SCHATTEN'S THEOREMS ON FUNCTIONALLY DEFINED SCHUR ALGEBRAS

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 r() of all matrices i[ajk] over such that [A[r]]:=[(|ajk|r)] defines a bounded operator from p to q for all s() is shown to be a Banach algebra under the Schur product operation, and the norm A=|A|p,q,r=sup{[A[r]]1/r: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.