On the zeroth L^2-homology of a quantum group

TL;DR

This paper investigates the zeroth L^2-homology of compact quantum groups, establishing conditions under which the zeroth L^2-Betti number vanishes or is non-trivial, linking it to coamenability and algebraic finiteness.

Contribution

It proves that the zeroth L^2-Betti number vanishes unless the C*-algebra is finite dimensional and characterizes non-trivial zeroth L^2-homology by coamenability.

Findings

Zeroth L^2-Betti number vanishes unless the C*-algebra is finite dimensional.

Zeroth L^2-homology is non-trivial exactly when the quantum group is coamenable.

Provides a characterization of the zeroth L^2-homology in terms of algebraic properties.

Abstract

We prove that the zeroth L^2-Betti number of a compact quantum group vanishes unless the underlying C*-algebra is finite dimensional and that the zeroth L^2-homology itself is non-trivial exactly when the quantum group is coamenable.