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arXiv · 1808.09952

Some Character Generating Functions on Banach Algebras

R. Brits·F. Schulz·C. Toure
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Abstract

We consider a multiplicative variation on the classical Kowalski-S\l{}odkowski Theorem which identifies the characters among the collection of all functionals on a Banach algebra $A$. In particular we show that, if $A$ is a $C^*$-algebra, and if $\phi:A\mapsto\mathbb C$ is a continuous function satisfying $\phi(\mathbf 1)=1$ and $\phi(x)\phi(y) \in \sigma(xy)$ for all $x,y\in A$ (where $\sigma$ denotes the spectrum), then $\phi$ generates a corresponding character $\psi_\phi$ on $A$ which coincides with $\phi$ on the principal component of the invertible group of $A$. We also show that, if $A$ is any Banach algebra whose elements have totally disconnected spectra, then, under the aforementioned conditions, $\phi$ is always a character.

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R. Brits, F. Schulz, C. Toure. 2018-08-29. Some Character Generating Functions on Banach Algebras. https://arxiv.org/abs/1808.09952

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