We introduce the notion of orbit equivalence directed graphs, following Matsumoto's continuous for topological Markov shifts. show that two graphs in which every cycle has an exit are equivalent if and only there is a diagonal-preserving isomorphism between their $C^*$-algebras. it necessary to assume forward implication, but reverse implication holds arbitrary graphs. As part our analysis $E$ we construct groupoid $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ from graph algebra $C^*(E)$ its...

Motivated by Williams' problem of measuring novel differences between shift equivalence (SE) and strong (SSE), we introduce three relations that provide new ways to obstruct SSE while merely assuming SE.Our arise from studying graph C*-algebras, where a variety intermediary naturally arise.As consequence realize goal sought after Muhly, Pask Tomforde, measure delicate difference SE in terms Pimsner dilations for C*-correspondences adjacency matrices, use this distinction refute proof previous paper.

We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only they stably isomorphic in an appropriate sense, relate this to Matui's notion of Kakutani equivalence. use result show diagonal-preserving stable isomorphisms graph C*-algebras or Leavitt path algebras give rise the associated stabilised graphs. deduce $L_Z(E_2)$ $L_Z(E_{2-})$ not *-isomorphic.

Let X be a product system over quasi-lattice ordered group. Under mild hypotheses, we associate to C*-algebra which is co-universal for injective Nica covariant Toeplitz representations of preserve the gauge coaction. appropriate amenability criteria, this coincides with Cuntz-Nica-Pimsner algebra introduced by Sims and Yeend. We prove two key uniqueness theorems, indicate how use our theorems realise number reduced crossed products as instances algebras. In each case, it an easy corollary...

We point out incorrect lemmas in some papers regarding the $C^*$-algebras associated with subshifts written by second named author. To recover and affected main results, we will describe an alternative construction of subshifts. The resulting are generally different from originally constructed they fit mentioned including results. simplicity conditions K-theory formulae for described. also introduce a condition called $(*)$ such that under this new original canonically isomorphic to each...

We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. reconstruct (graded) and use this to characterise when there is a diagonal-preserving isomorphism between two algebras. apply characterisation of directed graphs in order isomorphisms Leavitt path graph $C^*$-algebras.

By using C*-correspondences and Cuntz-Pimsner algebras, we associate to every subshift (also called a shift space) $X$ C*-algebra $O_X$, which is generalization of the Cuntz-Krieger algebras. We show that $O_X$ universal generated by partial isometries satisfying relations given $X$. also one-sided conjugacy invariant

In this paper we give a formula for the $K$ -theory of $C^{\ast }$ -algebra weakly left-resolving labelled space. This is done by realizing space as Cuntz–Pimsner algebra -correspondence. As corollary, obtain gauge-invariant uniqueness theorem any order to achieve this, must modify definition We also establish strong connections between various classes -algebras that are associated with shift spaces and graph algebras. Hence, computing algebra, providing common framework algebras, ultragraph...

Abstract We prove a sandwiching lemma for inner‐exact locally compact Hausdorff étale groupoids. Our says that every ideal of the reduced ‐algebra such groupoid is sandwiched between ideals associated to two uniquely defined open invariant subsets unit space. obtain bijection ‐algebra, and triples consisting nested sets an in subquotient they determine has trivial intersection with diagonal subalgebra full support. then introduce generalisation groupoids Ara Lolk's relative strong...

Let $C^*(E)$ be the graph $C^*$-algebra associated to a E and let J gauge invariant ideal in $C^*(E)$. We compute cyclic six-term exact sequence $K$-theory of extension terms adjacency matrix $E$. The ordered is complete stable isomorphism for several classes $C^*$-algebras, instance those containing unique proper nontrivial ideal. Further, many other cases, infinite collections such sequences comprise invariants. Our results allow explicit computation invariant, giving an kernels cokernels...

Abstract A one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ . On other hand, groupoid ${\mathcal{G}}_{A}$ together homomorphism $\unicode[STIX]{x1D716}_{A}$ We show that each these two sets data characterizes conjugacy class $\mathsf{X}_{A}$ This strengthens result Cuntz...

Abstract We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with partially defined local homeomorphism. Important examples such include self-covering maps, one-sided shifts finite type and, more generally, the boundary-path spaces directed and topological graphs. characterize conjugacy these in terms isomorphisms their associated groupoids C*-algebras. This significantly generalizes recent work Matsumoto second- third-named authors.