THE MEASURE ALGEBRA AS AN OPERATOR ALGEBRA.
Abstract:
Let G be a locally compact abelian group MG the algebra of bounded, Borel measures on G and MG the algebra of Fourier-Stieltjes transforms. In Chapter I, we show how the bounded linear functionals on MG can be represented as the semigroup of bounded operators on MG which commute with translation. We say that MG is an operator algebra. Let MG denote the topological dual of MG M sub MG the multiplicative linear functionals on MG and P the closed linear span of M sub MG in MG, P M sub MG tc MG. Since MG is an operator algebra, we may induce in P a natural multiplication. In Chapter II, it is shown that P is a commutative B - algebra with 1. Thus P CB, where B is a compact, Hausdorff space. In Chapter III, we show that B is a compact abelian semi-group and that MG is topologically embedded in MB. B is the Taylor structure semi-group for MG. This gives a simplified construction of the Taylor structure semi-group for MG. Author