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

Schur Algebras over C*-Algebras

Pachara Chaisuriya,1 Sing-Cheong Ong,2 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

Received 15 November 2006; Accepted 21 May 2007

Academic Editor: Aloys Krieg

Copyright © 2007 Pachara Chaisuriya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.