ALGEBRAS OF ONE-SIDED SUBSHIFTS OVER ARBITRARY ALPHABETS

. We introduce two algebras associated with a subshift over an arbitrary alphabet. One is unital, and the other not necessarily. We focus on the unital case and describe a conjugacy between Ott-Tomforde-Willis subshifts in terms of a home-omorphism between the Stone duals of suitable Boolean algebras, and in terms of a diagonal-preserving isomorphism of the associated unital algebras. For this, we realise the unital algebra associated with a subshift as a groupoid algebra and a partial skew group ring.


Introduction
The study of subshifts is at the heart of symbolic dynamics.Describing various types of equivalences between subshifts is one of the main arteries of research in the field.For example, for subshifts of finite type, conjugacy is equivalent to strong shift equivalence.However, it is not known for which classes of subshifts conjugacy is equivalent to shift equivalence, which is easier to decide.This question is usually referred to as the Williams' problem, see [9,30,31].
Recent work allows a complete recast of Williams' problem in operator algebras since isomorphism and Morita equivalence of two graph C*-algebras given by shifts of finite type are connected to equivalences of the subshifts, such as topological conjugacy and shift equivalence (see, for instance, [7] for recent results and further details).These results follow the celebrated interaction between dynamics and C*-algebras, which includes the work of Giordano-Matui-Putnam-Skau on describing orbit equivalence of Cantor minimal system via C*-algebraic invariants, [16,17], the seminal work of Cuntz-Krieger on C*algebras associated with shifts of finite type (given by matrices), [10], Matsumoto and Matui's work on orbit equivalence and continuous orbit equivalence of topological Markov chains [22,23], Carlsen's work on subshift algebras over finite alphabets, [6], and more recently the work by Brix-Carlsen on the description of conjugacy of subshifts over finite alphabets via an isomorphism of associated algebras, [5], to mention a few.
The aim of this paper is in the same spirit as the previous paragraph.We connect the theory of subshifts over infinite alphabets with operator algebras.For an infinite alphabet, there is no standard definition of a subshift.The usual approach for the full shift is to equip a countable alphabet A with the discrete topology.Then consider the product n∈N A with the product topology.This topology is induced by the product metric where k is such that x i = y i for i = 0, . . .k − 1 and x k = y k .In this setting, a subshift X is defined as a shift-invariant closed subspace.Alternatively, a subshift can be defined in terms of a family of forbidden blocks (see [11,24] and Section 2 for details).With the relative topology, the space obtained is Polish, but might not even be locally compact.The lack of local compactness is an obstacle to using analytical techniques to study such subshifts.
In [25], Ott, Tomforde, and Willis propose an alternative definition of a subshift over an infinite alphabet.They first consider a compact Hausdorff space as a new version for the full subshift.Then, they define a subshift as a shift-invariant closed subspace with an extra condition, which they call the infinite extension property (see [25,Definition 3.3]).We call these OTW-subshifts.Since OTW-subshifts are compact Hausdorff spaces, analytical tools are more readily available.Moreover, starting with a subshift X, there is a corresponding OTW-subshift X OT W .In Theorem 6.11, we show that isometric conjugacy for subshifts implies conjugacy of the correspondent OTW-subshifts.
Recently, the authors of this paper introduced algebras associated with a subshift in a purely algebraic setting, see [4].These algebras deepened the connection between OTWsubshifts and noncommutative algebras, which in the original work of Ott-Tomforde-Willis is restricted to the case of edge subshifts associated with a certain class of graphs.In [4], the authors describe conjugacy of OTW-subshifts in terms of isomorphisms of algebras associated with X.
We do not know of a C*-algebra associated with a general subshift over an infinite alphabet that forms an invariant for notions of equivalence of subshifts.In this paper we remedy this.We overcome the lack of local compactness of a subshift X by using combinatorial aspects of it to define a C*-algebra O X associated with X.We apply the results of [1] to show that O X together with additional data is a complete invariant for conjugacy of OTW-subshifts (Theorem 6.9).This is done by identifying an OTW-subshift with the spectrum of a certain commutative C*-subalgebra of O X .Moreover, using the connection between X and X OT W , we obtain one of the main results of this paper, which says that our C*-algebras form an invariant for isometric conjugacy of subshifts over countable alphabets (Theorem 6.13).
