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On non-self-adjoint operator algebras. (English) Zbl 0718.46020
An operator algebra is a norm closed algebra of bounded operators on a Hilbert space. The general structure of non-selfadjoint operator algebras has been largely studied in the recent years mainly by the authors, D. Blecher, V. Paulsen, P. Power and A. Sinclair. This paper studies the Arens’ products in \(A^{**}\) when A is an operator algebra. It is proved that the left and right Arens’ products coincide on \(A^{**}\) and that the resulting algebra is completely isometric to a \(\sigma\)-weakly closed operator algebra. This result completes a previous one by M. Hamana [Publ. Res. Inst. Math. Sci. Kyoto Univ. 15, 773-785 (1979; Zbl 0436.46046)]. The paper contains a characterization of M-ideals of a unital operator algebra as closed two-sided ideals with an approximate identity. There are also examples which show that isometric isomorphisms need not be completely isometric and that spatial tensor product norm on a tensor product need not be minimal among the matricial cross-norms.

MSC:
46K50 Nonselfadjoint (sub)algebras in algebras with involution
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
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