# Fourier-Stieltjes algebra

Let be an arbitrary locally compact group. For , let , where is the set of all equivalence classes of unitary continuous representations of (cf. also Unitary representation). The completion of with respect to this norm is a Banach algebra, denoted by and called the full -algebra of . If is Abelian and its dual group, then is isometrically isomorphic to the Banach algebra of all complex-valued continuous functions on vanishing at infinity.

Let be the complex linear span of the set of all continuous positive-definite functions on .

1) The -vector space is isomorphic to the dual space of . With the dual norm and the pointwise product on , is a commutative Banach algebra [a4].

This Banach algebra is called the Fourier–Stieltjes algebra of . If is Abelian, then is isometrically isomorphic to the Banach algebra of all bounded Radon measures on .

2) On the boundary of the unit ball of (i.e. on ) the weak topology coincides with the compact-open topology on ([a3]; see also [a9], [a6]).

3) The following properties are satisfied ([a4]):

a) The Fourier algebra is a closed ideal of ;

b) ;

c) coincides with the closure in of ;

d) , with equality of the corresponding norms. Here, is the algebra of functions of compact support on . In [a14], M.E. Walter showed that (and also ) completely characterizes . More precisely, assume that and are locally compact groups; then the following assertions are equivalent:

the locally compact groups and are topologically isomorphic;

the Banach algebras and are isometrically isomorphic;

the Banach algebras and are isometrically isomorphic.

He also gave a description of the dual of .

For a connected semi-simple Lie group , M. Cowling [a1] has given a description of the spectrum of ; surprisingly, if is Abelian, then the spectrum of seems to be much more complicated than in the non-Abelian case!

If is amenable, then ([a3])

 (a1)

V. Losert [a12] proved the converse assertion: if (a1) holds, then must be amenable!

In a difficult paper [a7], C.S. Herz tried to extend the preceding results, replacing unitary representations by representations in Banach spaces. He partially succeeded in the amenable case. See also [a2], [a5].

M. Lefranc generalized Paul Cohen's idempotent theorem to for arbitrary locally compact groups ([a10], [a8]; see also [a11] for detailed proofs).