Journal of Operator Theory

Volume 34, Issue 2, Fall 1995 pp. 315-346.

On quotients of function algebras and operator algebra structures on $\ell_p$

Authors: David P. Blecher (1) and Christian Le Merdy (2)
Author institution: (1) Department of Mathematics, University of Houston, Houston, Texas 77204-3476, U.S.A.
(2) Equipe de Mathematiques, URA CNRS 741, Universite de Franche-Comte, F-25030 Besancon Cedex, FRANCE

Summary: Recently, the first author gave a characterization of operator algebras up to complete isomorphism. We give here some characterizations of quotients of function algebras (Q-algebras), again up to complete isomorphism. Using these, we examine which operator space structures on $\ell_p$ (with pointwise product) correspond to operator algebras, and which to Q-algebras. We also give a new approach to the long outstanding similarity problem of Halmos, studying operator space structures on the disc algebra. Finally, we show that the Banach algebra of von Neumann-Schatten p-class operators on a Hilbert space is an operator algebra for all $1 \le p \le \infty$ (with either the usual or the Schur product). That is, these algebras are bicontinuously and algebraically isomorphic to a norm closed algebra of operators on some Hilbert space. However, with the usual operator space structures they are not completely bicontinuously isomorphic to any closed algebra of operators.

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