Schatten's theorems on functionally defined schur algebras

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 ℓr (ℬ) of all matrices A = [ajk] over ℬ such that φ [A[r]]:= [φ( ajkr)] defines a bounded operator from ℓp to ℓq for all φ ε 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 ε 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 ℓr (ℬ) setting. Copyright © 2005 Hindawi Publishing Corporation.