Pseudo-differential extension for graded nilpotent Lie groups

Classical pseudo-differential operators of order zero on a graded nilpotent Lie group $G$ form a $^*$-subalgebra of the bounded operators on $L^2(G)$. We show that its $C^*$-closure is an extension of a noncommutative algebra of principal symbols by compact operators. As a new approach, we use the generalized fixed point algebra of an $\mathbb{R}_{>0}$-action on a certain ideal in the $C^*$-algebra of the tangent groupoid of $G$. The action takes the graded structure of $G$ into account. Our construction allows to compute the $K$-theory of the algebra of symbols.


Introduction
A homogeneous Lie group is a nilpotent Lie group G with a dilation action of R >0 by group automorphisms.The dilation action allows to scale with different speed in different tangent directions.A slightly less general class are graded nilpotent Lie groups.A prominent example is the Heisenberg group whose Lie algebra is generated by {X, Y, Z} with [X, Y ] = Z and [X, Z] = [Y, Z] = 0. Then A λ (X) = λX, A λ (Y ) = λY and A λ (Z) = λ 2 Z for λ > 0 define dilations on the Heisenberg algebra.The dilations induce a new notion of order and homogeneity for differential operators on G.For example, in the case of the Heisenberg group, one would assign order 2 to Z and order 1 to X and Y .
Certain hypoelliptic operators, like Hörmander's sum of squares or Kohn's Laplacian b , can be analysed using homogeneous convolution operators on homogeneous Lie groups [Fol77].Therefore, it is desirable to have a pseudo-differential calculus that takes the homogeneous structure into account.In the 80s, a kernel-based pseudo-differential calculus for homogeneous Lie groups was developed in [CGGP92].Recently, Fischer and Ruzhansky introduced in [FR16] a symbolic calculus for graded nilpotent Lie groups.Instead of functions on the cotangent bundle as in the Euclidean case, the symbols are given here by fields of operators using operator valued Fourier transform.This uses that the representation theory of graded nilpotent Lie groups is well-known and the abstract Plancherel Theorem [Dix77] applies.In [FFK20] homogeneous expansions, classical pseudo-differential operators and their principal symbols were defined for this calculus.Graded nilpotent Lie groups are also instances of filtered manifolds, for which a pseudo-differential calculus was developed in [vEY19].
This article describes a different approach to pseudo-differential operators on homogeneous Lie groups using generalized fixed point algebras.Generalized fixed point algebras were introduced by Rieffel [Rie04,Rie90] to generalize proper group actions on spaces to the noncommutative setting.If a locally compact group H acts properly on a locally compact Hausdorff space X, the orbit space H\X is again locally compact.The generalized fixed point algebra is in this case C 0 (H\X), which can be viewed as a subalgebra of the H-invariant multipliers of A = C 0 (X).Moreover, R = C c (X) can be completed into an imprimitivity bimodule between an ideal in the reduced crossed product C * r (H, C 0 (X)) and the generalized fixed point algebra.In [Mey01] it is investigated for which group actions α : H A on a C * -algebra A, one can build a generalized fixed point algebra which is Morita-Rieffel equivalent to an ideal in C * r (H, A).The crucial step is to find a dense subset R ⊂ A that is continuously square-integrable.As it turns out, such R can fail to exist or to be unique.If R satisfies the requirements, the generalized fixed point algebra Fix H (A, R) is generated by averages H α x (a * b) dx for a, b ∈ R, understood as H-invariant multipliers of A.
A classical pseudo-differential operator of order k on a manifold M is determined up to operators of lower order by its principal symbol.The principal symbol is a k-homogeneous function on T * M \(M × {0}).Hence, for k = 0 the principal symbol is a generalized fixed point of the scaling action of H = R >0 on T * M \(M × {0}) in the cotangent direction.Therefore, the C * -closure of the 0-homogeneous symbols C 0 (S * M ) is a generalized fixed point algebra.As it turns out, not only the principal symbols, but also the pseudo-differential operators of order zero themselves are generalized fixed points.A special case of the results in [DS14] is that the classical pseudodifferential calculus for a manifold M can be recovered from Connes' tangent groupoid [Con94].Moreover, they observed that each pseudo-differential operator of order zero can be written as an average ∞ 0 f t dt t , where (f t ) t∈[0,∞) is an element of the C * -algebra of the tangent groupoid of M satisfying certain conditions.Elements of a generalized fixed point algebra are obtained in exactly this fashion.
It was shown in [Mil17] that the C * -closure of classical pseudo-differential operators of order zero on R n inside the bounded operators on L 2 (R n ) is a generalized fixed point algebra.In fact, it is the generalized fixed point algebra of a zoom action of R >0 on an ideal in the C * -algebra of the tangent groupoid.In this article, we generalize this result to graded nilpotent Lie groups G.We describe a variant of Connes' tangent groupoid where the operation on the tangent bundle T G is given by group multiplication in the fibres.This is a special case of the tangent groupoid of a filtered manifold which was considered before in [vEY17,CP19,HSSH18].It is equipped with a zoom action of R >0 , which is induced by the dilations on G.
Let J be the ideal in C * (G) that consists of all elements whose restriction to (x, 0), which is an element of C * (T x G) ∼ = C * (G), lies in the kernel of the trivial representation of G for all x ∈ G.In the commutative case, this corresponds under Fourier transform to taking out the zero section in T * R n , which is necessary to obtain a proper action.We show that there is a subset R ⊂ J such that Fix R>0 (J, R) is defined.Let J 0 and R 0 be the restriction of J and R to t = 0, respectively.Then we obtain another generalized fixed point algebra Fix R>0 (J 0 , R 0 ).These fit into an extension (1) K(L 2 G) Fix R>0 (J, R) Fix R>0 (J 0 , R 0 ).
We call this the pseudo-differential extension of G.We justify the name by showing that this extension is the C * -completion of the order zero pseudo-differential extension defined in [FFK20] The C * -algebra Fix R>0 (J 0 , R 0 ) of principal symbols is, in general, noncommutative.However, as it is a generalized fixed point algebra, it is Morita-Rieffel equivalent to an ideal in C * r (R >0 , J 0 ).Using the representation theory of nilpotent Lie groups and, in particular, Kirillov theory [Kir62] and Pukanszky's stratification [Puk67], we show that it is actually Morita-Rieffel equivalent to the whole crossed product.Furthermore, Fix R>0 (J, R) is Morita equivalent to C * r (R >0 , J), which was observed before in [DS14] for the case of dilations given by scalar multiplication.
The Morita equivalence allows us to prove that Fix R>0 (J 0 , R 0 ) is KK-equivalent to C 0 (S * R n ).Hence, although the symbols in the homogeneous and Euclidean case differ, the resulting C * -algebras have the same K-theory.Moreover, our approach can be used to recover the computation of the spectrum of C * ( Ṡ0 c ) in [FFK20].The article is organized as follows.Section 2 introduces generalized fixed point algebras and examines their behaviour for extensions of C * -algebras.Section 3 compiles some facts about analysis on homogeneous Lie groups and their representation theory.The tangent groupoid G of a homogeneous Lie group and its C * -algebra are defined in Section 4. In Section 5 we build the pseudo-differential extension (1) using generalized fixed point algebras.Section 6 relates Fix R>0 (J 0 , R 0 ) with operators of type 0 on G.In Section 7 we compare the generalized fixed point algebra extension to the calculus of Fischer-Ruzhansky-Fermanian-Kammerer.In Section 8 we show the mentioned Morita-Rieffel equivalences.Moreover, we compute the spectrum and K-theory of Fix R>0 (J 0 , R 0 ).The results in this article are also contained in the author's PhD thesis [Ewe20].
Acknowledgements.The author thanks her PhD advisors Ralf Meyer and Ryszard Nest for their suggestions and advice.Furthermore, she would like to thank Véronique Fischer for discussions during her visit in Göttingen.This research was supported by the RTG 2491 "Fourier Analysis and Spectral Theory".

Generalized fixed point algebras and extensions
Rieffel proposes a notion for proper group action on C * -algebras in [Rie04,Rie90], which generalizes proper actions on locally compact Hausdorff spaces.This leads to the construction of generalized fixed point algebras.We follow the approach taken in [Mey01].In this section, we recall the notions used there and prove some results regarding the behaviour of generalized fixed point algebras under extensions of C * -algebras, which will be needed in the later chapters.
2.1.The construction.For this section, let H be a locally compact group and A a C * -algebra with a strongly continuous action α : H → Aut(A).
If H acts properly on a locally compact Hausdorff space X, the generalized fixed point algebra is given by C 0 (H\X), where H\X denotes the orbit space.It is Morita-Rieffel equivalent to an ideal in the reduced crossed product C * r (H, C 0 (X)).A feature of the generalized fixed point algebra construction is that this property carries over to noncommutative A: the generalized fixed point algebra is Morita-Rieffel equivalent to an ideal in C * r (H, A).The following definitions and results are taken from [Mey01].
We recall first the definition of the crossed product C * r (H, A).There are covariant representation (ρ A , ρ H ) of the C * -dynamical system (A, H, α) on the right Hilbert A-module L 2 (H, A) defined by The representation (ρ A , ρ H ) integrates to the * -representation ρ of C c (H, A) with are H-equivariant and adjoint to each other with respect to the pairings a Let χ i : H → [0, 1], i ∈ I, be a net of continuous, compactly supported functions with [Mil17,1.13,1.15]that the convergence of (χ i f ) and its limit do not depend on the chosen net.
In this case, we understand a| as an operator A → L 2 (H, A).By [Mey01], a ∈ A is squareintegrable if and only if |a extends to an adjointable operator L 2 (H, A) → A. We also denote it by |a .Its adjoint is a|.Let A si be the vector space of all square-integrable elements in A. It becomes a Banach space with respect to the norm Definition 2.4.A continuously square-integrable H-C * -algebra (A, R) consists of a C * -algebra A with a strongly continuous action of H and a subset R ⊂ A which is relatively continuous, complete and dense in A.
If H acts properly on a locally compact Hausdorff space X, (C 0 (X), C c (X) si ) is a continuously square-integrable H-C * -algebra.Here, C c (X) is completed with respect to the • si -norm above.For an arbitrary C * -algebra A, a subset R ⊂ A satisfying the requirements above can fail to exist or to be unique as shown in [Mey01].However, there is a sufficient condition that guarantees the existence of a unique such R. Let the primitive ideal space of A be equipped with the Jacobson topology.There is a continuous H-action on Prim(A) defined by x • P = α x (P ) for x ∈ H and P ∈ Prim(A).The H-C * -algebra A is called spectrally proper, if the action on the primitive ideal space is proper.Theorem 2.5 ([Mey01,9.4]).Let A be spectrally proper H-C * -algebra.Then there is a unique relatively continuous, complete and dense subset.
lies again in the generalized fixed point algebra.As (|a b|) Now, we describe the elements of Fix H (A, R) more explicitly.In the commutative case H X, functions on the orbit space can be obtained by averaging functions in C c (X) over the action: Example 2.7.For H X proper and f ∈ C c (X) there is a function F ∈ C 0 (H\X) defined by The following lemma suggests to also think of elements of Fix H (A, R) for a noncommutative A as averages over the group action of certain elements of A.
Lemma 2.8 ([Mey01, (19)]).Let (χ i ) i∈I be a net of continuous, compactly supported functions on H that converges uniformly to 1 on compact subsets as above.Let a, b ∈ R. The net converges to |a b| with respect to the strict topology as multipliers of A.

