W∗-algebras and nonabelian harmonic analysis

Abstract

The Fourier-Stieltjes algebra of an arbitrary locally compact group G is the set of all finite, complex-linear combinations of continuous, positive definite functions on G, where addition and multiplication are defined pointwise and a Banach algebra norm (unique up to equivalence) can be specified. Thus, B( G) is a commutative, semisimple Banach algebra with unit. The main result is that B( G 1) and B( G 2) are isometrically isomorphic as Banach algebras if and only if G 1 and G 2 are topologically isomorphic as groups. The spectrum of B( G) is characterized as a ∗-semigroup of operators on Hilbert space, and its subgroup of invertible elements (being precisely those unitary elements which “preserve tensor products”) is topologically isomorphic to G. The Fourier algebra A( G) is also shown to characterize G. ( A( G) can be defined as the closure in B( G) of the functions in B( G) with compact support.) The representation theory of the lattice of subgroups of G is also studied. The main techniques of investigation come from the theory of C ∗ and W ∗-algebras.