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Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$$^\ast $$-algebra $$B$$, which includes the quantum permutation group $$S_N^+$$, where $$N = \dim B$$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $$\Gamma $$ lead to isomorphic $$\ast $$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $$1$$-boundedness between the various algebras associated with these quantum groups.
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Quantum Cuntz-Krieger algebras
Motivated by the theory of Cuntz-Krieger algebras we define and study C ∗ C^\ast -algebras associated to directed quantum graphs. For classical graphs the C ∗ 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 K K 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 ∗ C^\ast -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of K K 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.
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- Award ID(s):
- 2000331
- PAR ID:
- 10413969
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society, Series B
- Volume:
- 9
- Issue:
- 26
- ISSN:
- 2330-0000
- Page Range / eLocation ID:
- 782 to 826
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/btran/88
