Schatten's theorems on functionally defined Schur algebras

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