L^2-Betti numbers of coamenable quantum groups

Details L^2-Betti numbers of coamenable quantum groups L^2-Betti numbers of coamenable quantum groups

2007-04-12

arxiv.org/abs/0704.1582v4

Mathematics

Operator Algebras

Mistake in the proof of Theorem 6.1 is corrected. To appear in Munster Journal of Mathematics. 42 pages

Scientific paper

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.

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