Representations of Compact Groups on Banach Algebras

Let a compact group U act by automorphisms of a commutative regular and Wiener Banach algebra A. We study representations Ru of U on quotient spaces A/I(üj), where u> is an orbit of U in the Gelfand space X of A and L(uj) is the minimal closed ideal with hull w C X. The main result of the paper is: if A = AP(X) is a weighted Fourier algebra on a LCA group X = A with a subpolynomial weight p on A, and U acts by affine transformations on X, then for any orbit w C X the representation Ru has finite multiplicity. Precisely, the multiplicity of tc £ U in Rw is estimated as k(tc; Rw) < c ■ deg(7r) W € U with a constant c depending on A and p. Applications of this result are given to topologically irreducible representations of motion groups and primary ideals of invariant subalgebras. Introduction. Let (A,U) be a pair of commutative Banach algebra A and a compact group U acting by automorphisms on A. We shall always assume that A is regular and Wiener (Tauberian) on its Gelfand space X = A (see [11, Chapter 10]), and the representation a —► au (a E A) of U on A is strongly continuous. There are two other algebras associated with any such pair: the crossed product, or generalized (twisted) L^algebra L = £(U; A) and the subalgebra of [/-invariants, Z = AU. Algebras £ = i(U;A) naturally appear in the study of group extensions [1, 13]. Namely, if a locally compact group G = A xi U is a semidirect product of a normal subgroup A and a subgroup U acting on A by automorphisms, then the group algebra Ll(G) is isomorphic to the crossed product C(U;L1(A)). In some cases (for instance, if U is compact) more general weighted group algebras Lp(G) can also be realized as crossed products, Lp(G) ~ £(U;Lpl(A)), where p' = p \ A (see [10]). For abelian groups A and compact groups U semidirect products G = A x U are called motion groups. Correspondingly, crossed products L = Z(U,A) of commutative Banach algebras A by compact groups U were called motion algebras in [14]. In harmonic analysis of motion algebras £ and subalgebras Z = Au an important role is played by U-invariant ideals of A (i.e., Iu = I, Vu G U). Thus in many cases Received by the editors March 11, 1982 and, in revised form, January 17, 1983 and August 23, 1983. 1980 Mathematics Subject Classification. Primary 43A20, 43A65, 22D12; Secondary 22C05, 22B10.