Invertibility-preserving maps of $C^*$-algebras with real rank zero

Abstract

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between $C^{\ast }$-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 $\Phi :A\to B$ is a linear map onto $B$ that preserves the spectrum of elements, then $\Phi$ is a Jordan isomorphism if either $A$ or $B$ is a $C^{\ast }$-algebra of real rank zero. We also generalize a theorem of Russo.