A completely bounded characterization of operator algebras
Let \(B(H)\) denote the Banach algebra of all bounded operators on a Hilbert space \(H\). An operator algebra is a (not necessarily closed) subalgebra of \(B(H)\) for some Hilbert space \(H\). \textit{D. Blecher, Z-j. Ruan} and \textit{A. Sinclair} [J. Funct. Anal. 89, No. 1, 188-201 (1990; Zbl 0714.46043)] have given an abstract characterization of operator algebras up to completely isometric isomorphism as follows. Let \({\mathcal A}\) be a unital algebra which is also an \(L^{\infty}\)-matrix normed space. Then \({\mathcal A}\) may be represented completely isometrically isomorphically as an operator algebra if and only if the multiplication on \({\mathcal A}\) is contractive. An algebra which is also an \(L^{\infty}\)-matrix normed space such that the multiplication is completely bounded is called an \(h\)-algebra. Clearly every operator algebra is an \(h\)-algebra. The author proves that the converse is also true. He shows that every \(h\)-algebra is completely boundedly isomorphic to an operator algebra. Applications and examples are also presented.
scientific article; zbMATH DE number 840183
scientific article