Schatten's theorems on functionally defined Schur algebras

Pachara Chaisuriya and Sing-Cheong Ong

International Journal of Mathematics and Mathematical Sciences, 2005, vol. 2005, 1-19

Abstract:

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 𝒮 r ( ℬ ) of all matrices A = [ a j k ] over ℬ such that Ï• [ A [ r ] ] : = [ Ï• ( | a j k | r ) ] defines a bounded operator from â„“ p to â„“ q for all Ï• ∈ s ( ℬ ) is shown to be a Banach algebra under the Schur product operation, and the norm ‖ A ‖ = ‖ | A | ‖ p , q , r = sup { ‖ Ï• [ A [ r ] ] ‖ 1 / r : Ï• ∈ 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 𝒮 r ( ℬ ) setting.

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:198974

DOI: 10.1155/IJMMS.2005.2175

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