The C*-algebras we introduce in this paper are the analytical counterparts of the algebras introduced in [4].We use the results of [4] and of [15] on core subalgebras to obtain results about the structure of O X .In particular, we show that O X can be described using labelled spaces, groupoids and partial actions.
The K-theory of C*-algebras provides a standard invariant for isomorphism of C*algebras.The description of a unital subshift C*-algebra O X using labelled spaces and [2, Theorem 4.4] provide a concrete way to compute the K-theory of O X .We illustrate this by computing the K-theory of two examples.In the first, we use K-theory to show that there is a C*-algebra of a graph that is not isomorphic to any of our C*-algebras associated with the edge subshift of the graph.In the second example, we compute the K-theory of the C*-algebra of a 2-step subshift that is not conjugate to any 1-step subshift (example taken from [18]).In particular, it is not conjugate to any graph edge subshift.Lastly, from our theorem about isometric conjugacy, we deduce that the K-theory of subshift C*-algebras is an invariant for isometric conjugacy of subshifts.
We use our groupoid model for O X to prove Theorem 6.9, which gives a full invariant for topological conjugacy of OTW-subshifts.The unit space of this groupoid is homeomorphic to the spectrum of another commutative C*-subalgebra of O X .Under an extra condition, isomorphism of subshift C*-algebras preserving these commutative C*subalgebras is described both as an isomorphism of the corresponding groupoids and as a continuous orbit equivalency between certain topological partial actions (Theorem 6.7).
The paper is organized as follows.Section 2 contains basic elements of symbolic dynamics and the definitions of subshifts and OTW-subshifts.
In Section 3, we define two C*-algebras, O X and O X , associated with a subshift X, where O X is unital and O X is not necessarily unital.The latter algebra is mostly used to compare with non-unital graph and ultragraph C*-algebras.The main focus of this paper is on O X .We show that the purely algebraic subshift algebras of [4] are core subalgebras of our C*-algebras.We realize O X and O X as C*-algebras of labelled spaces.Finally, we prove a gauge invariance uniqueness theorem.
In Section 4, we provide examples of known C*-algebras that can be seen as C*-algebras of a subshift.First, we show that the C*-algebra of a graph is isomorphic to the edge subshift C*-algebra, provided the graph has no sinks and no vertex that is simultaneously a source or an infinite emitter.Our second example shows that for ultragraphs with only regular vertices, the C*-algebra of the ultragraph is the C*-algebra of the one-sided edge subshift of the ultragraph.Our last example of this section shows that every unital Exel-Laca algebra is isomorphic to the unital C*-algebra of a subshift.
We dedicate Section 5 to the K-theory of the unital subshift C*-algebra O X .Adapting [2,Theorem 4.4] to O X , we obtain a concrete way to compute the K-theory of O X and apply it to two examples.
Finally, Section 6 contains the main results of this paper, Theorems 6.7, 6.9, 6.11 and 6.13, which connect the dynamical and algebraic aspects as outlined above.

Symbolic Dynamics
In this paper, we consider N = {0, 1, 2, 3, . ..}.Let A be a non-empty set, called an alphabet.The shift map on A N is the map σ : A N → A N given by σ(x) = (y n ), where x = (x n ) and y n = x n+1 .Elements of A * := ∞ k=0 A k are called blocks or words, and ω stands for the empty word.Set A + = A * \ {ω}.Given α ∈ A * ∪ A N , |α| denotes the length of α and for 1 ≤ i, j ≤ |α|, we define α i,j := α i • • • α j if i ≤ j, and α i,j = ω if i > j.If β ∈ A * , then βα denotes concatenation of β and α.A subset X ⊆ A N is invariant for σ if σ(X) ⊆ X.For an invariant subset X ⊆ A N , we define L n (X) as the set of all words of length n that appear in some sequence of X, that is, Clearly, L n (A N ) = A n and we always have that L 0 (X) = {ω}.The language of X is the set consisting of all finite words that appear in some sequence of X.
