Doubly stochastic right multipliers
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, wh...
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| Format: | Article |
| Language: | English |
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Wiley
1984-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S016117128400051X |
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| _version_ | 1850163518906564608 |
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| author | Choo-Whan Kim |
| author_facet | Choo-Whan Kim |
| author_sort | Choo-Whan Kim |
| collection | DOAJ |
| description | 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)]. |
| format | Article |
| id | doaj-art-e7b1949b2b464ba8b4a407b24988a748 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1984-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e7b1949b2b464ba8b4a407b24988a7482025-08-20T02:22:15ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251984-01-017347748910.1155/S016117128400051XDoubly stochastic right multipliersChoo-Whan Kim0Department of Mathematics, Simon Fraser University, B.C., Burnaby V5A 1S6, CanadaLet 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)].http://dx.doi.org/10.1155/S016117128400051Xcompact groupregular Borel measuresdoubly stochastic measuresmultipliers. |
| spellingShingle | Choo-Whan Kim Doubly stochastic right multipliers International Journal of Mathematics and Mathematical Sciences compact group regular Borel measures doubly stochastic measures multipliers. |
| title | Doubly stochastic right multipliers |
| title_full | Doubly stochastic right multipliers |
| title_fullStr | Doubly stochastic right multipliers |
| title_full_unstemmed | Doubly stochastic right multipliers |
| title_short | Doubly stochastic right multipliers |
| title_sort | doubly stochastic right multipliers |
| topic | compact group regular Borel measures doubly stochastic measures multipliers. |
| url | http://dx.doi.org/10.1155/S016117128400051X |
| work_keys_str_mv | AT choowhankim doublystochasticrightmultipliers |