Bimeasure algebras on LCA groups

Abstract

For locally compact abelian groups Gλ and G2, with character groups Γ, and Γ2, respectively, let BM(GU G2) denote the Banach space of bounded bilinear forms on CO(GX) X C0(G2). Using a consequence of the fundamental inequality of A. Grothendieck, a multiplication and an adjoint operation are introduced on BM{Gλ9 G2) which generalize the convolution structure of M(G X H) and which make BM(GU G2) into a Λ^-Banach *-algebra, where KG is Grothendieck's universal constant. The Fourier transforms of elements of BM(GUG2) are defined and characterized in terms of certain unitary representations of Γ, and Γ2. Various aspects of the harmonic analysis of the algebras BM(GX, G2) are studied.