Invertibility-preserving maps of \(C^*\)-algebras with real rank zero (Q2495032)

Summary: In [Proc.\ Am.\ Math.\ Soc.\ 124, No.~8, 2415--2422 (1996; Zbl 0864.46035)], \textit{L.~A.\ Harris} and \textit{R.~V.\ 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 the present paper, we show that if \(A\) and \(B\) are semisimple Banach algebras and \(\Phi: A\rightarrow B\) 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 \(C^*\)-algebra of real rank zero. We also generalize a theorem of \textit{B.~Russo} [Proc.\ Am.\ Math.\ Soc.\ 17, 1019--1022 (1966; Zbl 0166.40003)].