Representations of compact groups on Banach algebras

Abstract

Let a compact group U U act by automorphisms of a commutative regular and Wiener Banach algebra A \mathcal {A} . We study representations R ω {R^\omega } of U U on quotient spaces A / I ( ω ) \mathcal {A}/I(\omega ) , where ω \omega is an orbit of U U in the Gelfand space X X of A \mathcal {A} and I ( ω ) I(\omega ) is the minimal closed ideal with hull ω ⊂ X \omega \subset X . The main result of the paper is: if A = A ρ ( X ) \mathcal {A} = \,{\mathcal {A}_\rho }(X) is a weighted Fourier algebra on a LCA group X = A ^ X = \hat A with a subpolynomial weight ρ \rho on A A , and U U acts by affine transformations on X X , then for any orbit ω ⊂ X \omega \subset X the representation R ω {R^\omega } has finite multiplicity. Precisely, the multiplicity of π ∈ U ^ \pi \in \hat U in R ω {R^\omega } is estimated as k ( π ; R ω ) ≤ c ⋅ deg ⁡ ( π ) ∀ π ∈ U ^ k(\pi ;{R^\omega }) \leq c \cdot \deg (\pi )\;\forall \pi \in \hat U with a constant c c depending on A A and ρ \rho . Applications of this result are given to topologically irreducible representations of motion groups and primary ideals of invariant subalgebras.