Schur Algebras over C-Algebras

Let be a C * -algebra with identity 1 , and let s ( ) denote the set of all states on . For p , q , r ∈ [ 1 , ∞ ) , denote by r ( ) the set of all infinite matrices A = [ a j k ] j , k = 1 ∞ over such that the matrix ( ϕ [ A [ 2 ] ] ) [ r ] : = [ ( ϕ ( a j k * a j k ) ) r ] j , k = 1 ∞ defines a bounded linear operator from ℓ p to ℓ q for all ϕ ∈ s ( ) . Then r ( ) is a Banach algebra with the Schur product operation and norm ‖ A ‖ = sup { ‖ ( ϕ [ A [ 2 ] ] ) r ‖ 1 / ( 2 r ) : ϕ ∈ s ( ) } . Analogs of Schatten's theorems on dualities among the compact operators, the trace-class operators, and all the bounded operators on a Hilbert space are proved.