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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |