Let G be a compact quantum group. We show that given a G-equivariant $\textrm {C}^*$-correspondence E, the Pimsner algebra $\mathcal {O}_E$ can be naturally made into a G-$\textrm {C}^*$-algebra. We also provide sufficient conditions under which it is guaranteed that a G-action on the Pimsner algebra $\mathcal {O}_E$ arises in this way, in a suitable precise sense. When G is of Kac type, a KMS state on the Pimsner algebra, arising from a quasi-free dynamics, is G-equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is G-equivariant, under a natural condition. We apply these results to the situation when the $\textrm {C}^*$-correspondence is obtained from a finite, directed graph and draw various conclusions on the quantum automorphism groups of such graphs, both in the sense of Banica and Bichon.

We investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underline {m}))$, introduced by Brzeziński and Szymański as graph $C^*$-algebras. We give a new description of $C(L_q^{2n+1}(r;\underline {m}))$ as graph $C^*$-algebras amending an error in the original paper by Brzeziński and Szymański. Furthermore, for $n\leq 3$, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz, and Sørensen.

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta )$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta )$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ .

We address the classification problem for graph ${{C}^{*}}$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition $\left( K \right)$, so that the graph ${{C}^{*}}$-algebras may come with uncountably many ideals.

We find that in this generality, stable isomorphism of graph ${{C}^{*}}$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C*-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph ${{C}^{*}}$-algebras are in fact classifiable by $K$-theory, providing, in particular, complete classification when the ${{C}^{*}}$- algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs.

Our results are applied to discuss the classification problem for the quantumlens spaces defined by Hong and Szymański, and to complete the classification of graph ${{C}^{*}}$-algebras associated with all simple graphs with four vertices or less.

We show that filtered K-theory is equivalent to a substantially smaller invariant for all real-rank-zero C*-algebras with certain primitive ideal spaces—including the infinitely many so-called accordion spaces for which filtered K-theory is known to be a complete invariant. As a consequence, we give a characterization of purely infinite Cuntz–Krieger algebras whose primitive ideal space is an accordion space.

Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extension

in terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph C*-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences constitute complete invariants.

Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.

For a locally finite directed graph E, it is known that the graph C*-algebra C*(E) has real rank zero if and only if the graph E satisfies the loop condition (K). In this paper we extend this to an arbitrary directed graph case using the desingularization of a graph due to Drinen and Tomforde