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International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 3, Pages 477-489

Doubly stochastic right multipliers

Department of Mathematics, Simon Fraser University, B.C., Burnaby V5A 1S6, Canada

Received 13 June 1983

Copyright © 1984 Hindawi Publishing Corporation. 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.


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 sG, and A and B are Borel subsets of G. We show that there exists a bijection μλ between P(G) and Dr such that ϕ1=mμ, where m is normalized Haar measure on G, and ϕ(x,y)=(x,xy1) for x,yG. 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 μTμ defined by Tμf=μf 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)].