Spectrally Bounded Operators on Simple $C^*$-Algebras, II

Abstract

Abstract. A linear mapping T from a Banach algebra intoanother Banach algebra is called spectrally bounded if thereis a constant M ‚ 0 such that r ( Tx ) • M r ( x ) for all x in the domain, where r ( ¢ ) denotes the spectral radius. Weinvestigate to what extent a unital spectrally bounded oper-ator from a simple unital C* -algebra of real rank zero onto aunital semisimple Banach algebra is a Jordan epimorphism. 1. IntroductionLet A and B be unital semisimple Banach algebras over the complexnumbers C. An open conjecture of Kaplansky states that every linearsurjective mapping T : A ! B which is spectrum-preserving , that is, ¾ ( Tx ) = ¾ ( x ) for all x 2 A (where ¾ ( x ) stands for the spectrumof x ), must be a Jordan algebra isomorphism. This conjecture hasbeen confirmed in the case of von Neumann algebras by Aupetitin [2]. An inspection of his proof shows that, in fact, no furtherassumption is needed on B and that it suffices that the spectraltheorem holds in A . Thus, an immediate extension of his result isthe following.Theorem 1.1.