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
Pachara Chaisuriya1
and Sing-Cheong Ong2
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
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.