# Reiter's properties (P 1) and (P 2) for locally compact quantum groups - Mathematics > Operator Algebras

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Abstract: A locally compact group $G$ is amenable if and only if it has Reiter-sproperty $(P p)$ 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 $\lim \alpha \sup {x \in K} \| L {x^{-1}} m \alpha -m \alpha \| p = 0$ 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 thedefinitions of properties $(P 1)$ and $(P 2)$ from locally compact groups tolocally compact quantum groups in the sense of J. Kustermans and S. Vaes. Weshow that a locally compact quantum group has $(P 1)$ if and only if it isamenable and that it has $(P 2)$ if and only if its dual quantum group isco-amenable. As a consequence, $(P 2)$ implies $(P 1)$.

Author: ** Matthew Daws, Volker Runde**

Source: https://arxiv.org/