Let S be a topological
semigroup with a continuous involution. We study a subalgebra F(S) of the algebra
of continuous weakly almost periodic functions on S. F(S) is translation invariant,
closed under conjugation and contains constants. When S has an identity, then F(S)
is the linear span of the cone of continuous positive definite functions on S. We
show that there exists a norm ∥⋅∥Ω on F(S) such that (F(S),∥⋅∥Ω) is a
commutative Banach algebra which can be identified with the predual of
a W∗-algebra W∗(S). When S is a locally compact group, then F(S) is
precisely the Fourier Stieltjes algebra of S. We also show that σ(F(S)), the
spectrum of F(S), is a ∗-semigroup in W∗(S), and study the relation of
σ(F(S1)) and σ(F(S2)) when F(S1) and F(S2) are isometric isomorphic Banach
algebras.
