International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 14, Pages 2175-2193
doi: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 48859, MI, USA
Received 25 January 2005; Revised 20 July 2005
Copyright © 2005 Pachara Chaisuriya and Sing-Cheong Ong. 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,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.