Diagonal-preserving Isomorphism Research Articles

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 homeomorphism 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.

Reconstruction of groupoids and C⁎-rigidity of dynamical systems

We show how to construct a graded locally compact Hausdorff étale groupoid from a C⁎-algebra carrying a coaction of a discrete group, together with a suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid. When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees with Renault's Weyl groupoid. We prove that if G is a second-countable locally compact étale groupoid carrying a grading of a discrete group, and if the interior of the trivially graded isotropy is abelian and torsion free, then the extended Weyl groupoid of its reduced C⁎-algebra is isomorphic as a graded groupoid to G. In particular, two such groupoids are isomorphic as graded groupoids if and only if there is an equivariant diagonal-preserving isomorphism of their reduced C⁎-algebras. We introduce graded equivalence of groupoids, and establish that two graded groupoids in which the trivially graded isotropy has torsion-free abelian interior are graded equivalent if and only if there is an equivariant diagonal-preserving Morita equivalence between their reduced C⁎-algebras. We use these results to establish rigidity results for a number of classes of dynamical systems, including all actions of the natural numbers by local homeomorphisms of locally compact Hausdorff spaces.

TENSOR PRODUCTS OF STEINBERG ALGEBRAS

AbstractWe prove that$A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$if$G$and$H$are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between$L_{2,R}\otimes L_{3,R}$and$L_{2,R}\otimes L_{2,R}$. In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every$\ast$-isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that$L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$(as$\ast$-rings).

Diagonal-preserving isomorphisms of étale groupoid algebras

Work of Jean Renault shows that, for topologically principal étale groupoids, a diagonal-preserving isomorphism of reduced C⁎-algebras yields an isomorphism of groupoids. Several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses. In this paper, we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids. Our approach has the additional feature that we only need to impose conditions on one of the two groupoids. Applications are given to Leavitt path algebras.

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph C⁎-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.

Diagonal-preserving gauge-invariant isomorphisms of graph C⁎-algebras

We study graph C⁎-algebras equipped with generalised gauge actions, and characterise in terms of groupoids and groupoid cocycles when two graph C⁎-algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the generalised gauge actions. We apply this characterisation to show that two Cuntz–Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding one-sided subshifts are eventually conjugate, and that the stabilisation of two Cuntz–Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding two-sided subshifts are conjugate.

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their$C^{\ast }$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs$E$we construct a groupoid${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$from the graph algebra$C^{\ast }(E)$and its diagonal subalgebra${\mathcal{D}}(E)$which generalises Renault’s Weyl groupoid construction applied to$(C^{\ast }(E),{\mathcal{D}}(E))$. We show that${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$recovers the graph groupoid${\mathcal{G}}_{E}$without the assumption that every cycle in$E$has an exit, which is required to apply Renault’s results to$(C^{\ast }(E),{\mathcal{D}}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.