Given F ⊆ A * , we define the subshift X F ⊆ A N as the set of all sequences x in A N such that no word of x belongs to F .Usually, the set F will not play a role, so we will say X is a subshift with the implication that X = X F for some F .We point out that subshifts are also called shift spaces in the literature.Given a subshift X over an alphabet A and α, β ∈ L X , we define C(α, β) := {βx ∈ X : αx ∈ X}.
We briefly recall the construction of OTW-subshifts (OTW stands for Ott-Tomforde-Willis), see [25].Let A be an alphabet and define Ã := A if A is finite and Ã When the alphabet is infinite, the set Σ fin A is identified with all finite sequences over A via the identification The sequence (∞∞∞ . ..) is denoted by 0. Let F ⊆ A * .We define there are infinitely many a ∈ A for which there exists y ∈ A N such that xay ∈ X inf F }.
The OTW-subshift associated with F is and n ≥ 2 0, if x = x 0 ∈ X f in F or x = 0.In the case that F = ∅, we have that X OT W F = Σ A , which we call the OTW full shift.Notice that X inf F coincides with the subshift X F defined above.As in the case of a subshift, we omit the subscript F and write X OT W for X OT W F .Let X OT W be an OTW-subshift and α ∈ L X .Then the follower set of α is defined as the set For a finite set F ⊆ A and α ∈ L X , we define the generalised cylinder set as To simplify notation, we denote Z(α, ∅) by Z(α).Endowed with the topology generated by the generalised cylinders, X OT W is a compact totally disconnected Hausdorff space and, in this topology, the generalised cylinders are compact and open.The shift map is continuous everywhere, except possibly at 0 (if 0 ∈ X OT W ). If the alphabet is finite, then X OT W is the usual subshift and the topology given by the generalised cylinders is the relative product topology (see [25,Remark 2.26]).

C*-algebras of subshifts
In this section, we define a unital and a not necessary unital C*-algebra associated with a subshift.Fix a subshift X over an alphabet A.

Unital C*-algebras of subshifts.
As in [4], let U to be the Boolean algebra of subsets of X generated by all C(α, β) for α, β ∈ L X , that is, each element of U is a finite union of elements of the form Definition 3.1.The unital subshift C*-algebra O X associated with X is the universal unital C*-algebra generated by projections {p A : A ∈ U} and partial isometries {s a : a ∈ A} subject to the relations: Remark 3.2.In [6], Carlsen defines a C*-algebra for subshifts over finite alphabets and proves it has a universal property.The definition above adapts the universal property described by Carlsen and generalises his definition to include infinite alphabets.
The following result is proved in [4,Proposition 3.6] for the unital algebra of a subshift.Since the same proof holds in the C*-algebraic case, we omit it here.
Next, we show that the unital algebra A R (X) defined in [4] can be seen as a dense subalgebra of O X .For that, we first build a representation that shows that each projection p A , with A ∈ U \ {∅}, is non-zero.Then, we consider the canonical gauge action on O X , in order to obtain a Z-grading on O X .Proposition 3.4.Consider the family of operators {S a } a∈A and {P A } A∈U on ℓ 2 (X) defined by where {δ x } x∈X is the canonical orthonormal basis of ℓ 2 (X).Then, there exists a *representation π : Proof.It is straightforward to check that P A is a projection, P X = 1, P A∩B = P A P B , P A∪B = P A + P B − P A∩B and P ∅ = 0, for every A, B ∈ U. Notice that, for every a ∈ A, S a is an isometry between the subspaces span{δ x : x ∈ F a } and span{δ x : x ∈ Z a }.In particular, S a is a partial isometry, whose adjoint is given by The existence of the representation comes from the universal property of O X .
Corollary 3.5.For all nonempty A ∈ U, we have that p A = 0.
