Choo-Whan Kim

Copyright © 1984 Choo-Whan Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let P(G) be the set of normalized regular Borel measures on a compact group G. Let Dr be the set of doubly stochastic (d.s.) measures λ on G×G such that λ(As×Bs)=λ(A×B), where s∈G, and A and B are Borel subsets of G. We show that there exists a bijection μ↔λ between P(G) and Dr such that ϕ−1=m⊗μ, where m is normalized Haar measure on G, and ϕ(x,y)=(x,xy−1) for x,y∈G. Further, we show that there exists a bijection between Dr and Mr, the set of d.s. right multipliers of L1(G). It follows from these results that the mapping μ→Tμ defined by Tμf=μ∗f is a topological isomorphism of the compact convex semigroups P(G) and Mr. It is shown that Mr is the closed convex hull of left translation operators in the strong operator topology of B[L2(G)].