Returning to the construction, F
The ideal J H (A, R) need not be the whole reduced crossed product.The following definition is due to Rieffel [Rie90].
) is saturated if the action of H on X is free.We will argue in Lemma 2.18 that the converse is true as well.
The next lemma, proved in [Mil17], gives a criterion when a set R ⊂ A si can be completed to a relatively continuous, complete and dense subset of A.
Lemma 2.11.Let R ⊂ A be a dense, linear subspace.Suppose R consists of square-integrable elements, is relatively continuous and H-invariant, and satisfies R • R ⊂ R. Denote by R the closure of R ⊂ A si with respect to the • si -norm.Then (A, R) is a continuously square-integrable H-C * -algebra.The generalized fixed point algebra Fix H (A, R) is the closed linear span of |R R|.

Proof. The inclusion
As a| = |a ≤ a si for all a ∈ A si , elements of R | R can be approximated with respect to the operator norm on L 2 (H, A) by elements of R | R .This shows that R is relatively continuous as well.
It remains to verify that R is complete.First, we show that R • A ⊂ R holds.Let r ∈ R and a ∈ A and choose sequences (r n ), (a n ) in R such that r − r n si → 0 and a − a n → 0. Note that ra ∈ A si because |ra = |r • ρ A a and r is square-integrable.By assumption r n a n ∈ R holds for all n ∈ N. We estimate using [Mey01,(17)] that ra − r n a n si ≤ r si a n − a + r − r n si a n , which converges to zero.Furthermore, R is also H-invariant, which follows from the invariance of R and [Mey01,(18)].This implies that |R (C c (H, A)) ⊂ R.
Using similar arguments as for the relative continuity of R, one obtains that any |a b| with a, b ∈ R is a norm limit of elements of |R R|.
Remark 2.12.Suppose R ⊂ A is a dense, H-invariant * -subalgebra such that a|b is bounded with respect to the I-norm for all a, b ∈ R as required in the original definition in [Rie90].Then by [Mey01,6.8]R is relatively continuous and square-integrable.Therefore, Lemma 2.11 shows that (A, R) is a continuously square-integrable H-C * -algebra.

Extensions of generalized fixed point algebras. Let I be an H-invariant ideal in
exact group, this is true for all H-invariant ideals I A. For example, this holds in our applications in the later chapters where H = (R >0 , • ) ∼ = (R, +).
Let R ⊂ A be a subset such that (A, R) is a continuously square-integrable H-C * -algebra.Consider R ∩ I ⊂ I and the image of R under the projection q : A → A/I.We will show that the generalized fixed point algebra construction can be applied to (I, R ∩ I) and (A/I, q(R)), and relate the respective generalized fixed point algebras to each other.
In particular, we are interested in what can be said about saturatedness in this case.This is inspired by the simple observation that if an H-space X can be partitioned into two H-invariant subsets X = X 1 X 2 , then the action on X is free if and only if it is free on X 1 and X 2 .Lemma 2.13 ([Mil17]).Let R ⊆ A be a relatively continuous, complete subspace of A.
and ( 7) is exact, we have r | r ∈ C * r (H, I).Now, let (u λ ) λ∈Λ be an approximate unit for I, satisfying u * λ = u λ and u λ ≤ 1 for all λ ∈ Λ.One computes By Lemma 2.1 this converges to zero.Furthermore, r − ru λ → 0 holds.Hence, r ∈ R • I follows from Cohen's Factorization Theorem applied to (R, • si ) as a right I-module.
Lemma 2.14.Let (A, R) be a continuously square-integrable H-C * -algebra and let I A be an H-invariant ideal such that the sequence in (7) is exact.Let q : A → A/I be the quotient map.
Then the following holds: (1) (I, R ∩ I) is a continuously square-integrable H-C * -algebra.
Proof.We prove (1).The linear subspace R ∩ I = R • I is dense in I because any element i ∈ I can be factorized as i = a • j for some a ∈ A and j ∈ I. Since R is dense in A, there is a net (r λ ) λ∈Λ ⊂ R with r λ → a and hence i = lim λ r λ • j.The square-integrability of elements in R ∩ I is inherited from R, and H, I) follows from the same argument as in the proof of Lemma 2.13 using that (7) is exact.Note that To prove (2) we show that Lemma 2.11 can be applied to q(R) ⊂ A/I.As R ⊂ A is a dense linear subspace, the same holds for q(R) ⊂ A/I.For a ∈ R and i ∈ I their product ai ∈ R • I = R ∩ I lies in R. All elements q(a) for a ∈ R are square-integrable because the quotient map The relative continuity of q(R) follows as Q maps C * r (H, A) to C * r (H, A/I).By [Mey01, 6.7], R is H-invariant and an essential right A-module, that is, R • A = R.This implies that q(R) is also H-invariant and satisfies q(R) • q(R) ⊂ q(R).Therefore, the claim follows from Lemma 2.11.
Remark 2.15.The restricted map q : A si → (A/I) si is continuous with respect to the respective • si -norms as for a ∈ A si q(a) + q(a) | q(a) In particular, for R ⊂ A si one has q R = q(R) with respect to the • si -norms.
In the situation of Lemma 2.14, F H (I, R ∩ I) is a closed Fix H (A, R)-J H (A, R) submodule of F H (A, R).Under the Rieffel correspondence (see for example [RW98,3.22]),F H (I, R ∩ I) corresponds to the ideals Fix H (I, R ∩ I) in Fix H (A, R) and J H (I, R ∩ I) in J H (A, R).
To study saturatedness, we relate the corresponding ideals in the reduced crossed products for I, A and A/I.
for a, b ∈ R, we see that J H (A, R) maps to J H (A/I, q(R)).Moreover, the linear span of elements of this form is dense in J H (A/I, q(R)) so that the restriction is onto.Hence, the claim follows from exactness of the bottom row in (8) once we show that J H (I, The converse inclusion is clear.Corollary 2.17.Let (A, R) be a continuously square-integrable H-C * -algebra and I A an H-invariant ideal such that (7) is exact.Then (A, R) is saturated if and only if (I, R ∩ I) and (A/I, q(R)) are saturated.
Proof.Suppose first that (A, R) is saturated.In the proof of Lemma 2.16 we showed that J H (I, R ∩ I) = J H (A, R) ∩ C * r (H, I).Hence (I, R ∩ I) is saturated.Because (8) has exact rows, this implies that (A/I, q(R)) is saturated as well.If (I, R ∩ I) and (A/I, q(R)) are saturated, (A, R) is saturated because (8) is exact.
As an application we show the following result for actions on spaces.
Lemma 2.18.Let H act properly on a locally compact Hausdorff space X and assume that (C 0 (X), C c (X)) is saturated.Then the action H X is free.
Proof.Let x ∈ X and let Hx ⊆ X be its orbit.On the other hand, Fix H (C 0 (Hx), C c (Hx)) is isomorphic to the functions on the orbit space.As Hx consists of a single H-orbit, this generalized fixed point algebra is isomorphic to C. Hence, C and C * (H x ) are Morita-Rieffel equivalent.This can only be true if H x = {e}.Therefore, the H-action on X is free.
Not only the ideals in the crossed product algebras fit into an exact sequence.The same is true for the corresponding generalized fixed point algebras.The surjective homomorphism q : A → A/I has a unique strictly continuous extension M(A) → M(A/I).Let q be its restriction to Fix H (A, R).
Proposition 2.19.Let (A, R) be a continuously square-integrable H-C * -algebra and I A an H-invariant ideal such that (7) is exact.There is an extension of generalized fixed point algebras Denote by β the induced H-action on A/I.Strict continuity of q and Lemma 2.8 imply for a, b ∈ R.This shows that the image of q is contained in Fix H (A, q(R)).Moreover, the linear span of elements of this form is dense in Fix H (A, q(R)).So q is onto.It remains to show that the kernel of q is Fix H (I, R ∩ I).The computation above yields q(|a b|) = |q(a) q(b)| = 0 for a, b ∈ R ∩ I. Thus, Fix H (I, R ∩ I) is contained in ker( q).Let T ∈ Fix H (A, R) be such that q(T ) = 0.By the C * -identity in Fix H (A, R)/Fix H (I, R ∩ I) it will suffice to show that T * T ∈ Fix H (I, R ∩ I).By [Mey01, (13)], T * |a b| = |T * a b| holds for a, b ∈ R. As T * a is square-integrable and . By definition of the generalized fixed point algebra, T is the limit of a sequence in the linear span of |R R|.Hence, it follows that T * T ∈ Fix H (I, R ∩ I).
2.3.Generalized fixed point algebras and C 0 (X)-algebras.In this section, we consider generalized fixed point algebras of H-C 0 (X)-algebras.For the definition of C 0 (X)-algebras and their relation to continuous fields of C * -algebras see [Nil96].
Definition 2.20 ([Kas88]).Let H act on a locally compact Hausdorff space X.Denote by τ h (f )(x) = f (h −1 • x) for h ∈ H and x ∈ X the induced action on C 0 (X).A C 0 (X)-algebra A with θ : C 0 (X) → ZM(A) and an H-action α : H A is called an H-C 0 (X)-algebra if the actions are compatible in the following sense In this case, one can relate the spectra of the H-C 0 (X)-algebra and the corresponding generalized fixed point algebra as follows.If H A is a strongly continuous action, there is a continuous action of H on the spectrum A given by ( Suppose now (A, R) is a continuously square-integrable H-C * -algebra.Every non-degenerate representation π of A can be extended to M(A).Denote its restriction to Fix H (A, R) by π.
In the commutative case, this procedure allows to completely describe the representation theory of the generalized fixed point algebra.By Gelfand duality, the spectrum of the commutative C * -algebra C 0 (X) is X.If H X is a proper action and (A, R) = (C 0 (X), C c (X)), the map π → π induces a homeomorphism This generalizes to H-C 0 (X)-algebras as follows.We summarize some results concerning H-C 0 (X)-algebras proved in [Mey01,aHRW00,Rie04,aHRW02].
Proposition 2.21.Let H X be a proper action on a locally compact Hausdorff space X and A an H-C 0 (X)-algebra with θ : C 0 (X) → ZM(A).
(1) The subset R := θ(C c (X))A is dense, relatively continuous and complete.Moreover, it is the unique such subset.
(2) If the action of H on X is free, the map π → π induces a homeomorphism H\ A → Fix H (A, R) and (A, R) is saturated.
Proof.An H-C 0 (X)-algebra with a proper action H X is spectrally proper as argued in [Mey01, Sec.9].The R above is the unique, dense, relatively continuous and complete subset constructed in [Mey01,9.4].It is proved in [aHRW00, 3.9] that the map π → π induces a homeomorphism.That (A, R) is saturated is shown in the preprint version of [Rie04] and in [aHRW02,4.1].
The following result applies to trivial continuous fields C 0 (X, A), where the action is taking place on the base space X.
There is an isomorphism In particular, for A = C the generalized fixed point algebra is isomorphic to C 0 (H\X) as claimed before.In contrast to the situation above, consider now what happens if the H-action only takes place in the fibres of a field of C * -algebras.
Remark 2.24.The above theorem in [Rie90] requires that the field A is Hilbert continuous and that C * r (H, A x ) = C * (H, A x ) for all x ∈ M .These assumptions are only needed to show that C * r (H, A) defines a continuous field of C * -algebras over X with fibres C * r (H, A x ).But this is true for any σ-compact, exact group H by [KW99, 4.2].

