Quantum symmetry of graph $C^{\ast }$-algebras at\cr critical inverse temperature

Abstract

We give a notion of quantum automorphism group of graph C * -algebra without sink at critical inverse temperature. This is defined to be the universal object of a category of CQG's having a linear action in the sense of [11] and preserving the KMS state at critical inverse temperature. We show that this category for a certain KMS state at critical inverse temperature coincides with the category introduced in [11] for a class of graphs. We also introduce an orthogonal filtration on Cuntz algebra with respect to the unique KMS state and show that the category of CQG's preserving the orthogonal filtration coincides with the category introduced in this paper.

Equivariant -correspondences and compact quantum group actions on Pimsner algebras

“…In that case, the graph C * -algebra coincides with the Pimsner algebra, allowing us to apply the above results. In particular, we recover the results obtained in [JM21a,JM21b,SW18]. This also enables us to gain a more concrete understanding of most of the results concerning the interaction between quantum symmetries of graphs and its graph C * -algebras.…”

“…Similar richness of quantum symmetries has been observed in other contexts as well. For example, compact quantum groups have been found to preserve fewer KMS states on certain graph C * -algebras as opposed to compact group actions [JM21a]. As a necessarily incomplete list of references for the reader interested in this direction, we mention [Gab14,GW16,Kat17,Pao97].…”

Invariance of KMS states on graph C*-algebras under classical and quantum symmetry

“…For , we have . So By the same argument as given in the proof of the Theorem 3.5 of [8], for , and hence the last expression equals to Observe that any state restricts to a state on so that by Lemma 2.32, for , . Using this, the last summation reduces to Using the same arguments used in the proof of Theorem 3.13 in [7] repeatedly, it can be shown that the last summation actually equals to .…”

“…In light of the Theorem 3.1, for strongly connected graphs, we can relax the condition on the graph in [8]. In that paper regularity of the underlying graph was assumed to ensure that belongs to the category (see [8] for notation) for the unique KMS state on a strongly connected graph . Now we have for a strongly connected graph (regular or not) with its unique KMS state , the category contains .…”

“…For the converse, given for all , we need to show that for all . The proof is similar to that of Theorem 3.5 of [8]. It is easy to see that for , .…”