Proof.For A ∈ U such that A = ∅, there exists x ∈ A. Using P A as defined in Proposition 3.4, we have that P A (δ x ) = δ x , so that P A = 0.It then follows that p A = 0.
We obtain an action of the circle T on O X as follows.For each z ∈ T, we define γ z : O X → O X on the generators by γ z (p A ) = p A for all A ∈ U, and γ z (s a ) = zs a for all a ∈ A.
Using the universal property of O X , we get that γ z is a *-homomorphism.It is clear that γ 1 is the identity, γ zw = γ z • γ w for every z, w ∈ T, and that γ z is an automorphism with γ −1 z = γ z −1 .We call the action γ the gauge action and observe that it induces a C * -grading on O X by Z, with fibers given by Remark 3.6.In several of the following results, we consider an algebraic version A C (X) of O X and its subalgebras, as defined in [4,Definition 3.3].The algebras in [4] are defined over a general ring R. When we specialize to the case R = C, we can consider the natural involution on A C (X) defined by (λs α p A s * β ) * = λs β p A s * α .In the next result, we show that there is an injective homomorphism from the unital subshift algebra A C (X) into O X .
Proposition 3.7.The canonical homomorphism A C (X) → O X that sends each generator to its corresponding generator with the same name is injective and has a dense image.The second part is due to the fact that the copy of A C (X) inside O X contains the generators of O X .
Let X OT W be as in Section 2 and U the Stone dual of U. To characterize C(X OT W ) and C( U) we will use the results in [4] and the theory of core subalgebras introduced in [15], which we briefly recall below.Definition 3.8.[15, Definition 3.1] Let A be a C*-algebra and B ⊆ A a (not necessarily closed) *-subalgebra.We say that B is a core subalgebra of A when every representation of B is continuous relative to the norm induced from A. By a representation of a *algebra B we mean a multiplicative, *-preserving, linear map π : B → B(H), where H is a Hilbert space.Remark 3.9.By [15,Example 3.3(v)], whenever A is a universal C*-algebra generated by generators and relations and B is the free *-algebra with the same generators and relations, then the image of B inside A is a core subalgebra.In particular, this applies to both graph algebras, ultragraph algebras and subshift algebras.
For examples of core subalgebras, we refer the reader to [15].The following result, which will be useful to us, is proved in [15].Proposition 3.10.[15, Proposition 3.4] Suppose that A 1 and A 2 are C*-algebras and that B i is a dense core subalgebra of A i , for i = 1, 2. If B 1 and B 2 are isomorphic as *-algebras, then A 1 and A 2 are isometrically *-isomorphic.Lemma 3.11.Let A be a C*-algebra and B a commutative *-algebra generated by projections of A. Then B is a core subalgebra of A.
For a topological space X, let Lc(X, C) denote the locally constant functions from X to C with compact support.By a Stone space, we mean a locally compact, totally disconnected Hausdorff space.Lemma 3.12.Let X be a Stone space.Then, Lc(X, C) is a dense core subalgebra of C 0 (X).
Proof.Since Lc(X, C) is generated by the characteristic functions of compact-open sets, which are projections in C 0 (X), it follows from Lemma 3.11 that Lc(X, C) is a core subalgebra of C 0 (X).As X is a Stone space, the characteristic functions of compactopen sets generate C 0 (X) as a C*-algebra, which implies Lc(X, C) is dense in C 0 (X).Proposition 3.13.Let X ⊆ A N be a subshift.Then, By [4, Proposition 3.17] and Remark 3.6, span{s α s * α : α ∈ L X } and Lc(X OT W , C) are *-isomorphic.Observe that Lc(X OT W , C) is a dense core subalgebra of C(X OT W ) by Lemma 3.12, since X OT W is a Stone space.Also, by Proposition 3.7 and Lemma 3.11, span{s α s * α : α ∈ L X } is a dense core subalgebra of span{s α s * α : α ∈ L X }.Thus, the first isomorphism follows from Proposition 3.10.
