INVERTIBILITY-PRESERVING MAPS OF C∗-ALGEBRAS WITH REAL RANK ZERO

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 Jor-dan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ: A → B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-algebra of real rank zero. We also generalize a theorem of Russo. 1. Notation In what follows, the term Banach algebra will mean a unital complex Banach algebra and a C∗-algebra will mean a unital complex C∗-algebra. The unit is denoted by 1 and the spectrum of an element x by σ(x). The set of invertible elements of a Banach algebra A is denoted by Ainv and the closed unit ball of A by A1. The density of a subset of a Banach algebra in another subset is meant to be in the norm topology. A linear map Φ from a Banach algebra A to a normed algebra B is a Jordan homomorphism if Φ(a2) = Φ(a)2 for every a ∈ A. Properties of Jordan homomorphisms are given in [7] or [9]. For C∗-algebras A and B, a C∗-homomorphism in the sense of Kadison is a selfadjoint linear mapping of A into B which is a Jordan homomorphism, that is, Φ(a∗) = Φ(a) ∗ and Φ(a2)=Φ(a)2 for all a∈A [13]. 2