Now we study what can be said about saturatedness in this case.
Lemma 2.25.Let A be an upper semi-continuous field of C * -algebras over X with fibre projections Proof.By Lee's Theorem (see [Lee76] or [Nil96,3.3])there is a continuous map ψ : Prim As I can be written as the intersection of primitive ideals, it follows that there is a primitive ideal P ∈ Prim(A) with and p x (P ) = A x as otherwise p x (P ) would correspond to A under this homeomorphism.

Corollary 2.26. In the situation of Theorem 2.23, (A, R) is saturated if and only if
Proof.Suppose first that (A, R) is saturated.By Corollary 2.17 (A x , p x (R)) is saturated for all x ∈ X. Assume now that all (A x , p x (R)) are saturated.By assumption on H,

Homogeneous Lie groups
In the following, we will consider homogeneous Lie groups, which are Lie groups that are equipped with a dilation action of R >0 .They allow to define homogeneity with respect to the dilations.A detailed discussion of homogeneous Lie groups can be found in [FS82] or [FR16].We recall some notions used there, which proved to be convenient to do analysis on these groups.
3.1.Homogeneous and graded Lie groups.Definition 3.1.A homogeneous Lie group is a simply connected Lie group G whose Lie algebra g is equipped with a family of dilations {A λ : g → g} λ>0 .That is, there is a diagonalizable, linear map D : g → g with positive eigenvalues q 1 ≤ q 2 ≤ . . .≤ q n , such that all A λ := Exp(D ln(λ)) are Lie algebra homomorphisms.Here, Exp denotes the matrix exponential.The eigenvalues q 1 , . . ., q n are called weights.
Folland and Stein assume in [FS82] that q 1 = 1.This can be achieved by scaling appropriately.We shall also assume this in the following, in particular, all weights satisfy q j ≥ 1. Fix a corresponding basis of eigenvectors ] is an eigenvector of D to the eigenvalue q i + q j .From that one deduces that g, and therefore G, is nilpotent.Consequently, the exponential map exp : g → G is a diffeomorphism.In the following, we often identify (x 1 , . . ., with [X, Y ] ∈ g j+k for all X ∈ g j and Y ∈ g k , where g j = {0} for j > N . For a graded Lie group G with grading as above, A λ (X) = λ j X for X ∈ g j and λ > 0 defines a family of dilations.So G becomes a homogeneous Lie group.However, homogeneous Lie groups are slightly more general.If all weights of a homogeneous Lie group are rational numbers, it is a (scaled) graded nilpotent Lie group (see [FR16, 3.1.9]).Also note that there are nilpotent Lie groups that do not admit a family of dilations as above (see [Dye70]).
Example 3.3.A famous example of a homogeneous Lie group is the Heisenberg group.Its Lie algebra g is generated by {X, Y, Z} and [X, Y ] = Z, [X, Z] = 0 and [Y, Z] = 0. Hence, g 1 = span{X, Y }, g 2 = span{Z} and g j = 0 for j > 2 defines a grading on g.
Example 3.4.A Lie algebra g may be equipped with different dilations.Choose a basis {X 1 , . . ., X n } for the Lie algebra of the Abelian group G = R n .Then for all (q 1 , . . ., q n ) ∈ R n >0 there is a dilation defined by DX j = q j X j .The standard dilation action on R n is given by scalar multiplication, that is, q j = 1 for all j = 1, . . ., n.

Analysis on homogeneous Lie groups.
Definition 3.5.The homogeneous dimension of a homogeneous Lie group G with weights 1 = q 1 ≤ q 2 ≤ . . .≤ q n is defined as for all λ > 0 and x = 0.

Lemma 3.6. Let G be a homogeneous Lie group of homogeneous dimension Q. The pullback of the Lebesgue measure under the exponential map defines a Haar measure on G. The group G is unimodular and the Haar measure is
For simply connected nilpotent Lie groups it is true in general that the pullback of the Lebesgue measure defines a left and right Haar measure, see [FS82, 1.2].The Q-homogeneity follows from the behaviour of the Lebesgue measure under scaling.Definition 3.7.For a multi-index α ∈ N n 0 its homogeneous degree is defined as Example 3.8.The polynomials x α for α ∈ N n 0 are [α]-homogeneous functions on G.The group law of a homogeneous Lie group is of a triangular form.Using the Baker-Campbell-Hausdorff formula and the homogeneity of the coordinate functions, the following is proved in [FS82,p. 23].
Proposition 3.9.For a homogeneous Lie group G with weights q 1 ≤ . . .≤ q n and a basis of eigenvectors X 1 , . . ., X n ∈ g there are constants c j,α,β for j = 1, . . ., n such that for all x, y ∈ G with respect to this basis

The basis of eigenvalues fixed above induces left-and right
The triangular group law allows to express these in terms of the partial differential operators as follows.
Proposition 3.10 ([FR16, 3.1.28]).Let G be a homogeneous Lie group with weights q 1 ≤ . . .≤ q n .For j = 1, . . ., n and k > j there are (q k − q j )-homogeneous polynomials P jk and Q jk such that the vector fields X j and Y j defined above can be written as The polynomials P jk and Q jk only depend on x 1 , . . ., x k−1 because otherwise they would be homogeneous of a higher order than q k − q j .Hence, they commute with the partial derivatives ∂ ∂x k .Because the Euclidean norm does not behave well with respect to the dilations, homogeneous quasi-norms are used instead.

Definition 3.11 ([FR16, 3.1.33]). A homogeneous quasi-norm on a homogeneous Lie group G is a continuous function
In the following, we fix a homogeneous quasi-norm on G.For instance, let In fact, by [FR16,3.1.35]all homogeneous quasi-norms on a given homogeneous Lie group are equivalent.There is an analogue of the triangle inequality and its consequences for a homogeneous quasi-norm.
Lemma 3.12 ([FS82, 1.8, 1.10]).Let G be a homogeneous Lie group.There is a constant Identifying G with R n one can consider the Schwartz space S(G).The following family of seminorms will be useful later on.Definition 3.13.For the fixed homogeneous quasi-norm Polynomials in x E for the Euclidean norm can be estimated by polynomials in x for a homogeneous quasi-norm and vice versa.Thus, a sequence ( The following integrability criterion for functions on a homogeneous Lie group is used later on. Lemma 3.14 ([FS82, 1.17]).Let α ∈ R and let f be a measurable function on a homogeneous Lie group 3.3.Representation theory of homogeneous Lie groups.Now, we recall some facts about the representation theory of nilpotent Lie groups G and their group C * -algebras.For homogeneous Lie groups the dilations induce actions on the respective spaces of representations.The continuous compactly supported functions C c (G) become a *-algebra when equipped with the convolution and involution defined by Denote by G the set of equivalence classes of irreducible, unitary representations π : G → U(H π ).For such a representation π and f ∈ C c (G) define the operator This defines a *-representation π : C c (G) → B(H π ).The full group C * -algebra C * (G) is defined as the closure of C c (G) with respect to f = sup π∈ G π(f ) .By [Dix59] homogeneous Lie groups are liminal so that all representations π map onto the compact operators K(H π ).
The homogeneous structure allows to define an R >0 -action on G.For an irreducible, unitary representation π set (λ • π)(x) = π(α λ (x)) for λ > 0 and x ∈ G.It is easy to see that λ • π is again an irreducible, unitary representation and that the action is well-defined on the equivalence classes.
Furthermore, define an action on C * (G) by . Using Lemma 3.6 one checks that each σ λ is a * -homomorphism and an isometry with respect to the C * -norm.This action induces in turn an action on the representations of C * (G) by (λ Proof.It is well-known that the map above is a homeomorphism for each locally compact group G.The equivariance under the R >0 -action follows from the Q-homogeneity of the Haar measure as Kirillov's orbit method [Kir62] allows to describe G as the orbit space of the coadjoint action of G on g * , the dual of its lie algebra g.Recall that the adjoint representation Ad : G → Aut(g) is given by Ad(x) = d(α x ) 0 : T 0 G → T 0 G, where α x (y) = xyx −1 is given by conjugation.The coadjoint action is defined by co-Ad(x)l, X := l, Ad(x −1 )X for l ∈ g * , x ∈ G and X ∈ g.
The corresponding infinitesimal representation co-ad of g on g * is given by co-ad(X)l, Y = l, [Y, X] for l ∈ g * and X, Y ∈ g.The orbit space G\g * is equipped with the quotient topology.For each l ∈ g * , one can construct a unitary representation π l of G as described in the following.Define a skew-symmetric bilinear form b l : A polarizing subalgebra for l is a subalgebra h l ⊂ g that is an isotropic subspace of codimension 1 2 dim(g/g l ).Such a polarizing subalgebra always exists (see [CG90, 1.3.3,1.3.5]).
The formula χ l (exp X) = e i l,X for X ∈ h l defines a one-dimensional representation of then Z is given by the Baker-Campbell-Hausdorff formula as Let R >0 act on g * by λ • l, X = l, A λ (X) for λ > 0, l ∈ g * and X ∈ g.This action descends to the orbit space of the co-adjoint action as Let H be a subgroup of a locally compact group G and let α be an automorphism of G and π a unitary representation of H. Then α −1 (H) is also a subgroup and Proof.Kirillov proved in [Kir62] that the map is a well-defined, so in particular, the equivalence class of π l does not depend on the choice of the polarizing subalgebra h l .Two representations π l1 and π l2 are equivalent if and only if l 1 and l 2 lie in the same co-adjoint orbit.Moreover, he proved that the map is continuous and onto.The continuity of the inverse map is due to [Bro73].To see that the map is equivariant, note that χ λ.l = χ l • α λ and that α λ −1 (H l ) is a polarizing algebra for λ • l.Hence, Lemma 3.16 yields that π λ•l λ • π l .
All l ∈ g that vanish on [g, g] induce one-dimensional representations π l .In particular, l = 0 induces the trivial representation on C. If the polarizing algebra is not all of g, the corresponding Hilbert space is infinite-dimensional.
3.4.Stratification.The goal of this section is to use Kirillov theory and the coarse stratification by Pukanszky [Puk67] to find a sequence of increasing, open, R >0 -invariant subsets such that all Λ i := V i \ V i−1 are Hausdorff and the R >0 -action on each of these subsets is free and proper.This sequence will play an essential role in Section 8.Note that the following construction to find such a sequence of open subsets 1works for all simply connected nilpotent Lie groups.However, a dilation action is only defined for homogeneous Lie groups.
We start by describing Pukánszky's stratification of g * .Recall that we fixed a basis {X 1 , . . ., X n } of g such that D λ (X j ) = λ qj X j for all 1 ≤ j ≤ n.By the triangular form of the group law (9) all form an ideal in g.In particular, {X 1 , . . ., X n } is a strong Malcev basis of G as defined in [CG90, 1.1.13],which is also called a Jordan-Hölder basis in [Puk67].Note that they require span{X 1 , . . ., X i } to be ideals.We stick to the reversed ordering of the basis as this is standard for homogeneous Lie groups.Let {X * 1 , . . ., X * n } denote the corresponding dual basis of g * and let As the k i are ideals and are, therefore, invariant under the adjoint action, the k ⊥ i are invariant under the coadjoint action.Hence G acts on each g * /k ⊥ i .Write p i : g * → g * /k ⊥ i for the projections.By [CG90, 3.1.4]the orbits G • p i (l) of p i (l) under the coadjoint action are closed.Hence, they define submanifolds of g * /k ⊥ i .Following [CG90], make the following definition.
The entries of d(l) are decreasing.The corresponding stabilizer subgroups G pi(l) form an increasing sequence The same is true for their Lie algebras g pi (l) .By [CG90, 3.1.1]they are given by Example 3.19.The computation in [CG90, 3.1.11]of the coadjoint action on the 3-dimensional Heisenberg group H 1 yields co-Ad(x, y, z)(αX With the help of the next lemma an argument by Pukánszky [Puk71,p. 70] shows that the definition of d(l) as above coincides with the one given, for example, in [BBL16], by jump indices.Lemma 3.20.Let b : V × V → R be a skew-symmetric bilinear form, V ⊥ its radical and W ⊂ V a subspace.Then Lemma 3.21.The dimensions in d(l) decrease by steps of zero or one.There is a jump, that is, Proof.The orthogonal complement of g l + span{X i+1 , . . ., X n } with respect to the bilinear form b l is g pi(l) .Hence, by Lemma 3.20 there is a change of dimension if and only if the dimension of the orthogonal complement decreases.This is the case if and only if X i / ∈ g l + span{X i+1 , . . ., X n }.
Let D denote the finite set of all dimension sequences that occur for G. Assemble all l ∈ g * \{0} with the same sequence to for all λ ∈ R >0 , Lemma 3.21 implies that they are also invariant under the dilation action.For The following theorem is due to Pukánszky [Puk71] and is also proved in [CG90].

