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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