Linear maps preserving the minimum and reduced minimum moduli. (English) Zbl 1187.47028

A problem posed by I.Kaplansky in [“Algebraic and analytic aspects of operator algebras” (CBMS Reg.Conf.Ser.Math.1; AMS) (1970; Zbl 0217.44902) (\({}^2\)1980; Zbl 0475.46043)] asks whether a spectrum-preserving linear map from a semisimple unital Banach algebra \({\mathcal A}\) onto another \({\mathcal B}\) is a Jordan isomorphism. M.Mbekhta [Extr.Math.22, No.1, 45–54 (2007; Zbl 1160.47033)] described surjective unital maps on \({\mathcal L}({\mathcal H})\) preserving the minimum, surjective and reduced minimum modulus, and these results were extended to \(C^*\)-algebras of real rank zero by the first two authors of the present paper in [Oper.Matrices 4, No.2, 245–256 (2010)] and [“Linear maps preserving regularity in \(C^*\)-algebras”, Ill.J.Math., in press]. Let \(c(\cdot)\) stand for any of the following quantities related to an element in a \(C^*\)-algebra: minimum modulus, reduced minimum modulus, maximum modulus, surjectivity modulus. In this paper, the authors study maps among \(C^*\)-algebras preserving these spectral quantities. Among the results they obtain, we mention here two that imply partial answers to Kaplansky’s problem:
Let \({\mathcal A}\) and \({\mathcal B}\) be \(C^*\)-algebras. If \(\Phi:{\mathcal A}\to\mathcal B\) is a unital linear map such that \(c(x)=c(\Phi(x))\) for all \(x\in {\mathcal A}\), then \(\Phi\) is an isometric Jordan \(*\)-homomorphism.
Let \({\mathcal A}\) be a semisimple Banach algebra and let \({\mathcal B}\) be a \(C^*\)-algebra. If \(\Phi:{\mathcal A}\to\mathcal B\) is a surjective linear map such that \(c(x)=c(\Phi(x))\) for all \(x\in {\mathcal A}\), then \({\mathcal A}\) (for its norm and some involution) is a \(C^*\)-algebra and \(\Phi\) is an isometric Jordan \(*\)-isomorphism multiplied by a unitary element of \({\mathcal B}\).


47B48 Linear operators on Banach algebras
46L05 General theory of \(C^*\)-algebras
Full Text: DOI


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