Vol. 77, No. 1, 1978

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The Fourier-Stieltjes algebra of a topological semigroup with involution

Anthony To-Ming Lau

Vol. 77 (1978), No. 1, 165–181

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.

Mathematical Subject Classification 2000
Primary: 43A15
Secondary: 46L05
Received: 16 March 1977
Revised: 7 December 1977
Published: 1 July 1978
Anthony To-Ming Lau