Theorem 3.22 ([CG90, 3.1.14]). There is an ordering
This allows to find a sequence as in (12) using Kirillov's map by setting V i = G\W i for i = 1, . . ., m.It remains to check that the V i \ V i−1 = G\Ω di are Hausdorff and that the corresponding R >0 -action is free and proper.Proof.In [CG90, 3.1.14]it is proved that there is a birational, nonsingular map where π 1 denotes the projection to Λ d .Hence, it induces a continuous map G\Ω d → Λ d .It is inverse to the map induced by the inclusion.Thus, the two spaces are homeomorphic.As Ω d and g * T (d) are invariant under the dilation action, so is Λ d .Therefore, the inclusion is equivariant.Since 0 ∈ g * is not contained in any Ω d , the Λ d are subsets of some R l \{0} equipped with the Euclidean subspace topology.Hence they are Hausdorff and the R >0 -action, which is given for λ > 0 by multiplying the coordinate entries by different powers of λ, is free and proper.
In this case, the topological space G can be equipped with a certain Borel measure µ, which is called the Plancherel measure.For a simply connected nilpotent Lie group G, it is supported within the orbits with maximal dimension sequence, these are the orbits in Ω d1 ⊂ g * with notation as in Theorem 3. 22 Define L 2 ( G, HS(H π )) to be the direct integral of the Hilbert spaces HS(H π ).It consists of sections It is a Hilbert space with respect to x, y := G x(π), y(π) HS dµ(π).
The Plancherel Theorem [FR16, B.2.32] states that the Fourier transform defined above yields an isometric isomorphism : The left group von Neumann algebra VN L (G) of G consists of bounded, left-invariant operators on L 2 (G) (see [FR16, B.2.13]).The Plancherel Theorem yields that the Fourier transform extends to a * -isomorphism It is a von Neumann algebra with the pointwise operations and the norm given by This operator-valued Fourier transform is essential for the definition of the symbolic pseudodifferential calculus of Fischer-Ruzhansky described in Section 7.1.

The tangent groupoid and its C * -algebra
In this section, the tangent groupoid of a homogeneous Lie group G is defined as the transformation groupoid of an action of G.We explain how this groupoid can be understood as a variant of Connes' tangent groupoid [Con94].The homogeneous structure is taken into account by replacing addition of tangent vectors by multiplication in the group.Furthermore, the groupoid C * -algebra of the tangent groupoid can be described as a continuous field of C * -algebras.
G given by (x, t) Here, α t for t > 0 are the dilations on G and α 0 (v) = lim t→0 α t (v) = 0 for all v ∈ G.
The unit map u : G 0 := G × [0, ∞) → G, the range and source map r, s : G → G 0 , the inverse and the multiplication are given by Let θ : G → [0, ∞) denote the projection to the second coordinate.Recall that the pair groupoid of G is the groupoid G × G with unit space G with r(x, y) = x, s(x, y) = y, (x, y) −1 = (y, x) and (x, y) • (y, z) = (x, z) for x, y, z ∈ G. Lemma 4.2.Let G be a homogeneous Lie group.Then (G, [0, ∞), θ) defines a continuous field of locally compact, amenable groupoids.The subgroupoids θ −1 {t} for t > 0 are isomorphic to the pair groupoid of G.The subgroupoid T G := θ −1 {0} is the trivial field of groups over G with fibre G.
Proof.It is easy to check that (G, [0, ∞), θ) defines a continuous field of locally compact groupoids in the sense of [LR01] or [BBDN18].All subgroupoids θ −1 {t} for t ≥ 0 are transformation groupoids of actions of G on itself.The group G is amenable as a nilpotent group, for that reason all θ −1 {t} are amenable.
The subgroupoid T G = θ −1 {0} corresponds to a (noncommutative) version of the tangent bundle.For (x, v) ∈ T G we interpret x as the base point and v ∈ G ∼ = g ∼ = T x G as a tangent vector at x.The groupoid multiplication is defined if and only if two vectors lie in the same fibre and is, in this case, defined by the group multiplication.Let q : T G → G denote the projection to the base point, then (T G, G, q) defines itself a continuous field of locally compact groupoids.It is the trivial field over G with fibre G. Again, all fibres q −1 {x} ∼ = G for x ∈ G are amenable.
G is smooth.Consequently, the tangent groupoid G is a Lie groupoid.A graded nilpotent Lie group is a special case of a filtered manifold as considered in [vEY17].Therefore, one can define the tangent groupoid as a smooth field of groupoids over [0, ∞) as in [CP19,vEY17,HSSH18].Using the isomorphisms 14) and the definition of the smooth structure of TG, one obtains that G and TG are isomorphic as Lie groupoids.
4.2.The groupoid C * -algebra.Now, we recall how the groupoid C * -algebra of the tangent groupoid G is constructed.As the tangent groupoid of G is an action groupoid of an amenable group, C * (G) is isomorphic to the reduced crossed product the involution and convolution defined in (2) and (3) are Proof.As all groupoids θ −1 {t} are amenable, C * (G) defines a continuous field of C * -algebras with fibres C * (θ −1 {t}) by [LR01,5.6].The groupoid θ −1 {t} for t > 0 is isomorphic to the pair groupoid G × G by Lemma 4.2.The Haar measure on θ −1 {t} is taken under ϕ t to the left Haar measure There is a well-known isomorphism For t > 0 compose β t and the homomorphism induced by ϕ −1

There is an isomorphism
Proof.The subgroupoid G G×(0,∞) is isomorphic to the trivial field of groupoids over R >0 with the pair groupoid G × G as fibre via (x, t, v) → (t, ϕ t (x, v)).Composing the induced isomorphism of the corresponding groupoid C * -algebras with the respective β t for t > 0 gives the claim.
Denote by p 0 : C * (G) → C * (T G) the * -homomorphism induced by restricting to t = 0.There is a short exact sequence by [HS87] (16 If G = R n , the C * -algebra on the right is isomorphic to C 0 (T * R n ) via Fourier transform.In general, C * (T G) can be noncommutative.It is the trivial field of C * -algebras over G with fibres isomorphic to the group C * -algebra C * (G).

