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∗‐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 : A → B 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.