Schur Algebras over C * -Algebras

Let 𝒜 be a C * -algebra with identity 1 , and let s ( 𝒜 ) denote the set of all states on 𝒜 . For p , q , r ∈ [ 1 , ∞ ) , denote by 𝒮 r ( 𝒜 ) the set of all infinite matrices A = [ a j k ] j , k = 1 ∞ over 𝒜 such that the matrix ( Ï• [ A [ 2 ] ] ) [ r ] : = [ ( Ï• ( a j k * a j k ) ) r ] j , k = 1 ∞ defines a bounded linear operator from â„“ p to â„“ q for all Ï• ∈ s ( 𝒜 ) . Then 𝒮 r ( 𝒜 ) is a Banach algebra with the Schur product operation and norm ‖ A ‖ = sup { ‖ ( Ï• [ A [ 2 ] ] ) r ‖ 1 / ( 2 r ) : Ï• ∈ s ( 𝒜 ) } . 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.