Schur algebras over \(C^{\ast}\)-algebras (Q925410)

Summary: Let \(\mathcal A\) be a \(C^*\)-algebra with identity 1, and let \(s({\mathcal A})\) denote the set of all states on \(\mathcal A\). For \(p,q,r\in [1,\infty)\), denote by \({\mathcal S}^r({\mathcal A})\) the set of all infinite matrices \(A=[a_{jk}]^\infty_{j,k=1}\) over \(\mathcal A\) such that the matrix \((\varphi[A^{[2]})^{[r]}:= [(\varphi(a^*_{jk}a_{jk})^r]^\infty_{j,k=1}\), defines a bounded linear operator from \(\ell^p\) to \(\ell^q\) for all \(\varphi\in s({\mathcal A})\). Then \({\mathcal S}^r({\mathcal A})\) is a Banach algebra with the Schur product operation and norm \(\| A\|=\sup\{\|(\varphi[A^{[2]}])^r\|^{1/(2r)}: \varphi\in s({\mathcal A})\}\). 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.