Pseudo-differential extension using generalized fixed point algebras
In this section we use the dilations on G to define a certain R >0 -action on C * (G).We show that the generalized fixed point algebra construction can be applied when the action is restricted to an ideal J C * (G).In particular, we prove the existence of a continuously square-integrable subset in J.Moreover, we obtain the pseudo-differential extension.
5.1.The generalized fixed point algebra of the zoom action.In the Euclidean case, the principal symbol of a pseudo-differential operator of order zero can be understood as a generalized fixed point of the proper action of R >0 on T * R n \({0} × R n ) given by scaling in the fibres, that is, λ • (x, ξ) = (x, λ −1 ξ) for x ∈ R n , ξ ∈ T * x R n and λ > 0. Under Fourier transform this action corresponds to ( Using the dilations we can define an analogous action on C * (T G) for a homogeneous Lie group G and extend it to C * (G).
As described above, in the Euclidean case the scaling action on T * R n is restricted to T * R n \ ({0} × R n ).This is necessary as the zero section is fixed by the scaling action, hence, the action on T * R n is not proper.For an arbitrary homogeneous Lie group, we must also choose an ideal inside the C * -algebra of the tangent groupoid in order to obtain a continuously square-integrable R >0 -C * -algebra.For f ∈ S(R n ) the property f (0) = 0 is equivalent to f (x) dx = 0, where f is the Fourier transform of f .Moreover, C 0 (R n \{0}) corresponds under Fourier transform to ker( π triv ) C * (R n ) where π triv is the trivial representation of R n .
For a homogeneous Lie group G let q x : C * (T G) → C * (G) for x ∈ G be the * -homomorphism induced by restricting f ∈ C c (T G) to the fibre T x G. Definition 5.3.Let G be a homogeneous Lie group and π triv its trivial representation.Let J be the closed ideal in C * (G) defined by The ideal J is invariant under the action σ : R >0 C * (G) above.Now, we define a linear subspace R G ⊂ J for the generalized fixed point algebra construction.
Using the seminorms from Definition 3.13, set for N ∈ N 0 For f ∈ R G conditions (a) and (b) ensure that f (N ) < ∞ and ∂ t f (N ) < ∞ for all N ∈ N. Note that f * (0) = f (0) holds.Hence, by Lemma 3.14 f I ≤ D f (0) holds for a constant D > 0, so that the elements of R G lie indeed in the groupoid C * -algebra C * (G).Condition (c) forces them to be in the ideal J.The goal of this section is to show that the generalized fixed point construction can be applied to the R >0 -action σ on (J, R G ).
Lemma 5.5.For every k ∈ N there is a group constant C k > 0 and such that for all g ∈ C ∞ (G) with g (k) < ∞ and all x, v, w ∈ G and t ≥ 0 Proof.It suffices to show the claim for real-valued g.Define a function By the polynomial form of the group law there is a group constant D > 0 such that for all v, w ∈ G and s ∈ [0, 1].Using Lemma 3.12 we estimate As α j (w, h) = h qj w j , it follows that |∂ h α j (w, s)| ≤ Q|w j | ≤ Q w qj .Together, these estimates imply the claim.
Lemma 5.6.Consider the action σ : R >0 J.For f ∈ R * G the operator f |, defined as in (5), Proof.Let f, g ∈ R G .We show that f * |g * ∈ L 1 (R >0 , J).Because σ λ is an isometry with respect to the I-norm for λ > 0, the following equality holds (17) So let λ ≥ 1 in the following.Using the homogeneity of the Haar measure, we compute To estimate this integral, let As f satisfies condition (c), we get We start by estimating the first summand.As Let T > 0 be such that f (x, t, v) = 0 holds whenever t > T .Using Lemma 3.12 and λ ≥ 1, we find This integral converges by Lemma 3.14.The estimate is independent of (x, t) ∈ G 0 .For the second summand, apply Lemma 5.5 to g with k = 1.We obtain Consequently, we obtain using the rapid decay of f that Again, this converges by Lemma 3.14 and the estimate does not depend on (x, t) ∈ G 0 .For σ λ (f ) * g * I,2 , one can estimate analogously by replacing (x, v, t) by (x, v, t) −1 in the argument above.Thus, the convergence of ∞ 1 λ −2 dλ implies (18).Moreover, together with the respective estimate for λ < 1 using (17), we obtain a constant D > 0 such that for all f, g ∈ R G that vanish for t > T lies in J if and only if it satisfies the vanishing integral condition (20).Note that R is contained in R * G .Proposition 5.8.Let G be a homogeneous Lie group and J C * (G) be defined as in Definition 5.7.Denote by R the completion of R with respect to the • si -norm.Then (J, R) is a continuously square-integrable R >0 -C * -algebra.
Proof.First, we show that R is dense in J. Let f ∈ J and ε > 0. Then f can be approximated by As the Laurent symbol of f | g is given by f |g, Lemma 5.6 implies f | g ∈ C * r (R >0 , J).Now, by [Mey01,6.8]the set R is square-integrable and relatively continuous.It is also R >0 -invariant, and f * g ∈ R holds for f, g ∈ R. Hence, Lemma 2.11 gives the claim.
In the following, it will be useful to know that the • si -closure of R contains the space R * G .In particular, this implies R * G = R. Lemma 5.10.The linear space R * G is contained in the completion of R with respect to the that vanish for t > T , and satisfy ).Consequently, f lies in the closure of R with respect to the • si -norm.
Therefore, the generalized fixed point algebra Fix R>0 (J, R) of the R >0 -action on J is defined as in Definition 2.6.The elements |f g| for f, g ∈ R * G can be characterized more explicitly.We fix for the rest of this article a monotone increasing net (χ i ) i∈I of continuous compactly supported functions χ i : R >0 → [0, 1] with χ i → 1 uniformly on compact subsets to cut off at zero and infinity.We may assume that χ i (λ −1 ) = χ i (λ) for all λ > 0. As described in Lemma 2.8 converges to |f g| with respect to the strict topology as multipliers of J.

The pseudo-differential extension. For a homogeneous Lie group G, let J
In the following, we use generalized fixed point algebras to derive a pseudo-differential extension We will justify the name "pseudo-differential" extension in Section 7 by comparing it to the calculus defined by Fischer-Ruzhansky-Fermanian-Kammerer in [FR16,FFK20] for graded Lie groups.
The homomorphism The short exact sequence from (16) restricts to p0 Proposition 5.11.Let G be a homogeneous Lie group and let R 0 = p 0 (R).The zoom action σ on (21) induces an extension Proof.Proposition 2.19 gives an extension of generalized fixed point algebras p0 By Remark 2.15 the completion of p 0 (R) with respect to the • si -norm on the right hand side is R 0 .The isomorphism p from Lemma 4.5 induces an isomorphism where R := p(R∩ker(p 0 )).By the description of the R >0 -action on C 0 (R >0 , K(L 2 G)) in Lemma 4.5, it follows that R is the unique relatively continuous, complete and dense subset by Theorem 2.5.Lemma 2.22 gives an isomorphism Therefore, we obtain (22).The inclusion of By Theorem 2.23 the symbol algebra Fix R>0 (J 0 , R 0 ) is a continuous field of C * -algebras over G with fibres Fix R>0 (J G , R G ).We show it is in fact a trivial continuous field.
Lemma 5.12.The map Θ : Proof.By Proposition 2.19 and Remark 2.15 each q x maps Fix R>0 (J 0 , R 0 ) onto Fix R>0 (J G , R G ) for x ∈ G. Let f, g ∈ R 0 , we show that Θ(|f g|) is continuous.For ε > 0 and x ∈ G there is an open neighbourhood U of x such that q x (f ) − q y (f ) 5 < ε and q x (g) − q y (g) 5 < ε for all y ∈ U .The estimate of the norm in (19) implies that there is a constant C > 0 such that |h ≤ C h 5 for all h ∈ R G .Hence for y ∈ U we obtain As f and g are compactly supported in the x-direction it follows that Θ(|f g|) is again compactly supported.Extend Θ linearly to the span of |f g| for f, g ∈ R 0 and let T be inside the linear span.As q x (T ) ≤ T for all x ∈ G it follows that Θ(T ) ≤ T .Let ψ ∈ J 0 satisfy ψ = 1.As C * (T G) is a continuous field of C * -algebras over G with fibres C * (G) it follows that Hence, Θ is an isometry and extends by continuity to Fix R>0 (J 0 , R 0 ).As q x is a homomorphism As the image of Θ is closed, the claim follows.

Representation as bounded operators on L 2 (G)
. We show that the C * -algebra of "order zero pseudo-differential operators" Fix R>0 (J, R) admits a faithful representation as bounded operators on L 2 (G).
The * -homomorphisms p t : G → K(L 2 G) defined in (15) can be restricted to the ideal J.The restrictions are still surjective.Hence, they yield strictly continuous representations for all t > 0.
Lemma 5.13.The representation Proof.As seen in Remark 5.2 the representations p t of J are related by This equality still holds for the corresponding extensions to the multiplier algebras.As each T ∈ Fix R>0 (J, R) is invariant under σ, it follows that p t (T ) = p 1 (T ) for all t > 0. Therefore, T ∈ ker( p 1 ) implies p t (T ) = 0 for all t > 0. Then it follows for all f ∈ J that p t (T f ) = pt (T )p t (f ) = 0 for all t > 0. As C * (G) is a continuous field of C * -algebras this implies by continuity that T f = 0. Since this holds for all f ∈ J, it follows T = 0.

Lemma 5.14. Let f, g ∈ R *
G and h = f * * g.Then the operators T i (h) given by for ψ ∈ L 2 (M ), x ∈ M , converge strictly to p 1 (|f g|) as multipliers of K(L 2 M ).
Proof.Strict continuity of p 1 together with (23) imply that The operators T i (h) above are obtained by inserting the definition of p λ .
Lemma 5.15.The following diagram commutes, where the horizontal maps are the inclusions: Then by strict continuity of p 1 and ( 23) holds.The claim follows as the linear span of Lemma 5.16.Let h ∈ R * G ∩ ker(p 0 ) and let T i (h) be defined as in (24).Then (T i (h)) converges in norm in K(L 2 G).In particular, its strict limit as multipliers of K(L 2 G) exists and is contained in p 1 (Fix R>0 (J, R)).
Proof.As h ∈ R * G vanishes at t = 0, it can be written as . By definition of the representation p λ in (15), it follows that p λ (h) = λp λ (f ) for all λ > 0. Hence, for all λ > 0 p λ (h) ≤ λ p λ (f ) ≤ λ f .We show that (T i (h)) is Cauchy.Let T > 0 be such that h vanishes for t ≥ T .For j ≥ i, we estimate As χ i → 1 on compact subsets, the first claim follows.The second claim holds as convergence in norm implies strict convergence and K(L 2 G) is contained in p 1 (Fix R>0 (J, R)) by Lemma 5.15.

Operators of type 0 as generalized fixed points
Let G be a homogeneous Lie group.In this section we show that Fix R>0 (J G , R G ) is the C * -closure of the operators of type 0 on G.

Homogeneous distributions. Recall that the dilations yield an
Definition 6.1 ([FR16, 3.2.9]).Let G be a homogeneous Lie group and let ν ∈ R. A distribution u ∈ S (G) is called a kernel of type ν if it is smooth away from zero and σ λ * (u) = λ ν u for all λ > 0. Denote by K ν (G) the space of kernels of type ν.
For u ∈ K ν (G) the corresponding continuous operator T u : S(G) → S (G) given by T u (f ) = u * f is called an operator of type ν.Remark 6.2.Let Q be the homogeneous dimension of the group G.The kernels of type ν are also called (ν − Q)-homogeneous distributions in the literature.This is because they coincide with (ν − Q)-homogeneous functions outside zero.
Example 6.3.Let δ ∈ S (G) denote the delta distribution.For α ∈ N n 0 the distribution X α δ ∈ S (G) defines a kernel of type −[α].The corresponding operator T X α δ is the right-invariant differential operator Y α on G defined as in (10).6.2.Factorization.Define the following subspace of S(G) which is the Euclidean Fourier transform of the space of Schwartz functions that vanish with all derivatives at 0.

