For locally compact abelian
groups G1 and G2, with character groups Γ1, and Γ2, respectively, let BM(G1,G2)
denote the Banach space of bounded bilinear forms on C0(G1) × C0(G2). Using a
consequence of the fundamental inequality of A. Grothendieck, a multiplication and
an adjoint operation are introduced on BM(G1,G2) which generalize the convolution
structure of M(G×H) and which make BM(G1,G2) into a KG2-Banach ∗-algebra,
where KG is Grothendieck’s universal constant. The Fourier transforms of elements
of BM(G1,G2) are defined and characterized in terms of certain unitary
representations of Γ1, and Γ2. Various aspects of the harmonic analysis of the
algebras BM(G1,G2) are studied.
