any $α \in (0, \frac{1}{2}],$ there exists $C_{α} > 0$ such that for any measurable $A \subseteq 𝒢^{(0)}$ with $α \leq μ (A) \leq \frac{1}{2},$ then $μ (r (B_{1} \cdot A)) > (1 + C_{α}) μ (A) .$

Lemma 6.25 can be further refined as follows under an extra assumption.

Proposition 6.26.

Let $G$ be a uniformly locally finite connected graph with $𝒢, ℓ, μ$ as above. Assume there exists $C \geq 1$ such that $\frac{b_{w}}{C} \leq b_{v} \leq C b_{w}$ for any $v, w \in V$ connected by an edge, where ${(b_{v})}_{v}$ are the weights used to define $μ .$ Then $𝒢$ is asymptotically expanding in measure if and only if for any $0 < α \leq \frac{1}{2},$ there exists $C_{α} > 0$ such that for any measurable $A \subseteq 𝒢^{(0)}$ of the form $A = ⋃_{i = 1}^{N} Z (α_{i})$ with $α \leq μ (A) \leq \frac{1}{2}$ for $N \in ℕ$ and $α_{i} \in F (G),$ we have $μ (r (B_{1} \cdot A)) > (1 + C_{α}) μ (A) .$

Proof.

By Lemma 6.25, we only need to prove sufficiency. Since

𝒢^{(0)} = P (G)

is a locally compact Hausdorff space in which every open subset is

σ -

compact and

μ

is finite, it follows from [12

G. B. Folland, Real analysis. Modern techniques and their applications, 2nd edn., Pure Appl. Math., John Wiley & Sons, New York, 1999, 386 pp. Zbl 0924.28001 MR 1681462

, Theorem 7.8] that

μ

is regular. Take

C

as in the assumption and set

D = \max {{sup}_{v \in V} | s^{- 1} (v) |, {sup}_{v \in V} | r^{- 1} (v) |} .

We claim that for any measurable

A^{'} \subseteq P (G),

we have

(6.7)

μ (r (B_{1} \cdot A^{'})) \leq C D (D + 2) μ (A^{'}) .

Indeed, for any

α = (α_{1}, \dots, α_{k}) \in F (G)

with

k \geq 1,

set

F {(G)}_{α} ≔ {α} \cup {(α_{2}, \dots, α_{k})} \cup {(α_{0}, α_{1}, \dots, α_{k}) ∣ α_{0} \in E with r (α_{0}) = s (α_{1})} .

Here, we take the convention that if

k = 1,

then

(α_{2}, \dots, α_{k}) = r (α_{1}) .

By the choice of

D,

we have

| F {(G)}_{α} | \leq D + 2 .

Moreover, it follows from the definition of

μ

in (6.6) that

μ (Z (β)) \leq C D μ (Z (α))

for any

β \in F {(G)}_{α} .

Hence we obtain

μ (r (B_{1} \cdot Z (α))) \leq \sum_{β \in F {(G)}_{α}} μ (Z (β)) \leq C D (D + 2) μ (Z_{α}) .

Denote

ℳ ≔ {measurable A^{'} \subseteq P (G) ∣ (6.7) holds for A^{'}} .

Note that the set

𝒫 ≔ {Z (α) ∣ α \in F (G)}

is a semi-ring, and hence the analysis above implies that

ℳ

contains the ring

R (𝒫)

generated by

𝒫 .

Due to the monotone class theorem,

ℳ

contains the

σ -

algebra generated by

𝒫,

which coincides with the Borel

σ -

algebra on

P (G) .

Hence we conclude the claim.

Given

0 < α \leq \frac{1}{2},

we have a constant

C_{α / 2} > 0

from the assumption for

\frac{α}{2} .

Also given a measurable

A \subseteq P (G)

with

α \leq μ (A) < \frac{1}{2},

take

ε_{0} ≔ \min {\frac{α}{2}, \frac{1}{2} - μ (A), \frac{α C_{α / 2}}{2 + 2 C_{α / 2} + 2 C D (D + 2)}} > 0 .

Due to regularity, there is an open subset

\tilde{A} = ⋃_{i = 1}^{N} Z (α_{i})

for

α_{i} \in F (G)

such that

μ (A △ \tilde{A}) < ε_{0},

which implies that

\frac{α}{2} \leq μ (\tilde{A}) \leq \frac{1}{2} .

