A \(C^*\)-algebra on Schur algebras (Q624896)

For any \(\Lambda, \Sigma\in\{c_0\}\cup\{\ell_p\,|\,1\leq p<+\infty\}\), \(r\in\mathbb N\) and a Banach algebra \({\mathcal B}\), one has the Banach algebra \(S^r_{\Lambda,\Sigma}({\mathcal B})\) under the Schur product and the absolute Schur \(r\)-norm. In this paper, the author gives some relations between two different such algebras. In the case that \({\mathcal C}\) is a \(C^*\) algebra, it is shown that \(S^r_{\Lambda,\Sigma}({\mathcal C})\) becomes a \(C^*\) algebra under the induced involution operator if and only if \((\Lambda,\Sigma)=(\ell_1,c_0)\). A property of the relation of the absolute Schur \(r\)-norm and the Schur multiplier norm in \(S^r_{\Lambda,\Sigma}({\mathcal C})\) is also given.