Abstract and Applied Analysis
Volume 2005 (2005), Issue 6, Pages 685-689
Invertibility-preserving maps of -algebras with real rank zero
Department of Mathematics, Case Western Reserve University, Cleveland 44106, OH, USA
Received 1 December 2003
Copyright © 2005 Istvan Kovacs. 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.
In 1996, Harris and Kadison posed the following problem: show that a linear bijection between -algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if and are semisimple Banach algebras and is a linear map onto that preserves the spectrum of elements, then is a Jordan isomorphism if either or is a -algebra of real rank zero. We also generalize a theorem of Russo.