For the second isomorphism, we use the same idea and apply [4, Proposition 3.19] in place of [4, Proposition 3.17].
In [2] a normal labelled space (E, L, U) is associated with a subshift X as follows: the graph E is given by E 0 = X, E 1 = {(x, a, y) ∈ X × A × X : x = ay}, s(x, a, y) = x and r(x, a, y) = y.The labelling map is given by L(x, a, y) = a, and the accommodating family U is the Boolean algebra defined above.Then, the triple (E, L, U) is a normal labelled space [2, Lemma 5.5].
For A ∈ U, define L(AE 1 ) = {a ∈ A : Z a ∩ A = ∅} and let U reg = {A ∈ U : 0 < |L(AE 1 )| < ∞}.Then, A ∈ U reg if and only if Z a ∩ A = ∅ for finitely many a ∈ A. Note that, because E has no sinks, the definitions of L(AE 1 ) and U reg above coincide with the general definitions of these sets for labelled spaces (see [4]).In particular, if the alphabet is finite, then U reg = U.Also, since X = a∈A Z a , we have that if A ∈ U then A = a∈L(AE 1 ) Z a ∩ A.
This last equality implies that r(α) = r(X, α) = F α = C(α, ω).Following the same line of thought as in [4, Theorem 3.12], we obtain that the subshift C*-algebra may be realized as the C*-algebra of the labelled space (E, L, U).We state this as a theorem below.Theorem 3.15.Let X be a subshift and let (E, L, U) be the labelled space defined above.Then, O X ∼ = C * (E, L, U).
Using Theorem 3.15 and [2, Corollary 3.10] we obtain the following.Corollary 3.16 (Gauge-invariance uniqueness theorem).Let X be a subshift, B a C*algebra and Φ : O X → B a *-homomorphism.If Φ(p A ) = 0 for all A ∈ U \ {∅}, and there exists a strongly continuous action β : T → B such that Φ • γ z = β z • Φ for all z ∈ T, then Φ is injective.

3.2.
Non-unital C*-algebras of subshifts.In this section, we define a possibly nonunital C*-algebra O X associated with a subshift X.
Let B be the Boolean algebra of subsets of X generated by all C(α, β), for α, β ∈ L X not both simultaneously equal to ω.In particular, we do not require that X is a generator for the Boolean algebra B. In some cases B and U coincide [4, Example 4.16], and in others they do not [4,Example 4.11].
Definition 3.17.The subshift C*-algebra O X is the universal C*-algebra generated by projections {p A : A ∈ B} and partial isometries {s a : a ∈ A} subject to the relations:

Examples
In this section, we realize several known algebras as subshift algebras.
Graphs C*-algebras: Let E = (E 0 , E 1 , r, s) be a directed graph, and let C * (E) denote the graph C*-algebra of E. The one-side edge subshift associated with E is the set of all infinite paths, which is the subshift over the alphabet A = E 1 given by the family of forbidden words {ef ∈ A 2 : r(e) = s(f )}.
Proposition 4.1.Let E be a graph with no sinks and with no vertex that is simultaneously a source and an infinite emitter.Let X be the associated one-sided edge subshift of E.
Then, O X ∼ = C * (E).  is not isomorphic to A R (X) because the first is unital, whereas the latter is not.The same argument can be applied in the C*-algebraic setting.It is natural to compare the unital subshift algebra O X with C * (E).In this case, considering the elements q v := s * f s f , q w := 1 − q v , t en := s en for n ∈ N and t f := s f inside O X , we obtain a surjective *-homomorphism Φ : C * (E) → O X .Using the gauge-invariant uniqueness theorem for graph C*-algebras, it follows that Φ is injective, and hence, an isomorphism.
The isomorphism in Example 4.2 is not guaranteed in general.In Example 5.2, we construct a graph E with finitely many vertices and use K-theory to show that C * (E) is not isomorphic to O X .
