Approximation properties for free orthogonal and free unitary quantum groups

TL;DR

This paper proves that the reduced von Neumann and C*-algebras associated with free orthogonal and free unitary quantum groups possess the Haagerup approximation property and the metric approximation property, respectively, advancing understanding of their operator algebraic structure.

Contribution

It establishes the Haagerup approximation property for these quantum groups' von Neumann algebras and the metric approximation property for their C*-algebras, using new and existing inequalities.

Findings

Von Neumann algebras of free orthogonal and free unitary quantum groups have the Haagerup property.

Reduced C*-algebras of these quantum groups have the metric approximation property.

Results connect quantum group properties with operator algebra approximation properties.

Abstract

We show that the reduced von Neumann algebras of the free orthogonal and free unitary quantum groups have the Haagerup approximation property. Using this result and a Haagerup-type inequality for these quantum groups due to Vergnioux (J. Operator Theory 57, 303-324 (2007)), we also show that the reduced C$^\ast$-algebras of the free orthogonal and free unitary quantum groups have the metric approximation property.