By the claim above, we have

(6.8)

\begin{matrix} μ (r (B_{1} \cdot A)) & \geq μ (r (B_{1} \cdot \tilde{A})) - μ (r (B_{1} \cdot (A △ \tilde{A}))) \\ > (1 + C_{α / 2}) μ (\tilde{A}) - C D (D + 2) μ (A △ \tilde{A}) \\ \geq (1 + C_{α / 2}) (μ (A) - ε_{0}) - C D (D + 2) ε_{0} \\ \geq (1 + C_{α / 2}) μ (A) - \frac{α C_{α / 2}}{2} \geq (1 + \frac{C_{α / 2}}{2}) μ (A) . \end{matrix}

Finally, we consider measurable

A \subseteq P (G)

with

μ (A) = \frac{1}{2} .

Take

ε_{1} ≔ \min {\frac{3}{8}, \frac{C_{1 / 8}}{32 + 4 C_{1 / 8} + 32 C D (D + 2)}} > 0 .

Without loss of generality, assume

C_{1 / 8} \leq 1 .

Again due to regularity, we choose an open subset

\tilde{A} = ⋃_{i = 1}^{N} Z (α_{i})

for

α_{i} \in F (G)

such that

μ (A △ \tilde{A}) < ε_{1} .

This implies that

\frac{1}{2} - ε_{1} < μ (\tilde{A}) < \frac{1}{2} + ε_{1} .

We need to control the expansion of

\tilde{A} .

If

μ (\tilde{A}) \leq \frac{1}{2},

then

μ (r (B_{1} \cdot \tilde{A}) ∖ \tilde{A}) \geq C_{1 / 8} μ (A) .

Otherwise, set

\tilde{B} ≔ P (G) ∖ (r (B_{1} \cdot \tilde{A}))

and then

μ (\tilde{B}) < \frac{1}{2} .

If

μ (\tilde{B}) < \frac{1}{4},

then

μ (r (B_{1} \cdot \tilde{A})) = 1 - μ (\tilde{B}) > 1 - \frac{1}{4} \geq \frac{3}{2} μ (A) .

If

μ (\tilde{B}) \geq \frac{1}{4},

then by applying (6.8) to

\tilde{B},

we obtain

μ (r (B_{1} \cdot \tilde{B})) > (1 + \frac{C_{1 / 8}}{2}) μ (\tilde{B}) .

Hence

μ (r (B_{1} \cdot \tilde{A}) ∖ \tilde{A}) \geq μ (r (B_{1} \cdot \tilde{B}) ∖ \tilde{B}) > \frac{C_{1 / 8}}{2} μ (\tilde{B}) \geq \frac{C_{1 / 8}}{8} \cdot μ (\tilde{A}) .

In conclusion, we obtain

μ (r (B_{1} \cdot \tilde{A})) > (1 + \frac{C_{1 / 8}}{8}) μ (\tilde{A}) .

Therefore,

\begin{matrix} μ (r (B_{1} \cdot A)) & \geq μ (r (B_{1} \cdot \tilde{A})) - μ (r (B_{1} \cdot (A △ \tilde{A}))) \\ > (1 + \frac{C_{1 / 8}}{8}) μ (\tilde{A}) - C D (D + 2) μ (A △ \tilde{A}) \\ \geq (1 + \frac{C_{1 / 8}}{8}) (μ (A) - ε_{1}) - C D (D + 2) ε_{1} \geq (1 + \frac{C_{1 / 8}}{16}) μ (A), \end{matrix}

which concludes the proof.

Finally, we provide a concrete example.

Example 6.27.

Fix

k \in ℕ .

Consider the graph

G = (V, E),

where

V = ℕ

and

E = {(n, n + i) ∣ n \in ℕ and i = 1, 2, \dots, k} .

Let

𝒢

be the associated graph groupoid with the length function from (6.3), and equip

P (G)

with the Borel probability measure from (6.6) by setting

b_{n} = μ (Z (n)) ≔ \frac{1}{2^{n + 1}}

for

n \in ℕ .

More precisely, for

α = (α_{1}, \dots, α_{m}) \in F (G)

with

s (α_{1}) = n,

set

μ (Z (α)) = \frac{1}{2^{n + 1} k^{m}} .

We have the following:

(1)

If $k = 1,$ then $𝒢$ is expanding in measure.
(2)

If $k \geq 2,$ then $𝒢$ is not asymptotically expanding in measure.

