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Abstract and Applied Analysis
Volume 2005, Issue 6, Pages 685-689
http://dx.doi.org/10.1155/AAA.2005.685

Invertibility-preserving maps of C-algebras with real rank zero

Department of Mathematics, Case Western Reserve University, Cleveland 44106, OH, USA

Received 1 December 2003

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras and Φ:AB is a linear map onto B that preserves the spectrum of elements, then Φ is a Jordan isomorphism if either A or B is a C-algebra of real rank zero. We also generalize a theorem of Russo.