Definition 6.4 ([CGGP92]
).For a homogeneous Lie group G, let S 0 (G) consist of all functions f ∈ S(G) with G v α f (v) dv = 0 for all α ∈ N n 0 .Lemma 6.5.The space S 0 (G) has the following properties: (1 Proof.It is clear that S 0 (G) is a linear subspace of S(G).It is left to show for (1) that S 0 (G) is closed under involution and convolution.Note that for v ∈ G Moreover, by the polynomial group law in Proposition 3.9 one can write v • w for v, w ∈ G as a linear combination of v β w γ .To show (2), let f ∈ S(G) and α ∈ N n 0 .We estimate The integral converges by Lemma 3.14.It follows that for a sequence (f k ) k∈N in S 0 (G) which converges in S(G), the limit lies in S 0 (G) as well.Property (3) is proved in [Gel83, Lemma 4].For (4) write X ∈ g as X = n j=1 a j Y j with a j ∈ R. Then Xf = Xδ * f = n j=1 a j X j δ * f As the X j δ define kernels of type −q j by Example 6.3, the claim follows from (3). 6.3.Operators of type 0. We compare now the operators of type 0 to the generalized fixed points of the R >0 -actions on J G .Proposition 6.6 ([FS82, 1.65]).Let G be a homogeneous Lie group and We will show that, conversely, every u ∈ K 0 (G) can be written as an average over the dilation action as above.First, this is proved for u = δ, which is a kernel of type 0. Lemma 6.7.There are Proof.As noted in [CGGP92], one can find a φ ∈ S 0 (G) with For example, take a function , where • is a homogeneous quasi-norm on g * which is smooth outside zero.The invariance of the Haar measure on R >0 implies that ∞ 0 ψ(α λ −1 (ξ)) dλ λ = 1 for all ξ = 0. Therefore, φ ∈ S 0 (G) can be taken as the inverse Euclidean Fourier transform of ψ.Now, φ needs to be factorized appropriately.Dixmier and Malliavin proved in [DM78, 7.2] that one can find k, l ∈ S(G) such that φ = k * l.In the first step of the proof they show φ = θ * µ, where µ is a measure and θ is the limit of a sequence of polynomials in X α φ in S(G).It follows θ ∈ S 0 (G) by Lemma 6.5.Repeating this procedure with θ, the factorization φ = k * l is achieved after finitely many steps with k ∈ S 0 (G) and l ∈ S(G).By [FS82, 1.60], k can be written as k = n j=1 X j k j with k 1 , . . ., k n ∈ S 0 (G).Therefore, we obtain Using Proposition 3.10 one obtains that (Y j l)(x) dx = 0 holds.Consequently, the claim holds with f j := k * j ∈ S 0 (G) and g j := Y j l ∈ R G for j = 1, . . ., n. Corollary 6.8.Every u ∈ K 0 (G) can be written as a finite sum of Proof.Let f j ∈ S 0 (G) and g j ∈ R G for j = 1, . . ., n be functions as in Lemma 6.7.Then use the homogeneity of u to compute Here we used that u * (φ * ψ) = (u * φ) * ψ for all φ, ψ ∈ S(G).The last equality holds as u * f j lies in S 0 (G) for all j = 1, . . ., n by Lemma 6.5, so that the expression in the bottom line converges in S (G) by Proposition 6.6.
The description of the kernels of type 0 above is already very close to the generalized fixed point algebra construction.We compare now the corresponding convolution operators to elements of Fix R>0 (J G , R G ) ⊂ M R>0 (J G ).To understand this fixed point algebra better, we show that the restrictions of the left regular representation λ G : C * (G) → B(L 2 G) to J G is still non-degenerate.Lemma 6.9.Let G be a homogeneous Lie group G.The restriction of the left regular representation But as ψ(π) is Hilbert-Schmidt, this means that ψ(π) = 0 for π = π triv .The Plancherel measure is supported within the representations corresponding to orbits of maximal dimension sequence.In particular, {π triv } has measure zero and, therefore, ψ = 0 must hold.
Consequently, as the restriction is also faithful, the multiplier algebra M(J G ) can be identified with the idealizer of λ G (J G ) in B(L 2 G).Therefore, elements of the generalized fixed point algebra Fix R>0 (J G , R G ) can be viewed as bounded, right-invariant operators on L 2 (G).This allows us to reprove the following theorem of [KS71].Theorem 6.10.Let G be a homogeneous Lie group.Every operator T of type 0 extends uniquely to a bounded operator L 2 (G) → L 2 (G).
Example 7.4 ([FR16, 4.1.8]).Let G be a graded Lie group with weights q 1 ≤ q 2 ≤ . . .≤ q n and corresponding basis X 1 , . . ., X n of g.Let q be a common multiple of the weights.Then the following operator is a Rockland operator n j=1 (−1) q/qj X 2q/qj j .Remark 7.5.The existence of a positive Rockland operator on a homogeneous Lie group is equivalent to the group being (up to rescaling) graded (see [FR16,4.1.3,4.1.8]).
Remark 7.6.The Helffer-Nourrigat Theorem [HN79] states that a left-invariant homogeneous differential operator on a graded Lie group satisfies the Rockland condition if and only if it is hypoelliptic.
From now on, let R be a fixed positive Rockland operator of homogeneous degree q on G.It will plays the role of the Laplace operator on R n in the Euclidean calculus.
Using the positive Rockland operator and its functional calculus, the Sobolev spaces L 2 s (G) for s ∈ R are defined in [FR16,4.The derivatives in the cotangent direction in the Euclidean calculus are replaced with difference operators ∆ α for α ∈ N n 0 , which are defined in the following.This is based on the observation that the Euclidean Fourier transform intertwines ∂ α and with multiplication by x α .For u ∈ S (G) denote by x α u for α ∈ N n 0 the tempered distributions defined by The following symbol classes are adapted to the notion of order induced by the dilations.Hence, the homogeneous degree [α] for α ∈ N n 0 as in Definition 3.7 is used.Definition 7.7 ([FR16, 5.2.11]).A field {a(x, π) : ) is defined on the smooth vectors and satisfies sup Denote the class of symbols of order m by S m .For a ∈ S m and α, β The smoothing symbols are S −∞ = m∈R S m .
One can form asymptotic expansions of these symbols in the following sense.
Proposition 7.8 ([FR16, 5.5.1]).Let {a j } j∈N0 be a sequence of symbols a j ∈ S mj with m j strictly decreasing to −∞ as j → ∞.Then there is a symbol a ∈ S m0 , unique modulo S −∞ , such that In this case, one writes a ∼ ∞ j=0 a j .Proposition 7.9 ([FFK20, 5.2.12, 5.2.17]).The symbol classes have the following properties: (1) (3) For a ∈ S m and α, β ∈ N n 0 the symbol X β ∆ α a is contained in S m− [α] .For a ∈ S m the following quantization formula is well-defined and yields a continuous operator Op(a) : S(G) → S(G) by [FR16, 5.2.15] Let κ x ∈ S (G) be the convolution kernel of the left-invariant operator whose Fourier transform is {a(x, π) | π ∈ G}.Then Op(a)ϕ(x) = ϕ * κ x holds.Denote by K a ∈ S (G × G) the integral kernel of Op(a).It is formally given by K a (x, y) = κ x (y −1 x).
In the following, we will consider operators with compactly supported integral kernels.Let S m cp consist of all symbols a ∈ S m such that Op(a) has a compactly supported integral kernel.Set S −∞ cp = m∈R S m cp .The following properties for the symbols with compactly supported integral kernels follow from the respective properties for the symbol classes S m shown in [FR16, 5.5.8, 5.5.12,5.7.2, 5.4.9, 5.2.21].
Proposition 7.10.The pseudo-differential calculus has the following properties: (1 For the following lemma, see also [FFK20,4.24]. Lemma 7.11.Let A ∈ Op(S m cp ) for m < 0. Then A extends to a compact operator on L 2 (G).Proof.By Proposition 7.10, A extends to a bounded operator A : be constant 1 on the support of A in the x-direction and be supported in a compact subset K ⊂ G.The map f → χ • f extends to a bounded operator L 2 −m (G) → H −m/qn (K) by [FFK20, 2.17], where H −m/qn (K) denotes the Euclidean Sobolev space.By Rellich's Theorem the embedding Moreover, in [FFK20] classical pseudo-differential operators, which admit a homogeneous expansion of their symbol, are defined.Definition 7.12 ([FFK20, 4.1, 4.20]).Let m ∈ R. A field {a(x, π) : (1) a(x, λ • π) = λ m a(x, π) for all x ∈ G and almost all π and λ > 0, (2) for all α, β ∈ N n 0 , the field of operators X β x ∆ α a(x, π) is defined on smooth vectors and satisfies sup Denote by Ṡm the class of all regular m-homogeneous symbols and by Ṡm c the ones with compact support in the x-direction.
Example 7.13 ([FFK20,4.3,4.4]).For each c ∈ C ∞ c (G) and multi-index α ∈ N n 0 , the symbol c(x)π(X) α belongs to Ṡ[α] c .In the Euclidean case, homogeneous symbols are cut off in a neighbourhood of the zero section to obtain actual elements of the symbol classes.This corresponds to the following procedure for graded Lie groups.
This allows to define a homogeneous expansion of symbols.Proposition 7.15 ([FFK20,4.14]).Let {a j } j∈N0 be a sequence of homogeneous symbols a j ∈ Ṡmj with m j strictly decreasing to −∞ as j → ∞.Then there is a symbol a ∈ S m0 , unique modulo S −∞ , such that Moreover, if a ∈ S m for m < m 0 , it follows that a 0 = 0.
In this case, we also write a ∼ a j .There is a well-defined principal symbol for these operators.
Definition 7.16 ([FFK20, 4.17]).Let a ∈ S m be a symbol that admits a homogeneous expansion a ∼ ∞ j=0 a j with a j ∈ Ṡm−j as above.The principal part of Op(a) is defined as princ m (Op(a)) := a 0 .Definition 7.17 ([FFK20,4.20]).A symbol a ∈ S m cp is a classical pseudodifferential symbol of order m if it admits a homogeneous expansion a ∼ ∞ j=0 a j with a j ∈ Ṡm−j .Denote by S m cl the classical pseudo-differential symbols of order m and by Ψ m cl = Op(S m cl ) the corresponding operators.Proposition 7.18 ([FFK20,4.19]).Let A ∈ Op(S m ) and B ∈ Op(S l ) be operators whose symbols have a homogeneous expansion as above.Then the symbols of AB and A * admit homogeneous expansions as well.Moreover, the following holds On the right hand side, the pointwise operations are used.
In particular, princ 0 is a * -homomorphism.For m = 0, it is a short exact sequence of * -algebras.
Proof.Except for surjectivity of the principal symbol map, exactness is clear.Let a 0 ∈ Ṡm c and a(x, π) := a 0 (x, π)ψ(π(R)) ∈ S m as in Proposition 7.14.Then princ m (Op(a)) = a 0 holds.We need to adjust a in a way such that its integral kernel K a is compactly supported.Let r > 0 be such that the support of a in the x-direction in contained in B(0, r) with respect to the homogeneous quasi-norm.Choose c ∈ C ∞ c (G) which is constant 1 on B(0, 2Cr).Here, C is the constant from the homogeneous triangle inequality in Lemma 3.12.Denote by (1) a(x, λ • π) = a(x, π) for almost all π and λ > 0, (2) for all α ∈ N n 0 , the field of operators ∆ α a(π) is defined on smooth vectors and satisfies sup The C * -algebra of invariant 0-homogeneous symbols C * ( S0 ) is the closure of S0 with respect to a = sup π∈ G/R>0 a(π) .
Lemma 7.21.The * -algebra K 0 (G) of kernels of type 0 is isomorphic to S0 under Fourier transform.Moreover, Proof.By [FFK20, 5.3] the kernels of type 0 correspond exactly to the invariant, 0-homogeneous symbols under Fourier transform.By uniqueness of the C * -completion, C * ( S0 ) is isomorphic to the C * -algebra generated by kernels of type 0. This is Fix R>0 (J G , R G ) by Proposition 6.11.
By Lemma 5.12, Fix R>0 (J Identifying the space of 0-homogeneous symbols with compact support in the x-direction Ṡ0 c with C ∞ c (G, S0 ) as in the proof of [FFK20, 5.9], yields the following result.

