Quantum Cuntz-Krieger algebras
Michael Brannan, Kari Eifler, Christian Voigt, Moritz Weber
Abstract
Motivated by the theory of Cuntz-Krieger algebras we define and study <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras associated to directed quantum graphs. For classical graphs the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper K"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">KK</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K upper K"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">KK</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.