A Lambda-Graph System for the Dyck Shift and Its K-Groups

A property of subshifts is described that allows to associate to the subshift a distinguishied presentation by a compact Shannon graph. For subshifts with this property and for the resulting invariantly associated compact Shannon graphs and their A-graph systems the term 'Cantor horizon' is proposed. The Dyck shifts are Cantor horizon. The C*-algebras that are obtained from the Cantor horizon A-graph systems of the Dyck shifts are separable, unital, nuclear, purely infinite and simple with UCT. The K-groups and Bowen-Franks groups of the Cantor horizon A-graph systems of the Dyck shifts are computed and it is found that the K 0 -groups are not finitely generated.


Introduction
Let Σ be a finite alphabet.On the shift space Σ Z one has the left-shift that sends a point (σ i ) i∈Z into the point (σ i+1 ) i∈Z .In symbolic dynamics one studies the dynamical systems, called subshifts, that are obtained by restricting the shift to a shift invariant closed subset of Σ Z .For an introduction to symbolic dynamics see [Ki] or [LM].A finite word in the symbols of Σ is said to be admissible for the subshift X ⊂ Σ Z if it appears somewhere in a point of X.A subshift is uniquely determined by its set of admissible words.Throughout this paper, we denote by Z + and N the set of all nonnegative integers and the set of all positive integers respectively.A directed graph G whose edges are labeled by symbols in the finite alphabet Σ is called a Shannon graph if for every vertex u of G and for every α ∈ Σ, G has at most one edge with initial vertex u and label α.We say that a Shannon graph G presents a subshift X if every vertex of G has a predecessor and a successor and if the set of admissible words of X coincides with the set of label sequences of finite paths on G.To a Shannon graph G there is associated a topological Markov chain M (G).The state space of M (G) is the set of pairs (u, α), where u is a vertex of G and α is the label of an edge of G with initial vertex u.Here a transition from state (u, α) to state (v, β) is allowed if and only if v is the final vertex of the edge with initial vertex u and label α.For a vertex u of Shannon graph G we denote the forward context of u by Γ + (u).Γ + (u) is the set of sequences in Σ N that are label sequences of infinite paths in G that start at the vertex u.We say that a Shannon graph G is forward separated if vertices of G, that have the same forward context, are identical.The Shannon graphs that we consider in this paper are forward separated.We always identify the vertices of a forward separated Shannon graph G with their forward contexts, and then use on the vertex set of G the topology that is given by the Hausdorff metric on the set of nonempty compact subsets of Σ N .
There is a one-to-one correspondence between forward-separated compact Shannon graphs G such that every vertex has a predecessor and a class of λ-graph systems [KM].We recall that a λ-graph system is a directed labelled Bratteli diagram with an additional structure.We write the vertex set of a λ-graph system as Every edge with initial vertex in V −n , has its final vertex in V −n+1 , n ∈ N.
It is required that every vertex has a predecessor and every vertex except the vertex in V 0 has a successor.In this paper we consider λ-graph-systems that are forward separated Shannon graphs.Their additional structure is given by a mapping ι : that is compatible with the labeling, that is, if u is the initial vertex of an edge with label α and final vertex v, then ι(u) is the intial vertex of an edge with label α and final vertex ι(v).
Given a subshift X ⊂ Σ Z there is a one-to-one correspondence between the compact forward separated Shannon graphs that present X, and the forward separated Shannon λ-graph systems that present X.To describe this oneto-one correspondence denote for a vertex v of a Shannon graph by v n the set of initial segments of length n of the sequences in V, n ∈ N. The λ-graph system that corresponds to the forward separated Shannon graph has as its set V −n the set of v n , n ∈ Z + , v a vertex of G, and if in G there is an edge with initial vertex u and final vertex v and label α then in the corresponding λ-graph system there is an edge with initial vertex u n , final vertex and label α, the mapping ι of the corresponding λ-graph system deleting last symbols.λ-graph systems can be described by their symbolic matrix systems [Ma] (M −n,−n−1 , I −n,−n−1 ) n∈Z+ . Here there is an edge with initial vertex v and final vertex u with label α i , i = 1, . . ., k and by setting that is given by setting I −n,−n−1 (u, v) equal to one if ι(v) = u and by setting I −n,−n−1 (u, v) equal to zero otherwise.We remark that the time direction considered here is opposite to the time direction in [Ma].For symbolic matrix systems there is a notion of strong shift equivalence [Ma] that extends the notion of strong shift equivalence for transition matrices of topological Markov shifts [Wi] and of the symbolic matrices of sofic systems [BK,N].