For (1). For any

α \in F (G),

the set

Z (α)

consists of a single point, which is the infinite path starting at

s (α) .

Hence

Z (α) = Z (s (α)) .

Now for any measurable

A \subseteq P (G)

with

0 < μ (A) \leq \frac{1}{2},

take

n_{A} ≔ \min {s (α) ∣ α \in A} .

If

n_{A} = 0,

then

μ (A) \leq \frac{1}{2}

implies that

A = Z (0) .

Hence

Z (1) \subseteq r (B_{1} \cdot A),

which implies that

μ (r (B_{1} \cdot A)) \geq \frac{3}{2} μ (A) .

If

n_{A} \geq 1,

then by the choice of

μ,

we have

μ (Z (n_{A})) \geq \frac{1}{2} μ (A) .

Since

Z (n_{A} - 1) \subseteq r (B_{1} \cdot A),

then

μ (r (B_{1} \cdot A)) \geq μ (A) + 2 μ (Z (n_{A})) \geq 2 μ (A) .

Hence

𝒢

is expanding in measure, and we conclude (1).

For (2). Fix

k \geq 2 .

For

n_{1} \leq n_{2} \in ℕ,

denote

F {(G)}_{n_{1}, n_{2}} ≔ {α \in F (G) ∣ s (α) = n_{1}, r (α) = n_{2}}

and

Z_{n_{1}, n_{2}} ≔ ⋃ {Z (α) ∣ α \in F {(G)}_{n_{1}, n_{2}}} .

Direct calculations show that for

n \geq k,

we have

μ (Z_{0, n}) + \sum_{j = 1}^{k - 1} \frac{k - j}{k} μ (Z_{0, n - j}) = \frac{1}{2} .

Note that

1 + \sum_{j = 1}^{k - 1} \frac{k - j}{k} = \frac{k + 1}{2} .

Hence for any

p \in ℕ,

there exists an integer

n_{p} \in {p k + 1, p k + 2, \dots, (p + 1) k}

such that

μ (Z_{0, n_{p}}) \geq \frac{1}{k + 1} .

Now for positive integer

p \in ℕ,

define

F {(G)}_{n_{p}} ≔ ⨆_{m = 0}^{n_{p}} F {(G)}_{m, n_{p}} and A_{p} = ⋃ {Z (α α_{p}) ∣ α \in F {(G)}_{n_{p}}},

where

α_{p} ≔ (n_{p}, n_{p} + 1) .

Then we have

μ (A_{p}) > \frac{1}{k} μ (Z_{0, n_{p}}) \geq \frac{1}{k (k + 1)} and μ (A_{p}) \leq \frac{1}{k} μ (P (G)) = \frac{1}{k} .

By the construction of

F {(G)}_{n_{p}},

it is clear that for any

α \in F {(G)}_{n_{p}}

with

| α | \geq 1,

we have

r (B_{1} \cdot Z (α α_{p})) \subseteq A_{p} .

Hence

μ (r (B_{1} \cdot A_{p})) = μ (A_{p} ⊔ Z (n_{p} + 1)) = μ (A_{p}) + \frac{1}{2^{n_{p} + 2}},

which shows that

μ (r ((B_{1} \cdot A_{p}) ∖ A_{p})) = \frac{1}{2^{n_{p} + 2}} \to 0

as

p \to \infty .

This concludes (2).

Funding

Qin Wang is partially supported by NSFC (No. 12571135), Key Laboratory of MEA (Ministry of Education), the Science and Technology Commission of Shanghai (No. 22DZ2229014). Jiawen Zhang is supported by the National Key R&D Program of China 2022YFA1007000 and NSFC12422107.

Footnotes

¹
Since we are only interested in the existence of $K,$ we can always make it unital and symmetric.
²
This is equivalent to saying that the reduction $𝒢_{Y}^{Y} ≔ s^{- 1} (Y) \cap r^{- 1} (Y)$ is asymptotically expanding.
³
The extra requirement on the Radon–Nikodym derivatives will play an important role in proving our main results in Section 5.
⁴
Note that although we define the notion of (asymptotic) expansion in the setting of probability measure in Definitions 3.1 and 3.2, these can be naturally extended to all finite measure spaces as mentioned at the beginning of Section 3.

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Cite this article

Xulong Lu, Qin Wang, Jiawen Zhang, Asymptotic expansion for groupoids and Roe-type algebras. J. Noncommut. Geom. (2026), published online first

DOI 10.4171/JNCG/666