The associated one-sided edge subshift X G of G is the set of all infinite paths.This is the subshift over the alphabet A = G 1 given by the family of forbidden words {ef ∈ A 2 : s(f ) / ∈ r(e)}.We show that a certain class of ultragraph C*algebras can be realized as subshift C*-algebras.Proposition 4.3.Let G be an ultragraph such that every vertex is regular, that is, Proof.By Remark 3.9, the ultragraph Leavitt path algebra L C (G) is a core subalgebra of C * (G).Remark 3.9 also implies that A C (X G ) is core subalgebra of O X G .Hence, [4, Proposition 4.8] and Proposition 3.10 imply that C * (G) ∼ = O X G .
Unital Exel-Laca algebras: Let I be a set of indices and A = (A ij ) i,j∈I a {0, 1}-matrix with no rows identically zero.We let X A = {(x n ) ∈ I N : A xnx n+1 = 1 for all n ∈ N}.The unital Exel-Laca algebra O A is the unital C*-algebra generated by partial isometries {S i } i∈I such that (EL1) S * i S i S * j S j = S * j S j S * i S i , for all i, j ∈ I; (EL2) S i S * i S j S * j = 0, whenever i = j; (EL3) S * i S i S j S * j = A ij S j S * j for all i, j ∈ I; (EL4) for all X, Y ⊆ I finite such that is zero for all but a finite number of j's, we have that, Proof.We show that the elements {s i } i∈I satisfy the relations defining O A .(EL1) and (EL2) follows from Proposition 3.3.For (EL3), we observe that for i, j ∈ I we have that In particular, if A(X, Y, j) is zero for all but a finite number of j's, then the above union is finite and hence

K-Theory
In [2, Theorem 4.4], Bates, Carlsen and Pask describe the K-theory of a labelled space C*-algebra.Since every unital C*-algebra of a subshift may be realized as a labelled space C*-algebra by Theorem 3.15, we can reformulate [2,Theorem 4.4] for C*-algebras of subshifts as follows.
Theorem 5.1.Let X be a one-sided subshift over an arbitrary alphabet A. Let (1 − Φ) : span Z {χ A : A ∈ U reg } → span Z {χ A : A ∈ U} be the linear map given by We present two applications of this result that illustrate the computation of K-theory for subshift C*-algebras.In the first example, we present a graph such its C*-algebra is not isomorphic to the corresponding edge subshift algebra.In the second example, we consider a subshift over an infinite alphabet that is not conjugate to any 1-step OTWsubshift, and thus does not fall under any of the examples of Section 4.
Example 5.2.Let E be the graph with three vertices, say v 1 , v 2 , v 3 , infinitely many edges from v 1 to v 3 , infinitely many edges from v 2 to v 3 , and a loop in v 3 , see the picture below.
We use K-theory to show that the graph C*-algebra C * (E) is not isomorphic to the subshift algebra O X , where X is the edge subshift.Using [2, Corollary 5.1] (see also [14,Theorem 3.1]) to compute the K-theory of C * (E), we obtain that K 0 (C * (E)) ∼ = Z 3 and K 1 (C * (E)) ∼ = Z.
We compute the K-theory of O X .Denote the edges from v 1 to v 3 by {a n } n∈N , from v 2 to v 3 by {b n } n∈N and the loop on v 3 by c.
It is easy to see that the elements in U reg are the finite subsets of X, and the elements in U are the finite or cofinite subsets of X.Then, an element in span Taking Φ as in Theorem 5.1, we have Therefore, x ∈ ker(1 − Φ) if, and only if, λ n = 0 and µ n = 0.It follows from Theorem 5.

Define the surjective map Ψ : span
Clearly, y ∈ ker Ψ if, and only if, ρ = 0 and η = − , we conclude that C * (E) and O X are not isomorphic.

