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International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 14, Pages 2175-2193
http://dx.doi.org/10.1155/IJMMS.2005.2175

Schatten's theorems on functionally defined Schur algebras

1Department of Mathematics, Faculty of Science, Mahidol University, Rama VI Road, Bangkok 10400, Thailand
2Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA

Received 25 January 2005; Revised 20 July 2005

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For each triple of positive numbers p,q,r1 and each commutative C*-algebra with identity 1 and the set s() of states on , the set 𝒮r() of all matrices A=[ajk] over such that ϕ[A[r]]:=[ϕ(|ajk|r)] defines a bounded operator from p to q for all ϕs() 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:ϕs()}. 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() setting.