Doubly stochastic right multipliers

Choo-Whan Kim

International Journal of Mathematics and Mathematical Sciences, 1984, vol. 7, 1-13

Abstract:

Let P ( G ) be the set of normalized regular Borel measures on a compact group G . Let D r be the set of doubly stochastic (d.s.) measures λ on G × G such that λ ( A s × B s ) = 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 D r such that ϕ − 1 = m ⊗ μ , where m is normalized Haar measure on G , and ϕ ( x , y ) = ( x , x y − 1 ) for x , y ∈ G . Further, we show that there exists a bijection between D r and M r , the set of d.s. right multipliers of L 1 ( G ) . It follows from these results that the mapping μ → T μ defined by T μ f = μ ∗ f is a topological isomorphism of the compact convex semigroups P ( G ) and M r . It is shown that M r is the closed convex hull of left translation operators in the strong operator topology of B [ L 2 ( G ) ] .

Date: 1984
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:140352

DOI: 10.1155/S016117128400051X

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