Comparison of the operators.
To compare the sequence of generalized fixed point algebras to the order zero pseudo-differential extension from (26), we show that operators in p 1 (Fix R>0 (J, R)) can be written as in (25) in terms of a symbol.
Proof.Using the Dixmier-Malliavin Theorem as in the proof of Lemma 6.7 one can factorize that restrict to g i , h i at t = 0. Then we can write The first part converges strictly in p 1 (Fix R>0 (J, R)) by Lemma 5.16, whereas the second converges by Lemma 5.14.
We consider now a slightly different generalized fixed point algebra.The reason is that one has to understand the convolution kernels κ x as a family of left-invariant operators in order to take their Fourier transform.Let B := C * (T G) ⊗ C 0 ([0, ∞)) with R >0 -action given by For t ≥ 0 let ev t : B → C * (T G) be the homomorphism induced by the restriction.Define the following R >0 -invariant ideal J B with dense subset R B ⊂ J B : Similar arguments as in Lemma 5.6 and Proposition 5.8 show: Lemma 7.24.The * -subalgebra R B ⊂ J B is square-integrable for the action β of R >0 .Furthermore, (J B , R B ) is a continuously square-integrable R >0 -C * -algebra.
Hence, Fix R>0 (J B , R B ) is defined.The evaluations at t = 1 and x ∈ G composed with the right regular representation C * (G) → B(L 2 G), yield strictly continuous representations converges strictly to an element of Fix R>0 (J B , R B ).Its image under ρ x is given by Proof.The first claim is proved as in Lemma 7.23.This uses that there is an isomorphism which is induced by Ψ(h)(t) = t −Q ev t (h).The action β corresponds to τ ⊗1 under the isomorphism.Here, τ is induced by the action of R >0 on itself by scaling.The second claim follows from strict continuity of ρ x and the computation ev Proof.We show that one can write T (f )ϕ(x) = ϕ * κ x for a smooth family of kernels κ x ∈ S (G) such that ϕ → ϕ * κ x extends to a bounded operator on L 2 (G) for all x ∈ G.This implies that one can apply Fourier transform to κ x and one obtains a symbol as above with a f (x, π) = κ x (π).
For each i ∈ I, one can write We claim that all (κ i,x ) for x ∈ G converge to distributions κ x ∈ S (G) whose convolution operators are bounded.Note that f can be understood as an element of R B with f 0 ∈ S 0 (T G).It follows from Lemma 7.25 that κ i,x converges in S (G) to the convolution kernel of ρ x (f ).In particular, a f (x, π) is the strict limit of Let a 0 ∈ Ṡ0 c .As discussed in Proposition 7.14 a 0 ψ(π(R)) is in S 0 .Using Lemma 7.25 we obtain a different way to attach a symbol to a 0 .By Corollary 7.22, there is a h 0 ∈ S 0 (T G) such that a 0 is the Fourier transform of We compare now the symbols a 0 ψ(π(R)) and a h .As a preparation, the following lemma is proved.
Then for all m > 0, there exists a constant C m > 0 with for all x ∈ G and almost all π ∈ G.
Proof.The symbols can be written as strict limits as multipliers of K(H π ) for almost all π ∈ G.This implies that The integral converges to 0 as the dominated convergence theorem can be applied due to the assumptions on ω.Note that h 0 (x, π) is a smoothing symbol by [FR16, 5.2.21], so that h 0 S −m ,0,0 is finite for all m > 0. Using the same estimates there is a constant C m > 0 such that b i (x, π) ≤ C m h 0 S −m ,0,0 for all i ∈ I and (x, π) ∈ G × G.As b(x, π) is the norm limit of this net, the claim follows.
Theorem 7.30.Let G be a graded Lie group.The order zero pseudo-differential extension from Lemma 7.19 embeds into the generalized fixed point algebra extension for G such that the following diagram commutes Proof.Every operator in Ψ −1 cl extends to a compact operator on L 2 (G) by Lemma 7.11.Let A be a classical pseudo-differential operator of order zero with principal symbol a 0 ∈ Ṡ0 c .Let Q ∈ Ψ 0 cl be the element constructed in the proof of Lemma 7.19 with princ 0 In the following we show that there is an element T ∈ p 1 (Fix R>0 (J, R)) with p 0 (T ) = a 0 and Q − T ∈ K(L 2 G).Once is this established, writing shows that A lies in p 1 (Fix R>0 (J, R)) as A − Q has order −1 since its principal symbol vanishes.
In particular, A − Q is compact by Lemma 7.11.The above decomposition also shows that p0 (A) = p0 (T ) = a 0 = princ 0 (A) so that the diagram in (27) commutes.To construct T , let h 0 ∈ S 0 (T G) be such that a 0 is the Fourier transform of Let h ∈ R B and a h be as in Lemma 7.27 and g(x, v, t) This is a compact operator as its convolution kernel is smooth by Lemma 7.29 and compactly supported.
Denote by C * (Ψ 0 cl ) the closure of the * -algebra Ψ 0 cl in B(L 2 G).Corollary 7.31.The C * -algebra C * (Ψ 0 cl ) generated by classical order zero pseudo-differential operators on a graded Lie group G is isomorphic to Fix R>0 (J, R).There is an extension of C * -algebras such that p 0 extends the principal symbol map princ 0 : Ψ 0 cl → Ṡ0 c .
For the converse, note first that K(L 2 G) ⊂ C * (Ψ 0 cl ).This holds as Ψ 0 cl contains the kernels in C ∞ c (G × G) and these generate the compact operators on L 2 (M ).Now, let f, g ∈ R. Let a ∈ Ṡ0 c be the inverse of |f 0 g 0 | ∈ C 0 (G, K 0 (G)) under Fourier transform.Since the principal symbol map is surjective, there is a P ∈ Ψ 0 cl with princ 0 (P ) = a.Then the operator p 1 (|f g|) = p 1 (|f g|) − Op(P ) + Op(P ) is contained in C * (Ψ 0 cl ).This is because Op(P ) is and p 1 (|f g|) − Op(P ) ∈ K(L 2 G) as the diagram in (27) commutes.The C * -algebra Fix R>0 (J, R) is generated by |f g| with f, g ∈ R. Thus, the result follows.

Morita equivalence and K-theory
In this section, we will show that (J, R) and (J 0 , R 0 ) are saturated for the zoom action of R >0 .Therefore, for each homogeneous Lie group G the C * -algebras of order zero pseudodifferential operators Fix R>0 (J, R) and principal symbols Fix(J 0 , R 0 ) are Morita-Rieffel equivalent to C * r (R >0 , J) and C * r (R >0 , J 0 ), respectively.For the Euclidean scalings on G = R n this is a result of [DS14].
Hausdorff for all i = 1, . . ., m.Moreover, the induced R >0 -action on each Λ i is free and proper by Proposition 3.23.There is a corresponding increasing sequence of closed, dilation invariant ideals in C * (G) 0 = J 0 J 1 J 2 . . .J m = J G (28) which is given by In this section, it will be shown that the subquotients J i /J i−1 of the filtration in (28) define continuous fields of C * -algebras over Λ i , respectively.This will allow us to prove, using Corollary 2.17, that Fix(J G , R G ) is Morita-Rieffel equivalent to the crossed product C * r (R >0 , J G ). Remark 8.1.In [BBL16] Pedersen's fine stratification [Ped89] is used to obtain a similar sequence of increasing ideals, where the respective subquotients are even isomorphic to trivial fields C 0 ( Λi , K(H i )) for some finite-or infinite-dimensional Hilbert spaces H i .For our purposes the coarse stratification suffices.Proposition 8.2.Each subquotient J i /J i−1 for i = 1, . . ., m is isomorphic to a continuous field of C * -algebras over Λ i with a unique dense, relatively continuous and complete subset R i for the induced R >0 -action.Furthermore,

