The Baum-Connes conjecture for free orthogonal quantum groups

TL;DR

This paper proves an analogue of the Baum-Connes conjecture for free orthogonal quantum groups, showing they have a gamma element equal to one, which implies K-amenability and explicit K-theory computations.

Contribution

It establishes the Baum-Connes conjecture for free orthogonal quantum groups using triangulated categories and monoidal equivalence, providing explicit K-theory results.

Findings

Free orthogonal quantum groups have a gamma element equal to 1.

These quantum groups are K-amenable.

Reduced C*-algebras of these groups have no nontrivial idempotents in the unimodular case.

Abstract

We prove an analogue of the Baum-Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a $ \gamma $-element and that $ \gamma = 1 $. It follows that free orthogonal quantum groups are $ K $-amenable. We compute explicitly their $ K $-theory and deduce in the unimodular case that the corresponding reduced $ C^* $-algebras do not contain nontrivial idempotents. Our approach is based on the reformulation of the Baum-Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group $ SU_q(2) $. The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle\'s sphere.