To a symbolic matrix system there are invariantly associated K-groups and Bowen-Franks groups [Ma].To describe them, let be the nonnegative matrix that is given by setting M n,n+1 (u, v) equal to zero if M −n,−n−1 (u, v) is zero, and by setting it equal to the number of the symbols whose sum is Let Z I be the group of the projective limit lim , n ∈ Z + acts on it as an endomorphism, denoted by M − I.
The Bowen-Franks groups BF I (M, I), i = 0, 1 are defined by Given a subshift X ⊂ Σ Z we use for x = (x i ) i∈Z ∈ X notation like and we set using similar notations when indices range is infinite intervals.We denote the forward context of a point x − in X (−∞,0] by Γ + (x − ), } is the vertex set of a forward separated Shannon graph that presents X.The λ-graph system of its closure was introduced in [KM] as the canonical λ-graph system of the subshift X.It is canonically associated to the subshift in the sense that a topological conjugacy of subshifts induces a strong shift equivalence of their canonical λ-graph systems.
For a subshift X ⊂ Σ Z that is synchronizing [Kr] (or semisynchronizing [Kr] ) one has an intrinsically defined shift invariant dense subset P s (X) of periodic points of X, and one has associated to X the presenting forward separated Shannon graph whose vertex set is the set of forward contexts Γ + (x), where x is left asymptotic to a point in P s (X).These Shannon graphs are canonically associated to the synchronizing (or semisynchronizing) subshift, in the sense that a topological conjugacy of subshifts induces a block conjugacy of the topological Markov chains of the Shannon graphs and also a strong shift equivalence [Ma] of the λ-graph systems of their closures.Prototype examples of semisynchronizing subshifts are the Dyck shifts that can be defined via the Dyck inverse monoids.The Dyck inverse monoid is the inverse monoid (with zero) with generators α n , β n , 1 ≤ n ≤ N, and relations In section 1 we introduce another class of subshifts X ⊂ Σ Z with an intrinsically defined shift invariant dense set P Ch (X) of periodic points.Again the Dyck shifts serve here as prototypes.In the Dyck shift D N the points in P Ch (D N ) are such that during a period there appears an event that has the potential to influence even the most distant future.In other words, a point (x i ) i∈Z in One can view here the record of an infinite sequence of events as a point in a Cantor discontinuum.With this in mind, we call the subshifts in this class Cantor horizon subshifts.The presenting Shannon graph with vertex set the set of forward contexts Γ + (y − ), where y − is negatively asymptotic to a point in P Ch (X), is canonically associated to the Cantor horizon subshift X ⊂ Σ Z , and so is the λ-graph system of its closure, that we call the Cantor horizon λ-graph system of X.The Cantor horizon λ-graph system of a Cantor horizon subshift is a sub λ-graph system of its canonical λ-graph system.The K-groups and Bowen-Franks groups of the symbolic matrix system (M DN , I DN ) for the canonical λ-graph systems of the Dyck shifts D N , N ≥ 2 were computed in [Ma2].These are In section 3 we determine the symbolic matrix system (M Ch(D2) , I Ch(D2) ) of the Cantor horizon λ-graph system L Ch(D2) of the Dyck shift D 2 , and we compute its K-groups.Denoting the group of all Z-valued continuous functions on the Cantor discontinuum C by C(C, Z) one has and one has K 1 (M Ch(D2) , I Ch(D2) ) ∼ = 0.
A λ-graph system is said to be irreducible if for a sequence v and for a vertex u, there exists an N ∈ Z + such that there is a path from v −N to u.It is said to be aperiodic if for a vertex u, there exists an N ∈ Z + such that for all v ∈ V −N there exist paths from v to u.The Cantor horizon λ-graph system L Ch(D2) of the Dyck shift D 2 is irreducible and moreover aperiodic.Hence the resulting C * -algebra O L Ch(D 2 ) is simple and purely infinite whose K 0 -group and K 1 -group are the above groups K 0 (M Ch(D2) , I Ch(D2) ) and K 1 (M Ch(D2) , I Ch(D2) ) respectively (cf.[Ma3]).In section 4, we compute the Bowen-Franks groups of the symbolic matrix system (M Ch(D2) , I Ch(D2) ).
In section 5, we consider the K-groups and Bowen-Franks groups of the Dyck shifts D N , N ≥ 2.Here one has 1. Subshifts with Cantor horizon lambda-graph systems Denoting for a given subshift X ⊂ Σ Z , the right context of an admissible block we set Lemma 1.1.Let X ⊂ Σ Z , X ⊂ Σ Z be subshifts and let ψ : X → X be a topological conjugacy.Let for some L ∈ Z + ψ be given by a (2L + 1)-block map Ψ and ψ −1 be given by a (2L + 1)-block map Ψ.Let N ∈ N, and let x ∈ X be such that Then for x = ψ( x) and Proof.Let n ≥ N, and let One has confirming (1.2).We note that by (1.1) one has that (1.4) follows from which in turn follows from (1.3).
