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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Amenability of locally compact groups and subspaces of $ L\sp \infty(G)$


Author: Tianxuan Miao
Journal: Proc. Amer. Math. Soc. 111 (1991), 1075-1084
MSC: Primary 43A07
DOI: https://doi.org/10.1090/S0002-9939-1991-1045143-1
MathSciNet review: 1045143
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Abstract: If $ G$ is a locally compact group, let $ \mathcal{A}$ be the set of all the functions which left average to a constant, i.e. the function $ f \in {L^\infty }(G)$ such that there is a constant in the $ {\left\Vert \right\Vert _{{\infty ^ - }}}$ closed convex hull of $ {\{ _x}f:x \in G\} $. We prove in this paper that $ \mathcal{A}$ is a subspace of $ {L^\infty }(G)$ if and only if $ G$ is amenable as a discrete group. This answers a problem asked by Emerson, Rosenblatt and Yang, and Wong and Riazi. We also answer two other problems of Rosenblatt and Yang on whether the set $ \mathcal{U}$ of functions in $ {L^\infty }(G)$ admitting a unique left invariant mean value is a subspace of $ {L^\infty }(G)$ when $ G$ is not amenable and whether there is a largest admissible subspace of $ {L^\infty }(G)$ with a unique left invariant mean.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1045143-1
Keywords: Locally compact groups, amenable groups, invariant means, left averaging functions, functions admitting a unique left invariant mean value
Article copyright: © Copyright 1991 American Mathematical Society