The dilation action on J
) denote the non-degenerate homomorphism which is given by pointwise multiplication when J i /J i−1 is viewed as a continuous field.It satisfies the compatibility condition where τ denotes the R >0 -action on C 0 (Λ i ) given by τ λ (φ)(π) = φ(λ −1 • π).Therefore, J i /J i−1 is an R >0 -C 0 (Λ i )-algebra.The dilation action on Λ i is free and proper by Proposition 3.23.By Proposition 2.21 J i /J i−1 is saturated with respect to the subset It is the unique dense, complete, relatively continuous subset by Theorem 2.5 as J i /J i−1 is spectrally proper.
Using Corollary 2.17 and an inductive argument for the sequence in (28), we obtain as a consequence: Corollary 8.3.For a homogeneous Lie group G the R >0 -C * -algebra (J G , R G ) is saturated for the dilation action of R >0 .The generalized fixed point algebra Fix R>0 (J G , R G ) is Morita-Rieffel equivalent to C * r (R >0 , J G ). 8.2.Computation of the spectrum of the symbol algebra.Recall that it was shown in Proposition 6.11 that for a homogeneous Lie group G, Fix R>0 (J G , R G ) is the C * -algebra generated by kernels of type 0. As an application of the above results, we give a different proof of the description of its spectrum obtained in [FFK20, 5.5].
Proof.The ideals in (28) yield short exact sequences of generalized fixed point algebras for i = 1, . . ., m by Proposition 2.19: q Each quotient J i /J i−1 is an R >0 -C 0 (Λ i )-algebra with a free and proper R >0 -action on Λ i by Proposition 8.2.Therefore, the spectrum of Fix R>0 (J i /J i−1 , R i ) is Λ i /R >0 by Proposition 2.21.In particular, the spectrum is Hausdorff and, thus, T 0 .As Fix R>0 (J i /J i−1 , R i ) is separable, this implies that it is of type I by [Dix77, 3.1.6,9.1].If an ideal I of a C * -algebra A and the quotient A/I are of type I, it follows that A is of type I. Thus, one can use an inductive argument for the sequences above to show that Fix R>0 (J G , R G ) is of type I.We proceed by showing the second claim.The spectrum of J G is given by By Proposition 3.23, R >0 acts freely on Λ i for all i = 1, . . ., m.Hence, the action of R >0 on the spectrum of J G is free.As J G and Fix R>0 (J G , R G ) are of type I and separable, their spectra can be identified with their primitive ideal spaces.We show that there is a homeomorphism with ψ([ker(π)]) = ker( π) for π ∈ J G .By [KM20, 6.3, 6.4] there is a continuous, open and surjective quasi-orbit map ρ : Prim(C * r (R >0 , J G )) → Prim(J G )/R >0 .As the action of the amenable group R >0 on Prim(J G ) is free, the quasi-orbit map is also injective by [GR79,3.3].Hence, it is a homeomorphism.In particular, J G separates ideals in C * r (R >0 , J G ).We describe now the inverse of ρ.First, there is a homeomorphism φ : Prim(J G )/R >0 → Prime(I R>0 (J G )) by [KM20, 6.3], which is induced by mapping a primitive ideal P of J G to the largest R >0 -invariant ideal contained in P : It follows from [KM20, 2.11, 6.1, 6.3] that ρ −1 = i • φ with i : Prime(I R>0 (J G )) → Prim(C * r (R >0 , J G )), By Corollary 8.3, C * r (R >0 , J G ) is Morita-Rieffel equivalent to Fix R>0 (J G , R G ). Therefore, the Rieffel correspondence gives a homeomorphism r : Prim(C * r (R >0 , J G )) → Prim(Fix R>0 (J G , R G )). Together, we obtain a homeomorphism It is left to show that ψ([ker(π)]) = ker( π) for π ∈ J G .Let Q = φ([ker(π)]).Using that the action on J G is saturated, Lemma 2.16 implies It follows that T * a ∈ R G ∩ Q.Now the same argument as in Proposition 2.19 shows that As Fix R>0 (J 0 , R 0 ) is the trivial continuous field over G with fibre Fix R>0 (J G , R G ) we obtain the following result (compare [FFK20, 5.11]): Corollary 8.5.For each homogeneous Lie group G there is a homeomorphism Fix R>0 (J 0 , R 0 ) ∼ = G × ( G\{π triv })/R >0 .8.3.Morita equivalence.We deduce saturatedness for the respective ideals in the C * -algebras of T G and the tangent groupoid G. Proposition 8.6.Let G be a homogeneous Lie group.The C * -algebra of order 0 principal symbols Fix R>0 (J 0 , R 0 ) is Morita-Rieffel equivalent to C * r (R >0 , J 0 ).The C * -algebra of order 0 pseudo-differential operators Fix R>0 (J, R) is Morita-Rieffel equivalent to C * r (R >0 , J). Proof.As (J G , R G ) is saturated by Corollary 8.3, (J 0 , R 0 ) is saturated by Corollary 2.26.Therefore, the generalized fixed point algebra construction gives the Morita-Rieffel equivalence between Fix R>0 (J 0 , R 0 ) and C * r (R >0 , J 0 ).The second claim follows from Corollary 2.17 applied to the sequence in (21) if saturatedness for the ideal C 0 (R >0 ) ⊗ K(L 2 G) is shown.By Remark 5.2 the R >0 -action is given by τ ⊗ 1, where τ is induced by the action of R >0 on itself by multiplication.Then R ∩ (C 0 (R >0 ) ⊗ K(L 2 G)) is the unique dense, relatively continuous and complete subspace and τ is free and proper.Therefore, the action is saturated by Proposition 2.21.The Morita-Rieffel equivalence follows again from the generalized fixed point algebra construction.8.4.K-theory of the C * -algebra of 0-homogeneous symbols.The Morita-Rieffel equivalence between the C * -algebra of 0-homogeneous symbols and the crossed product C * r (R, J 0 ) allows us to compute its K-theory.We recover the same result as in the Euclidean setting.Theorem 8.7.Let G be a graded nilpotent Lie group with n = dim g.Then Fix R>0 (J G , R G ) is KK-equivalent to C(S n−1 ).The C * -algebra of principal symbols Fix R>0 (J 0 , R 0 ) is KK-equivalent to C 0 (S * R n ).
Proof.The Morita-Rieffel equivalences between Fix R>0 (J G , R G ) and C * r (R, J G ) obtained in Corollary 8.3 implies that they are KK-equivalent.By the Connes-Thom isomorphism, C * r (R, J G ) is in turn KK-equivalent to C 0 (R) ⊗ J G .
Let g be the Lie algebra of G and for each t ∈ [0, 1] define [X, Y ] t := t[X, Y ] for X, Y ∈ g.Note that here the usual scalar multiplication by t ∈ [0, 1] is used and not the dilation action.One checks that [ • , • ] t defines a Lie bracket for all t ∈ [0, 1].Denote by g t the corresponding Lie algebra and by G t its Lie group.All Lie algebras g t for t > 0 are isomorphic to g via X → tX.
Consider the groupoid D G = R n × [0, 1] ⇒ [0, 1], where source and range are given by the projection to the last coordinate and the multiplication in s −1 (t) = r −1 (t) = R n , identified with G t under the exponential map, is given by group multiplication in G t .This is a continuous field of groups over [0, 1] that deforms the graded nilpotent Lie group G into the Abelian group R n .Using Fourier transform at t = 0 one obtains the short exact sequence ev0 Consider the associated KK-element [ev 0 ] −1 ⊗ [ev 1 ] ∈ KK(C 0 (R n ), C * (G)), as described in [DL10].First, we shall prove as in [Nis03] that it is a KK-equivalence for any simply connected, nilpotent Lie group G by induction on the dimension of G.If G is one-dimensional, it must be Abelian, so that G t is the constant field and [ev 1 ] −1 ⊗ [ev 0 ] is the inverse class.If G has dimension greater than one, it can be written as a semidirect product The naturality of the Connes-Thom isomorphism shows that the bottom row defines a KK-equivalence by induction hypothesis, which yields that C * (G) and C 0 (R n ) are KK-equivalent.We show that it restricts to a KK-equivalence between J G and C 0 (R n \{0}).Consider the ideal I G ⊂ C * (D G ) that consists of all sections (a t ) ∈ C * (D G ) such that all a t ∈ C * (G t ) lie in the kernel of the trivial representation of G t .In the commuting diagram the associated KK-classes in the middle and on the right are KK-equivalences.The long exact sequences in KK-theory show that the deformation element on the left is also a KK-equivalence.

q.
ProofFor a, b ∈ R ∩ I, we can view |a b| as a multiplier of I or A. As |a b|(A) ⊂ I it follows that |a b| I = |a b| A .Hence, by extending continuously we obtain an injective * -homomorphism Fix H (I, R ∩ I) → Fix H (A, R).

Proposition 3. 23 .
For d ∈ D let Λ d := Ω d ∩ g * T (d) .The map Λ d → G\Ω d induced by the inclusion is an R >0 -equivariant homeomorphism.The corresponding R >0 -action on Λ d is free and proper.

4. 1 .
The tangent groupoid.Definition 4.1.For a homogeneous Lie group G the tangent groupoid is the continuous action groupoid 2 }, where f I,1 = sup (x,t) G |f (x, t, v)| dv and f I,2 = f * I,1 .The groupoid C * -algebra C * (G) is the closure of C c (G) under the representation ρ as in (4).In particular, the C * -norm of f ∈ C c (G) is bounded by f I .Lemma 4.4.The continuous field of groupoids (G, [0, ∞), θ) gives rise to a continuous field of C * -algebras C * (G) over [0, ∞) with fibres isomorphic to K(L 2 G) for t > 0 and C * (T G) at t = 0.
4.2].Moreover, the operator-valued Fourier transform extends to a Fourier transform between left-invariant operators in B(L 2 a (G), L 2 b (G)) for a, b ∈ R to a space of fields denoted by L ∞ a,b ( G), see [FR16, 5.1.21,5.1.24].The spaces of corresponding convolution kernels in S (G) are denoted by K a,b (G).

Lemma 7. 19 .
For m ∈ Z, there are short exact sequences the multiplication operator ϕ → c • ϕ.It belongs to Ψ 0 cl with symbol given by c(x)id Hπ for x ∈ G, π ∈ G. Let Q := Op(a)M c .Its integral kernel K a (x, y)c(y) is compactly supported.Moreover, Q belongs to Ψ m cl and princ m (Q) = princ m (Op(a)) − princ m (Op(a) − Q).We show that last term is zero.Let κ denote the convolution kernel of Op(a).The convolution kernel of Op(a) − Q is κ x (y)(1 − c(xy −1 )).This is only non-zero if xy −1 ≥ 2Cr and x < r.The homogeneous triangle inequality implies that y ≥ r.But the κ x are smooth outside zero by [FR16, 5.4.1].Therefore, Op(a) − Q is a smoothing operator and its principal symbol vanishes.7.2.Comparison of the symbol algebras.In this section, we identify Fix R>0 (J G , R G ) with the C * -algebra of invariant 0-homogeneous symbols defined in [FFK20, 5.1, 5.5].Definition 7.20.The * -algebra of invariant 0-homogeneous symbols S0 consists of all a ∈ L ∞ ( G, B(H π )) with
Since the action is proper, Hx is H-equivariantly homeomorphic to H/H x .Here H x is the stabilizer of x, which is a compact subgroup of H.
[Dix77,90, p. 154]).By[Dix77, 8.6.1]there is a subspace Γ ⊂ π∈ G H π that turns ((H π ) π∈ G , Γ) into a µ-measurable field of Hilbert spaces over G as defined in [FR16, Def.B.1.3].The elements of Γ are called the measurable sections.For a Hilbert space H denote by HS(H) the Hilbert space of Hilbert-Schmidt operators on H. Identifying H ⊗ H * with HS(H), one obtains for a simply connected nilpotent Lie group G the µ-measurable field ((HS(H π