Linear and Multiplicative Maps under Spectral Conditions

Abstract

The multiplicative version of the Gleason–Kahane–Żelazko theorem for \(C^*\)-algebras given by Brits et al. in [4] is extended to maps from \(C^*\)-algebras to commutative semisimple Banach algebras. In particular, it is proved that if a multiplicative map \(\phi\) from a \(C^*\)-algebra \(\mathcal{U}\) to a commutative semisimple Banach algebra \(\mathcal{V}\) is continuous on the set of all noninvertible elements of \(\mathcal{U}\) and \(\sigma(\phi(a)) \subseteq \sigma(a)\) for any \(a \in \mathcal{U}\), then \(\phi\) is a linear map. The multiplicative variation of the Kowalski–Słodkowski theorem given by Touré et al. in [14] is also generalized. Specifically, if \(\phi\) is a continuous map from a \(C^*\)-algebra \(\mathcal{U}\) to a commutative semisimple Banach algebra \(\mathcal{V}\) satisfying the conditions \(\phi(1_\mathcal{U})=1_\mathcal{V}\) and \(\sigma(\phi(x)\phi(y)) \subseteq \sigma(xy)\) for all \(x,y \in \mathcal{U}\), then \(\phi\) generates a linear multiplicative map \(\gamma_\phi\) on \(\mathcal{U}\) which coincides with \(\phi\) on the principal component of the invertible group of \(\mathcal{U}\). If \(\mathcal{U}\) is a Banach algebra such that each element of \(\mathcal{U}\) has totally disconnected spectrum, then the map \(\phi\) itself is linear and multiplicative on \(\mathcal{U}\). It is shown that a similar statement is valid for a map with semisimple domain under a stricter spectral condition. Examples which demonstrate that some hypothesis in the results cannot be discarded.