𝐶*-algebras, groupoids and covers of shift spaces

Abstract

To every one-sided shift space X we associate a cover X, a groupoid G X and a C *-algebra O X. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between X and Y in terms of isomorphism of G X and G Y , and diagonal-preserving *-isomorphism of O X and O Y. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces Λ X and Λ Y in terms of isomorphism of the stabilized groupoids G X × R and G Y × R, and diagonal-preserving *-isomorphism of the stabilized C *-algebras O X ⊗ K and O Y ⊗ K. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of X from the pair (O X , C(X)), and the flow equivalence class of Λ X from the pair (O X ⊗ K, C(X) ⊗ c 0). In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces. Contents 154 7. Two-sided conjugacy 166 8. Flow equivalence 170 References 182

Conjugacy of local homeomorphisms via groupoids and C*-algebras

“…It follows from [BC20b, Lemma 2.3] that ( X, σ X ) is a Deaconu-Renault system. If two one-sided shift spaces (X, σ X ) and (Y, σ Y ) are conjugate, then the corresponding Deaconu-Renault systems ( X, σ X ) and ( Y, σ Y ) are conjugate (see [BC20b,Lemma 4.1])l however, there are examples of non-conjugate onesided shift spaces (X, σ X ) and (Y, σ Y ) for which ( X, σ X ) and ( Y, σ Y ) are conjugate (for example, consider a one-sided strictly sofic shift (X, σ X ) and the one-sided edge shift (Y, σ Y ) of its left Krieger cover, cf. [Ki98, Exercise 6.1.9]).…”

“…A system consisting of a locally compact Hausdorff space X together with a local homeomorphism σ X between open subsets of X is called a Deaconu-Renault system (see, for example, [D95, Re00, CRST, ABS]). Examples of Deaconu-Renault systems include selfcovering maps ([D95, EV06]), one-sided shifts of finite type [Wi73,LM95,Ki98], the boundary-path space of a directed graph together with the shift map [We14,BCW17], and more generally, the boundary-path space of a topological graph together with the shift map [KL17], the one-sided edge shift space of an ultragraph together with the restriction of the shift map to points with nonzero length [GR19], the full one-sided shift over an infinite alphabet together with the restriction of the shift map to points with nonzero length [OMW14], the cover of a one-sided shift space constructed in [BC20b], and more generally, the canonical local homeomorphism extension of a locally injective map constructed in [Th11].…”

“…A C*-algebra is naturally associated to a Deaconu-Renault system via a groupoid construction (see, for example, [D95, Re00, CRST]), and the class of such C*-algebras includes crossed products by actions of Z on locally compact Hausdorff spaces, Cuntz-Krieger algebras [CK80], graph C*-algebras [Ra05], and, via Katsura's topological graphs [Ka04], all Kirchberg algebras (i.e. all purely infinite, simple, nuclear, separable C*-algebras) satisfying the universal coefficient theorem (see [Ka08]), C*-algebras associated with one-sided shift spaces [BC20b], and C*-algebras of locally injective surjective maps [Th11].…”

Conjugacy of local homeomorphisms via groupoids and C*-algebras

“…A system consisting of a locally compact Hausdorff space together with a local homeomorphism between open subsets of is called a Deaconu–Renault system (see, for example, [ABS, CRST, D95, Re00]). Examples of Deaconu–Renault systems include self-covering maps [D95, EV06], one-sided shifts of finite type [Ki98, LM95, Wi73], the boundary-path space of a directed graph together with the shift map [BCW17, We14], and, more generally, the boundary-path space of a topological graph together with the shift map [KL17], the one-sided edge shift space of an ultragraph together with the restriction of the shift map to points with non-zero length [GR19], the full one-sided shift over an infinite alphabet together with the restriction of the shift map to points with non-zero length [OMW14], the cover of a one-sided shift space constructed in [BC20b], and, more generally, the canonical local homeomorphism extension of a locally injective map constructed in [Th11].…”

“…A C*-algebra is naturally associated to a Deaconu–Renault system via a groupoid construction (see, for example, [CRST, D95, Re00]), and the class of such C*-algebras includes crossed products by actions of on locally compact Hausdorff spaces, Cuntz–Krieger algebras [CK80], graph C*-algebras [Ra05], and, via Katsura’s topological graphs [Ka04], all Kirchberg algebras (i.e. all purely infinite, simple, nuclear, separable C*-algebras) satisfying the universal coefficient theorem (see [Ka08]), C*-algebras associated with one-sided shift spaces [BC20b], and C*-algebras of locally injective surjective maps [Th11].…”