Quantum Cuntz-Krieger algebras

Motivated by the theory of Cuntz-Krieger algebras we define and study $C^\ast$-algebras associated to directed quantum graphs. For classical graphs the $C^\ast$-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to $KK$-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these $C^\ast$-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of $KK$-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.