Let X ⊂ Σ Z be a subshift and P (X) be its set of periodic points.Denote by P a (X) the set of x ∈ P (X) such that there is an Lemma 1.2.Let X ⊂ Σ Z , X ⊂ Σ Z be subshifts and let ψ : X → X be a topological conjugacy.Then ψ(P a ( X)) = P a (X).
By Lemma 1.1 the following property of a subshift X ⊂ Σ Z is invariant under topological conjugacy : For x ∈ X and N ∈ N such that For a subshift X ⊂ Σ Z with this property we consider the subgraph G Ch (X) of G(X) with vertices Γ + (u − ) where u − ∈ X (−∞,0] is negatively asymptotic to a point in P Ch (X) = P (X)\P a (X).If here G Ch (X) presents X then we say that X is a Cantor horizon subshift, and we call the λ-graph system of the closure of G Ch (X) the Cantor horizon λ-graph system of X.By Lemma 1.2 the Cantor horizon property is an invariant of topological conjugacy and the Cantor horizon λ-graph system is invariantly associated to the Cantor horizon subshift.

The Dyck shift
We consider the Dyck shift D 2 with alphabet Σ = Σ − ∪ Σ + where Σ − = {α 0 , α 1 }, Σ + = {β 0 , β 1 }.A periodic point x of D 2 with period p is not in P a (D 2 ) precisely if for some i ∈ Z the normal form of the word (x i+q ) 0≤q<p is a word in the symbols of Σ − , in other words, if the multiplier of x in the sense of [HI] is negative.We also note that periodic points with negative multipliers give rise to the same irreducible component of G Ch (D 2 ) precisely if they have the same multiplier.We describe the Cantor horizon λ-graph system L Ch(D2) of D 2 : The vertices at level l are given by the words of length l in the symbols of Σ − .The mapping ι deletes the first symbol of a word.A word (α i(n) ) 1≤n≤l accepts β i precisely if i(l) = i, i = 0, 1, effecting a transition to the word (α i(n) ) 1≤n<l , and it accepts α i , effecting a transition to the word (α i(n) ) 2≤n≤l .The forward context of the word a = (α i(n) ) 1≤n≤l contains precisely all words c = (γ n ) 1≤n≤l in symbols of Σ such that (a, c) is admissible for D 2 .In describing the Cantor horizon symbolic matrix system (M, I) of the Dyck shift and the resulting nonnegative matrix system (M, I) we use the reverse lexcographic order on the words in the symbols in Σ − , that is, we assign to a word (α One has then For l ∈ Z + and a ∈ {α 0 , α 1 , β 0 , β 1 }, let I l (a) be the 2 l × 2 l diagonal matrix with diagonal entries a, and S l (a) be the 2 l−1 × 2 l+1 matrix [S l (a)(i, j)] 1≤i≤2 l−1 ,1≤j≤2 l+1 where S l (a)(i, j) is a for j = 4i, 4i−1, 4i−2, 4i−3, and is otherwise zero.

and
(2.2) Proof.The first summand in (2.1) describes the transitions that arise when a vertex accepts a symbol in Σ + .The second summand arises from the transitions that arise when a vertex accepts a symbol in Σ − , the arrangement of the components of the matrix as well as (2.2) being a component of the ordering of the vertices at level l and l − 1.
We note that the λ-graph systems of the closures of the irreducible components of G Ch (X) are identical.
Proof.Let V −l , l ∈ Z + be the vertex set of the λ-graph system L Ch(D2) .For any vertex u of V −l , there are labeled edges from each of the vertices in V −2l to the vertex u.This implies that (M, I) is aperiodic.

