Reduced operator algebras of trace-preserving quantum automorphism groups

By Michael Brannan. Documenta Mathematica, 2012

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Let B be a finite dimensional C ∗ -algebra equipped with its canonical trace induced by the regular representation of B on it- self. In this paper, we study various properties of the trace-preserving quantum automorphism group G of B. We prove that the discrete dual quantum group b G has the property of rapid decay, the reduced von Neumann algebra L ∞ (G) has the Haagerup property and is solid, and that L ∞ (G) is (in most cases) a prime type II1-factor. As applica- tions of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced C ∗ -algebra Cr(G), and the existence of a multiplier-bounded approx- imate identity for the convolution algebra L 1 (G).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f20cd79e9c04bb41bdf677d9d1b58d7ae925eb84

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Citations in the corpus

CCAP for Universal Discrete Quantum Groups (2013, background): “Note that this was known as for the quantum groups above which are monoidally equivalent to a quantum group of Kac-type, by [8], [9] and [19].”
ON THE PARTITION APPROACH TO SCHUR-WEYL DUALITY AND FREE QUANTUM GROUPS (2014, background): “However, the conclusion is known to hold also in most of these cases by [42], [15] and [29].”
The Haagerup property for locally compact quantum groups (2013, background): “The papers [14, 15] study the Haagerup approximation property for L(1).”