L^2-Betti numbers of coamenable quantum groups
Abstract
Description
We prove that a compact quantum group is coamenable if and only if its corepresentation ring is amenable. We further propose a Foelner condition for compact quantum groups and prove it to be equivalent to coamenability. Using this Foelner condition, we prove that for a coamenable compact quantum group with tracial Haar state, the enveloping von Neumann algebra is dimension flat over the Hopf algebra of matrix coefficients. This generalizes a theorem of Lueck from the group case to the quantum group case, and provides examples of compact quantum groups with vanishing L^2-Betti numbers.
Mistake in the proof of Theorem 6.1 is corrected. To appear in Munster Journal of Mathematics. 42 pages
Mistake in the proof of Theorem 6.1 is corrected. To appear in Munster Journal of Mathematics. 42 pages