Consider the subsets of
We leave it to the reader to verify that Taking Φ as in Theorem 5.1, we have that This shows that x ∈ ker(1 − Φ) if, and only if, ρ = 0 and µ j = λ j for all j ≥ 1.Therefore, by Theorem 5.1, The calculations above also show that Im(1 − Φ) is equal to Clearly, Ψ is surjective and ker Ψ = Im(1 − Φ).Therefore, by Theorem 5.1, 6.The dynamical structure of the C*-algebra of a subshift Let X be a subshift.Two groupoid models for the purely algebraic version of the subshift algebra, A R (X), are given in [4] (a transformation groupoid and a Deaconu-Renault groupoid).The same groupoids give groupoid C*-algebra realizations of the subshift C*-algebra O X .We focus, first, on the groupoid constructed in [4, Section 5.2], which arises as the transformation groupoid induced by the action of the free group F, generated by the alphabet A, on the Stone dual U of the Boolean algebra U, defined in Section 3.1.The topology on U has a basis given by the sets O A = {ξ ∈ U : A ∈ ξ}, where A ∈ U.This groupoid is defined as where V βα −1 = O C(α,β) , and ϕ αβ −1 (ξ) = {A ∈ U : r(B, β) ⊆ r(A, α) for some B ∈ ξ} for every ξ ∈ V βα −1 , for every α, β ∈ L X such that αβ −1 is in reduced form, and V t = ∅ if t = βα −1 for some α, β ∈ L X .
Note that for j ∈ N large enough, we have that δ 1 β j , . . ., δ l β j / ∈ L X .Taking the relative range with respect to β j and using (3.14), we then see that from where it follows that X does not satisfy condition (L).
(ii)⇒(i) Suppose that X does not satisfy condition (L).In this case there exist P ⊆ L X finite and γ ∈ L X such that F P = {γ ∞ }.Since F P ∈ U, we found a periodic point x such that {x} ∈ U.
In the case that the alphabet is countable, the free group F is also countable, so (iv)⇔(v) follows from [ Next, we characterize continuous orbit equivalence of the partial actions associated with a pair of subshifts that satisfy Condition (L), in terms of their associated groupoids and C*-algebras.Theorem 6.7.Let X and Y be subshifts over countable alphabets satisfying condition (L).Let ϕ X and ϕ Y be the corresponding partial actions of F X and F Y on U X and U Y , respectively.Then, the following are equivalent: (i) ϕ X and ϕ Y are continuously orbit equivalent, (ii) the transformation groupoids F X ⋉ U X and F Y ⋉ U Y are isomorphic as topological groupoids, (iii) there exists an isomorphism Φ : Moreover, (ii) implies (i) holds even if condition (L) is not satisfied.
We now consider the problem of finding invariants for topological conjugacy for OTWsubshifts and how it relates to the isomorphism of subshift C*-algebras.We do this by adding some equivalent conditions to [4,Theorem 7.6].For this we use the Deaconu-Renault groupoid G( U, σ) associated with X ( [4]), where U is the Stone dual of U as before, and σ has domain dom( σ) := a∈A V a = a∈A O Za and is defined by σ(ξ) := ϕ a −1 (ξ), for ξ ∈ V a .See [4, Section 6] for more details.By Theorem 6.1 and [4, Theorem 6.4], we have that O X ∼ = C * (G( U, σ)).
We recall the definition of a conjugacy between OTW-subshifts.Definition 6.8.Let X OT W and Y OT W be OTW-subshifts over alphabets A 1 and A 2 , respectively.A map h : X OT W → Y OT W is a conjugacy if it is a homeomorphism, commutes with the shifts and is length-preserving.
In Theorem 6.9 we characterize conjugacy of OTW-subshifts.For that, the idea is to invoke [1,Theorem 3.1].However, the shift map of an OTW-subshift is not a local homomorphism in general.For instance, a subshift over a finite alphabet is a local homeomorphism if and only if it is of finite type [13, Proposition 2.5].Nevertheless, using [4, Proposition 7.5 and Theorem 7.6] we can rephrase the problem in terms of a conjugacy between the corresponding Deaconu-Renault system of their subshifts.
