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Invariant subspaces for algebras of linear operators and amenable locally compact groups


Authors: Anthony T. M. Lau and James C. S. Wong
Journal: Proc. Amer. Math. Soc. 102 (1988), 581-586
MSC: Primary 43A20; Secondary 47D05
DOI: https://doi.org/10.1090/S0002-9939-1988-0928984-8
MathSciNet review: 928984
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Abstract: Let $ G$ be a locally compact group. We prove in this paper that $ G$ is amenable if and only if the group algebra $ {L_1}\left( G \right)$ (respectively the measure algebra $ M\left( G \right)$) satisfies a finite-dimensional invariant subspace property $ T\left( n \right)$ for $ n$-dimensional subspaces contained in a subset $ X$ of a separated locally convex space $ E$ when $ {L_1}\left( G \right)$ (respectively $ M\left( G \right)$) is represented as continuous linear operators on $ E$. We also prove that for any locally compact group, the Fourier algebra $ A\left( G \right)$ and the Fourier Stieltjes algebra $ B\left( G \right)$ always satisfy $ T\left( n \right)$ for each $ n = 1,2, \ldots $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0928984-8
Keywords: Amenable locally compact groups, group algebra, measure algebra, Fourier algebra, Fourier Stieltjes algebra, finite-dimensional, invariant subspaces
Article copyright: © Copyright 1988 American Mathematical Society

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