In this paper, we establish logical equivalence between statements involving center dot the Cuntz C*-algebra O-infinity with its canonical diagonal; center dot graph C*-algebras with their canonical diagonals; center dot Leavitt path algebras over general fields with their canonical diagonals; center dot Leavitt path algebras over Z; center dot topological full groups; center dot groupoids; and center dot the automorphism x -> -x on certain K0-and homology groups equal to Z. Deciding whether these equivalent statements are true or false is of importance in studies of geometric classification of diagonal-preserving isomorphism between graph C*-algebras and Leavitt path algebras, mirroring a similar hindrance studied by Cuntz more than 40 years ago.

The quantum lens spaces form a natural and well-studied class of noncommutative spaces which can be subjected to classification using algebraic invariants by drawing on the fully developed classification theory of unital graph $C^*$-algebras. We introduce the problem of deciding when two quantum lens spaces are equivariantly isomorphic, and solve it in certain basic cases. As opposed to classification up to isomorphism, we can not appeal to a complete general classification theory in the equivariant case, but by combining existing partial results with an ad hoc analysis we can solve the case with dimension 3 completely, and the case with dimension 5 in the prime case. Our results can be formulated directly in terms of the parameters defining the quantum lens spaces, and here occasionally take on a rather complicated form which convinces us that there is a deep underlying explanation for our findings. We complement the fully established partial results with computer experiments that may indicate the way forward.

In this paper, we construct a class of ASH algebras of real rank zero and stable rank one which is not K-pure. Then we show the following: (i) There exists a real rank zero inductive limit of 1-dimensional noncommutative CW complexes which is not an Aℋ𝒟 algebra, when K_1 is torsion free or has bounded torsion. (ii) Total K-theory is not a complete invariant for ASH algebras of real rank zero. (iii) There are obstructions both in the total K-theory of ideals and quotients in the classification of C^*-algebras of real rank zero and stable rank one.

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

This workshop aimed to strengthen ties and foster collaborations between different communities working on combinatorial * -algebras, including C^{*} -and pure algebraists.

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

Let E be a countable directed graph that is amplified in the sense that whenever there is an edge from v to w, there are infinitely many edges from v to w. We show that E can be recovered from C^*(E) together with its canonical gauge-action, and also from L_K(E) together with its canonical grading.

We show that certain extensions of classifiable C*-algebra are strongly classified by the associated six-term exact sequence in K-theory together with the positive cone of K_{0}-groups of the ideal and quotient. We apply our result to give a complete classification of graph C*-algebras with exactly one ideal.

We geometrically describe the relation induced on a set of graphs by isomorphism of their associated graph C*-algebras as the smallest equivalence relation generated by five types of moves. The graphs studied have finitely many vertices and finitely or countably infinitely many edges, corresponding to unital and separable C*-algebras.

AbstractSince their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and$C^{*}$-algebras with additional$C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from$C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

In this paper, a G G -shift of finite type ( G G -SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G G . We reduce the classification of G G -SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G G . For a special case of two irreducible components with G = Z 2 G=\mathbb {Z}_2 , we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G G -SFT applications, including a new connection to involutions of cellular automata.

We give a complete $K$-theoretical description of when an extension of two simple graph $C^{*}$-algebras is again a graph $C^{*}$-algebra.

A group may be considered $C^*$-stable if almost representations of the group in a $C^*$-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are $C^*$-stable or only stable with respect to some subclass of $C^*$-algebras, e.g. finite dimensional $C^*$-algebras. We provide criteria and invariants for stability of groups and this allows us to completely determine stability/non-stability of crystallographic groups, finitely generated torsion-free step-2 nilpotent groups, surface groups, virtually free groups and certain Baumslag-Solitar groups.

We formalize eight different notions of isomorphism among (unital) graph C*-algebras, and initiate the study of which of these notions may be described geometrically as generated by moves. We propose a list of seven types of moves that we conjecture has the property that the collection of moves respecting one of six notions of isomorphism indeed generate that notion, in the sense that two graphs are equivalent in that sense if and only if one may transform one into another using only these kinds of moves. We carefully establish invariance properties of each move on our list, and prove a collection of generation results supporting our conjecture with an emphasis on the gauge simple case. In two of the six cases, we may prove the conjecture in full generality, and in two we can show it for all graphs defining gauge simple C*-algebras. In the two remaining cases we can show the conjecture for all graphs defining gauge simple C*-algebras provided that they are either finite or have at most one vertex allowing a path back to itself.

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.

We introduce filtered algebraic $K$-theory of a ring $R$ relative to a sublattice of ideals. This is done in such a way that filtered algebraic $K$-theory of a Leavitt path algebra relative to the graded ideals is parallel to the gauge invariant filtered $K$-theory for graph $C^*$-algebras. We apply this to verify the Abrams-Tomforde conjecture for a large class of finite graphs.

A group may be considered C∗-stable if almost representations of the group in a C∗-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are C∗-stable or only stable with respect to some subclass of C∗-algebras, e.g. finite dimensional C∗-algebras. We provide criteria and invariants for stability of groups and this allows us to completely determine stability/non-stability of crystallographic groups, surface groups, virtually free groups, and certain Baumslag-Solitar groups. We also show that among the non-trivial finitely generated torsion-free 2-step nilpotent groups the only C∗-stable group is Z.

We characterize when there exists a diagonal-preserving *-isomorphism between two graph C*-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of 'orbit equivalence' between the boundary path spaces of the directed graphs E and F and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving *-isomorphism between the graph C*-algebras C* (E) and C* (F).

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KK epsilon(e, e'), with one specified ideal, into Hom(Lambda) (K-epsilon(e), K-epsilon(e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K-1-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results of Kirchberg, we classify automorphisms of the stabilized purely infinite Cuntz-Krieger algebras with exactly one non-trivial ideal modulo asymptotically unitary equivalence. We also get a classification result modulo approximately unitary equivalence. The results in this paper also apply to certain graph algebras.