Reiter's properties (P_1) and (P_2) for locally compact quantum groups

Abstract

A locally compact group $G$ is amenable if and only if it has Reiter's property $(P_p)$ for $p=1$ or, equivalently, all $p\in [1,\infty )$, i.e., there is a net $(m_{\alpha }{)}_{\alpha }$ of non-negative norm one functions in $L^p(G)$ such that ${\lim }_{\alpha }{\sup }_{x\in K}\Vert L_{x^{-1}}{m}_{\alpha }-m_{\alpha }{\Vert }_p=0$ for each compact subset $K\subset G$ ($L_{x^{-1}}{m}_{\alpha }$ stands for the left translate of $m_{\alpha }$ by $x^{-1}$). We extend the definitions of properties $(P_1)$ and $(P_2)$ from locally compact groups to locally compact quantum groups in the sense of J. Kustermans and S. Vaes. We show that a locally compact quantum group has $(P_1)$ if and only if it is amenable and that it has $(P_2)$ if and only if its dual quantum group is co-amenable. As a consequence, $(P_2)$ implies $(P_1)$.

23 pages; some rewriting, references added