Before the next theorem, we briefly adapt a definition used in [1, Theorem 3.1] to our needs.Given a subshift X and f ∈ C( U, Z), we define the weighted action γ X,f : T O X on the generators by γ X,f z (p A ) = p A and γ X,f z (s a ) = z f s a for all A ∈ U, a ∈ A and z ∈ T, where z f is to be interpreted as an element of C( U) seen as a subalgebra of O X .More specifically, z f : U → C is defined by z f (ξ) = z f (ξ) for each z ∈ T and ξ ∈ U. Theorem 6.9.Let X and Y be subshifts over countable alphabets and h : X OT W → Y OT W a homeomorphism.The following are equivalent: (i) h is a conjugacy.(ii) There exists a homeomorphism h : such that |β| = |α| and h(Z α ) = Z β .Then Z(α) = Z(β).Because h OT W is a bijection, it preserves set operations, so that if F ⊆ A 1 is finite, then h(Z(α, F )) = Z(β, G) for some G ⊆ A 2 finite.Hence h is open, since sets of the form Z(α, F ) generate the topology on X OT W 1 .For the converse statement, note that a conjugacy of OTW subshifts is, by definition, length preserving.Therefore, since (X OT W ) inf = X for any subshift X, a conjugacy of OTW subshifts restricts to a conjugacy of subshifts with the product topology.Remark 6.12.Recall that for OTW-subshifts the definition of a conjugacy includes the length preserving condition, in addition to the usual requirement that the map is a shift commuting homeomorphism.The difference between the definitions of conjugacy for subshifts with the metric d given in (6.10) and OTW-subshifts may also be understood from the shift commuting condition, which is equivalent to the map being a sliding block code for subshifts with the metric given in (6.10), and a generalised sliding block code for OTW-subshifts, see [20,19,27].Theorem 6.13.If X 1 and X 2 are isometrically conjugate subshifts over countable alphabets, then there exists a diagonal-preserving gauge-invariant *-isomorphism between the C*-algebras O X 1 and O X 2 .
Proof.It follows from Theorems 6.9 and 6.11.
Remark 6.14.The K-theory of the corresponding C*-algebras of subshifts can be used as an invariant for isometric conjugacy.For instance, the subshifts considered in Examples 5.2 and 5.3 are not isometrically conjugate.

Remark 3 . 18 .
Similar to[4, Proposition 4.8], the C*-algebra O X coincides with O X when it is unital, and its unitization coincides with O X when it is not unital.Remark 3.19.Propositions 3.3, 3.4, 3.7, Theorem 3.15, Corollaries 3.5 and 3.16 hold for non-unital C*-algebras of subshifts.The proofs are similar and we omit them.

Proof. By Remark 3 . 9 ,
the Leavitt path algebra L C (E) is a core subalgebra of C * (E), and similarly the algebraic subshift algebra A C (X) (see [4, Definition 4.1]) is a core subalgebra of O X .By [4, Proposition 4.10], we have that L C (E) ∼ = A C (X).Notice that (s e ) * = s * e and (p A ) * = p A defines an involution on A C (X), and then the isomorphism L C (E) ∼ = A C (X) of [4, Proposition 4.10] is *-preserving.Hence, Proposition 3.10 implies that O X ∼ = C * (E).

Example 4 . 2 .
Let E be the graph such that E 0 = {v, w},E 1 = {e n } n∈N ∪ {f }, s(e n ) = v and r(e n ) = w = s(f ) = r(f ) for all n ∈ N. It was shown in [4, Example 4.11] that L R (E)

( 1 Proposition 4 . 4 .
− S * y S y ) = j∈I A(X, Y, j)S j S * j .In the context above, O A ∼ = O X A .
(X,Y,j)=1 Z j = j:A(X,Y,j)=1s j s * j = j∈I A(X, Y, j)s j s * j .By the universal property of O A , there exists a * -homomorphism ϕ :O A → O X A ,which is surjective by Proposition 3.3.Injectivity follows from the gauge-invariant uniqueness theorem [26, Theorem 2.7].Remark 4.5.An analogous result for the algebras that are not necessarily unital can be deduced from Proposition 4.3 and [29, Theorem 4.5].