Schatten's theorems on functionally defined Schur algebras
For each triple of positive numbers p,q,r≥1 and each commutative C*-algebra ℬ with identity 1 and the set s(ℬ) of states on ℬ, the set 𝒮r(ℬ) of all matrices A=[ajk] over ℬ such that ϕ[A[r]]:=[ϕ(|ajk|r)] defines a bounded operator from ℓp to ℓq for all ϕ∈s(ℬ) is shown to be a Banach alg...
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| Format: | Article |
| Language: | English |
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Wiley
2005-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.2175 |
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| author | Pachara Chaisuriya Sing-Cheong Ong |
| author_facet | Pachara Chaisuriya Sing-Cheong Ong |
| author_sort | Pachara Chaisuriya |
| collection | DOAJ |
| description | For each triple of positive numbers p,q,r≥1 and each
commutative C*-algebra ℬ with identity 1 and the
set s(ℬ) of states on ℬ, the set 𝒮r(ℬ) of all matrices A=[ajk] over ℬ such that ϕ[A[r]]:=[ϕ(|ajk|r)] defines a bounded operator from ℓp to
ℓq for all ϕ∈s(ℬ) is shown to be a Banach
algebra under the Schur product operation, and the norm ‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}.
Schatten's theorems about the dual of the compact
operators, the trace-class operators, and the decomposition of the
dual of the algebra of all bounded operators on a Hilbert space
are extended to the 𝒮r(ℬ) setting. |
| format | Article |
| id | doaj-art-208712e36da341438908b6b09c58f111 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-208712e36da341438908b6b09c58f1112025-02-03T01:23:54ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-012005142175219310.1155/IJMMS.2005.2175Schatten's theorems on functionally defined Schur algebrasPachara Chaisuriya0Sing-Cheong Ong1Department of Mathematics, Faculty of Science, Mahidol University, Rama VI Road, Bangkok 10400, ThailandDepartment of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USAFor each triple of positive numbers p,q,r≥1 and each commutative C*-algebra ℬ with identity 1 and the set s(ℬ) of states on ℬ, the set 𝒮r(ℬ) of all matrices A=[ajk] over ℬ such that ϕ[A[r]]:=[ϕ(|ajk|r)] defines a bounded operator from ℓp to ℓq for all ϕ∈s(ℬ) is shown to be a Banach algebra under the Schur product operation, and the norm ‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}. Schatten's theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the 𝒮r(ℬ) setting.http://dx.doi.org/10.1155/IJMMS.2005.2175 |
| spellingShingle | Pachara Chaisuriya Sing-Cheong Ong Schatten's theorems on functionally defined Schur algebras International Journal of Mathematics and Mathematical Sciences |
| title | Schatten's theorems on functionally defined
Schur algebras |
| title_full | Schatten's theorems on functionally defined
Schur algebras |
| title_fullStr | Schatten's theorems on functionally defined
Schur algebras |
| title_full_unstemmed | Schatten's theorems on functionally defined
Schur algebras |
| title_short | Schatten's theorems on functionally defined
Schur algebras |
| title_sort | schatten s theorems on functionally defined schur algebras |
| url | http://dx.doi.org/10.1155/IJMMS.2005.2175 |
| work_keys_str_mv | AT pacharachaisuriya schattenstheoremsonfunctionallydefinedschuralgebras AT singcheongong schattenstheoremsonfunctionallydefinedschuralgebras |