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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Inner amenable locally compact groups

Authors: Anthony To Ming Lau and Alan L. T. Paterson
Journal: Trans. Amer. Math. Soc. 325 (1991), 155-169
MSC: Primary 43A07
MathSciNet review: 1010885
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Abstract: In this paper we study the relationship between amenability, inner amenability and property $ P$ of a von Neumann algebra. We give necessary conditions on a locally compact group $ G$ to have an inner invariant mean $ m$ such that $ m(V) = 0$ for some compact neighborhood $ V$ of $ G$ invariant under the inner automorphisms. We also give a sufficient condition on $ G$ (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan's property $ T$, e.g., $ {\text{SL}}(n,\mathbb{Z})$, $ n \geq 3$) such that each linear form on $ {L^2}(G)$ which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach $ G$-modules is also obtained.

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Keywords: Amenable locally compact groups, inner amenable groups, $ [{\text{IN}}]$-groups, property $ P$, von Neumann algebra, free group, Kazhdan's property $ T$, automatic continuity, fixed point property
Article copyright: © Copyright 1991 American Mathematical Society

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