Multiplicative maps into the spectrum
Volume 239 / 2017
Studia Mathematica 239 (2017), 55-66
MSC: Primary 46H05, 46L05; Secondary 46L10.
DOI: 10.4064/sm8705-1-2017
Published online: 17 May 2017
Abstract
We consider the converse of a famous result of W. Żelazko et al. which characterizes multiplicative functionals amongst the dual space members of a complex unital Banach algebra $A$. Specifically, we investigate when a continuous multiplicative map $\phi :A\rightarrow \mathbb C$, with values $\phi (x)$ belonging to the spectrum of $x$, is automatically linear. Our main result states that if $A$ is a $C^\star $-algebra, then $\phi $ always generates a corresponding character $\psi _\phi $ of $A$. It is then shown that $\phi $ shares many linear properties with its induced character. Moreover, if $A$ is a von Neumann algebra, then it turns out that $\phi $ itself is linear, and that it corresponds to its induced character.
