THE MEASURE ALGEBRA AS AN OPERATOR ALGEBRA.
Abstract
Let G be a locally compact abelian group; M(G) the algebra of bounded, Borel measures on G; and M(G)' the algebra of Fourier-Stieltjes transforms. In Chapter I, we show how the bounded linear functionals on M(G) can be represented as the semigroup of bounded operators on M(G)' which commute with translation. We say that M(G)' is an operator algebra. Let M(G)* denote the topological dual of M(G); M sub M(G) the multiplicative linear functionals on M(G); and P the closed linear span of M sub M(G) in M(G)*, P = M sub M(G) tc M(G)*. Since M(G)' 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 = C(B), where B is a compact, Hausdorff space. In Chapter III, we show that B is a compact abelian semi-group and that M(G) is topologically embedded in M(B). B is the Taylor structure semi-group for M(G). This gives a simplified construction of the Taylor structure semi-group for M(G). (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1967
- Accession Number
- AD0651890
Entities
People
- Donald E. Rameriz
Organizations
- University of Washington