International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 63808, 15 pages
doi:10.1155/2007/63808

Schur Algebras over C*-Algebras

Pachara Chaisuriya1 , Sing-Cheong Ong2 and Sheng-Wang Wang3

1Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10700, Thailand
 2Department of Mathematics, Central Michigan University, Mt. Pleasant 48859, MI, USA
 3Department of Mathematics, Nanjing University, Nanjing 210029, China

Abstract

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 𝒮r(𝒜) the set of all infinite matrices A=[ajk]j,k=1∞ over 𝒜 such that the matrix (ϕ[A[2]])[r]:=[(ϕ(ajk*ajk))r]j,k=1∞ defines a bounded linear operator from ℓp to ℓq for all ϕ∈s(𝒜). Then 𝒮r(𝒜) is a Banach algebra with the Schur product operation and norm ‖A‖=sup{‖(ϕ[A[2]])r‖1/(2r):ϕ∈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.