Continuity and linearity of centralizers on a complemented algebra.

Abstract

Let A be a semisimple complemented algebra and let T be a mapping of A into itself such that either T(xy) = xTy or T(xy) = (Tx)y holds for all x, y E A. If T is defined everywhere on A then T is a bounded linear operator. 1. A right centralizer on an algebra A is a mapping T of A into A such that T(xy) = xTy for all x, y E A. A left centralizer is a mapping T: A -> A such that T(xy) = (Tx)y for all x, y E A. This terminology is somewhat different from the terminology of [5] and is due to B. Johnson, who developed the theory of centralizers in [3] and was able to show that for a certain class of Banach algebras centralizers are always linear and bounded. In [4] he showed that this is the case when the algebra has a certain type of bounded approximate identity. The purpose of this paper is to extend these results of Johnson to the case of complemented algebras. Results of [4] are not applicable to our case. In fact the authors are convinced that Johnson's condition on existence of a certain type of approximate identity in the case of complemented algebras would be equivalent to the assumption of finite-dimensionality of the algebra. Inasmuch as every proper right H*-algebra [8] is a complemented algebra, we have extension of Johnson's result to all types of H*-algebras, and in particular, to the algebra of Hilbert Schmidt operators. We developed our theory for right centralizers but it is obvious that the same theory could be developed for left centralizers. 2. In this section we recall some basic definitions and facts from the theory of complemented algebras. For a more complete background the reader is referred to [6] and [7]. Presented to the Society, October 31, 1970; received by the editors August 19, 1970 and, in revised form, January 10, 1971. AMS 1970 subject classifications. Primary 46K1 5; Secondary 47B10.