Journal of Operator Theory

Volume 48, Issue 3, Supplementary 2002 pp. 615-619.

Strongly reductive algebras are selfadjoint

Authors: Bebe Prunaru
Author institution: Institute of Mathematics of the Romanian Academy, PO Box 1--764, RO--70700 Bucharest, Romania

Summary: Let

$B(H)$ denote the algebra of all bounded linear operators on some complex Hilbert space

$H$ . A unital subalgebra

${\Cal A}\subset B(H)$ is said to be strongly reductive if, whenever

$\{P\sb\lambda\}$ is a net of orthogonal projections in

$B(H)$ such that

$\Vert(1-P\sb\lambda)TP\sb\lambda\Vert\to 0$ for all

$T\in\Cal A,$ then the same holds true for all

$T$ in the

$C\sp *$ -algebra generated by

$\Cal A$ in

$B(H).$ In this paper we prove that the norm-closure of every strongly reductive algebra is selfadjoint.

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