Classification of irreversible and reversible Pimsner operator algebras

Abstract

Since 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.

C*-dynamical invariants and Toeplitz algebras of graphs

“…This generalises a reconstruction theorem from [7] to the case of infinite graphs. This generalisation has also been obtained in [21] using completely different techniques.…”

“…We now show that our results can be used to obtain a generalisation of one of the main results from [7] to the case of infinite graphs. This generalisation can also be proved using entirely different techniques without mention of KMS or ground states, see [21,Section 6], but we emphasise that the approach using ground states allows one to recover the graph explicitly in terms of equilibrium states of the C*dynamical system. Given a countable directed graph E, we follow [7] and consider the vertex algebra…”

Toeplitz quotient $C^*$-algebras and ratio limits for random walks

“…Another instance of this is in graph theory and symbolic dynamics, where invariants of C*-algebras studied by Cuntz and Krieger coincide with invariants coming from subshifts of finite type [13,12]. After contributions and improvements by too many authors to list here, these works led to C*-algebraic interpretations of equivalence relations occurring naturally in symbolic dynamics [40,6], and provided a rich class of examples for classification of operator algebras [22,17]. A concrete way of constructing and studying C*-algebras of directed graphs is by realizing them as unique T-equivariant quotients of the Toeplitz C*-algebras of the graph.…”

Amplified graph C*-algebras II: Reconstruction

“…The geometric classification (that is, classification by the underlying graph modulo the equivalence relation generated by a list of allowable graph moves) of the C * -algebras of finite-vertex amplified graph C * -algebras was completed in [12], and was an important precursor to the eventual geometric classification of all finite graph C * -algebras [13]. But there has been increasing recent interest in understanding isomorphisms of graph C * -algebras that preserve additional structure: for example the canonical gauge action of the circle; or the canonical diagonal subalgebra isomorphic to the algebra of continuous functions vanishing at infinity on the infinite path space of the graph; or the smaller coefficient algebra generated by the vertex projections; or some combination of these (see, for example, [5][6][7][8][9][10]19]).…”