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Transactions of the American Mathematical Society

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



Representations of compact groups on Banach algebras

Author: David Gurarie
Journal: Trans. Amer. Math. Soc. 285 (1984), 1-55
MSC: Primary 22C05; Secondary 22D20, 43A20, 46J99
MathSciNet review: 748829
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Abstract: Let a compact group $ U$ act by automorphisms of a commutative regular and Wiener Banach algebra $ \mathcal{A}$. We study representations $ {R^\omega }$ of $ U$ on quotient spaces $ \mathcal{A}/I(\omega )$, where $ \omega $ is an orbit of $ U$ in the Gelfand space $ X$ of $ \mathcal{A}$ and $ I(\omega )$ is the minimal closed ideal with hull $ \omega \subset X$. The main result of the paper is: if $ \mathcal{A} = \,{\mathcal{A}_\rho }(X)$ is a weighted Fourier algebra on a LCA group $ X = \hat A$ with a subpolynomial weight $ \rho $ on $ A$, and $ U$ acts by affine transformations on $ X$, then for any orbit $ \omega \subset X$ the representation $ {R^\omega }$ has finite multiplicity. Precisely, the multiplicity of $ \pi \in \hat U$ in $ {R^\omega }$ is estimated as $ k(\pi ;{R^\omega }) \leq c \cdot \deg (\pi )\;\forall \pi \in \hat U$ with a constant $ c$ depending on $ A$ and $ \rho $.

Applications of this result are given to topologically irreducible representations of motion groups and primary ideals of invariant subalgebras.

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Keywords: Compact, locally compact abelian, motion groups, Banach, regular, Wiener, Beurling (weighted), Fourier, Lie algebras, Banach, induced, finitely multiplied, *-, topologically irreducible representations, Banach bundles, tangent and normal bundles, jets, Whitney functions, $ m$-smooth vectors, affine transformations, minimal and primary ideals, module, Whitney's Theorem, Frobenius reciprocity theorem
Article copyright: © Copyright 1984 American Mathematical Society