Computation of the K-groups
Let M l,l+1 for l ∈ Z + be the nonnegative matrix obtained from M −l,−l−1 by setting all the symbols of the components of M −l,−l−1 equal to 1.The matrix I l,l+1 for l ∈ Z + is defined to be I −l,−l−1 .For l > 1 and 1 ≤ i ≤ 2 l−2 let a i (l) = [a i (l) n ] 2 l n=1 be the vector that is given by Define for 1 < l ∈ N E l as the 2 l × 2 l -matrix whose i−th column vector and whose (2 l−1 + i)−th column vector are both equal to a i (l), 1 ≤ i ≤ 2 l−2 , the other column vectors being equal to zero vectors.For l > 1 and be the vector that is given by Define for 1 < l ∈ N F l as the 2 l × 2 l -matrix whose i−th column vector is equal to b i (l), 1 ≤ i ≤ 2 l−1 , the other column vectors being equal to zero vectors.One has Let I l denote the unit matrix of size 2 l .Define a 2 l × 2 l−1 matrix H l,l−1 by setting Lemma 3.1.For l > 1 one has Proof.One has , and define inductively for l > 2 vectors y i (l), 1 ≤ i ≤ 2 l , where where one defines the vectors by setting (3.4) and by setting Define for l > 2, T l as the 2 l × 2 l -matrix whose column vectors Here y 1 (l) has all components equal to 1, that is, y 1 (l) is the eigenvector of E l − F l for the eigenvalue 1.Also the vectors are linearly independent vectors in the kernel of E l − F l and one sees from (3.1), (3.2), (3.3), (3.4) and (3.5) that T l is invertible and that T −1 l (E l − F l )T l a matrix that is in a normal form.This normal form is a Jordan form in the sense that by conjugation with a suitable permutation matrix, followed by a conjugation with a suitable diagonal matrix whose entries are 1 or −1, the matrix assumes a Jordan form with Jordan blocks arranged along the diagonal.There will be one Jordan block of length 1 for the eigenvalue 1, and there will be 2 l−1 Jordan blocks for the eigenvalue 0, and if one lists these by decreasing length then the k−th Jordan block for the eigenvalue 0 has length l − µ(k) where µ(k) be given by 2 By an elementary column operation we will mean the addition or subtraction of one column vector from another or the exchange of two column vectors.Lemma 3.2.Let l ∈ N and let K be a 2 l × 2 l -matrix with column vectors and column vectors Then K can be converted into the unit matrix by a sequence of elementary column operations.
Proof.Let the vector c j = [c j,n ] n=1 , 1 ≤ j ≤ 2 l−1 , be given by c j,n = 1 (n = 2j), 0 elsewhere.and denote by K[j], 1 ≤ j ≤ 2 l−1 the matrix that is obtained by replacing in the matrix K the last j column vectors by the vectors c , can be obtained from K[j − 1] by subtracting from and adding to the (2 l − j)-th column appropriate selections of the the (2 l − i)-th columns, 1 ≤ i ≤ j.K[2 l−1 ] has as its first 2 l−1 column vectors the vectors b i (l), 1 ≤ i ≤ 2 l−1 , and as its last 2 l−1 column vectors the vectors c i (l), 1 ≤ i ≤ 2 l−1 , and can be converted into the unit matrix by elementary column operations.
Then K can be converted into the unit matrix by a sequence of elementary column operations.
Proof.For all i, 2 l−1 < i ≤ 2 l , one subtracts from the i-th column of K and adds to the i-th column of K appropriate selections of the first 2 l−1 columns of K to obtain a matrix to which Lemma 3.2 applies.
Proposition 3.4.The matrix T l is unimodular.
Proof.The matrix T 2 can be converted into the unit matrix by elementary column operations.The proof is by induction on l.Assume that the matrix T l−1 , l > 2 can be converted into the unit matrix by a sequence of elementary column operations.Then by (3.2) the matrix T l can be converted by a sequence of elementary column operations into a matrix whose first 2 l−1 column vectors are the vectors b i (l), 1 ≤ i ≤ 2 l−1 and whose last 2 l−1 column vectors are those of the matrix T l , and by (3.2),(3.3)and (3.4) Lemma 3.3 is applicable to this matrix.
By the preceding lemma, the matrix J l+1,l induces a homomorphism Jl+1,l : and by Proposition 3.4 and Lemma 3.6 the matrix U l , as Lemma 3.9.The diagram : is commutative.
Define a diagonal matrix D(l) by setting 2), we have Proposition 3.10.The following diagram is commutative: Proof.As the group of the inductive limit: lim − → l {I t l,l+1 : Z 2 l → Z 2 l+1 } is isomorphic to C(C, Z), we get the assertion.Theorem 3.12.
By Proposition 2.2, the Cantor horizon λ-graph system L Ch(D2) of D 2 is aperiodic so that the C * -algebra O L Ch(D 2 ) associated with the λ-graph system L Ch(D2) is simple and purely infinite ([Ma3;Proposition 4.9]).It satisfies the UCT by [Ma3;Proposition 5.6] (cf.[Bro], [RS]).By [Ma3;Theorem 5.5], the Kgroups K i (O L Ch(D 2 ) ) are isomorphic to the K-groups K i (M, I) so that we get We compute next the Bowen-Franks groups BF 0 (M, I) and BF 0 (M, I).
Lemma 4.1.Ext 1 Z (C(C, Z), Z) ∼ = 0. Proof.From [Ro] (cf.[Sch;Theorem 1.3]) one has that for an inductive sequence {G i } of abelian groups there exists a natural short exact sequence 0 → lim The lemma follows therefore from C(C, Z) = lim − → l {I t l,l+1 : Z 2 l → Z 2 l+1 } and As in [Ma;Theorem 9.6], one has the following lemma that provides a universal coefficient type theorem.