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Proceedings of the American Mathematical Society

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

 

 

Criteria for compactness and for discreteness of locally compact amenable groups


Author: Edmond Granirer
Journal: Proc. Amer. Math. Soc. 40 (1973), 615-624
MSC: Primary 43A07
DOI: https://doi.org/10.1090/S0002-9939-1973-0340962-8
MathSciNet review: 0340962
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Abstract: Let $ G$ be a locally compact group $ P(G) = \{ 0 \leqq \phi \in {L_1}(G);\int {\phi (x)dx = 1\} } $ and $ ({l_a}f)(x) = {}_af(x) = f(ax)$ for all $ a,x \in G$ and $ f \in {L^\infty }(G).0 \leqq \Psi \in {L^\infty }{(G)^ \ast },\Psi (1) = 1$ is said to be a [topological] left invariant mean ([TLIM] LIM) if $ \Psi {{\text{(}}_a}f) = \Psi (f)[\Psi (\phi \ast f) = \Psi (f)$] for all $ a \in G,\phi \in P(G),f \in {L^\infty }(G)$. The main result of this paper is the

Theorem. Let $ G$ be a locally compact group, amenable as a discrete group. If $ G$ contains an open $ \sigma $-compact normal subgroup, then LIM = TLIM if and only if $ G$ is discrete. In particular if $ G$ is an infinite compact amenable as discrete group then there exists some $ \Psi \in LIM$ which is different from normalized Haar measure. A harmonic analysis type interpretation of this and related results are given at the end of this paper.$ ^{2}$


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DOI: https://doi.org/10.1090/S0002-9939-1973-0340962-8
Article copyright: © Copyright 1973 American Mathematical Society