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arXiv · 2509.03663

The Multiplicative Kowalski-Slodkowski Theorem for Hermitian Algebras

Rudi Brits·Muhammad Hassen·Cheick Toure
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Abstract

We prove, for Hermitian algebras, the multiplicative version of the Kowalski-S\l{}odkowski Theorem which identifies the characters among the collection of all complex valued functions on a Banach algebra $A$ in terms of a spectral condition. Specifically, we show that, if $A$ is a Hermitian algebra, and if $\phi:A\mapsto\mathbb C$ is a continuous function satisfying $\phi(x)\phi(y) \in \sigma(xy)$ for all $x,y\in A$ (where $\sigma$ denotes the spectrum), then either $\phi$ or $-\phi$ is a character of $A$; of course the converse holds as well. Our proof depends fundamentally on the existence of positive elements and square roots in these algebras.

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Rudi Brits, Muhammad Hassen, Cheick Toure. 2025-09-03. The Multiplicative Kowalski-Slodkowski Theorem for Hermitian Algebras. https://arxiv.org/abs/2509.03663

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