Representations of compact groups on Banach algebras

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.