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.