# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## The spectral extension property and extension of multiplicative linear functionalsHTML articles powered by AMS MathViewer

by Michael J. Meyer
Proc. Amer. Math. Soc. 112 (1991), 855-861 Request permission

## Abstract:

Let $\mathcal {A}$ be a commutative Banach algebra. Denote the spectral radius of an element $a$ in $\mathcal {A}$ by ${\rho _\mathcal {A}}(a)$. An extension of $\mathcal {A}$ is a Banach algebra $\mathcal {B}$ such that $\mathcal {A}$ is algebraically, but not necessarily continuously, embedded in $\mathcal {B}$. We view $\mathcal {A}$ as a subalgebra of $\mathcal {B}$. If $\mathcal {B}$ is an extension of $\mathcal {A}$ then $S{p_\mathcal {B}}(a) \cup \{ 0\} \subseteq S{p_\mathcal {A}}(a) \cup \{ 0\}$ and thus ${\rho _\mathcal {B}}(a) \leq {\rho _\mathcal {A}}(a),\forall a \in \mathcal {A}$. Let us say that $\mathcal {A}$ has the spectral extension property if ${\rho _\mathcal {B}}(a) = {\rho _\mathcal {A}}(a)$ for all $a \in \mathcal {A}$ and all extensions $\mathcal {B}$ of $\mathcal {A}$, that $\mathcal {A}$ has the strong spectral extension property if $S{p_\mathcal {B}}(a) \cup \{ 0\} = S{p_\mathcal {A}}(a) \cup \{ 0\}$ for all $a \in \mathcal {A}$ and all extensions $\mathcal {B}$ of $\mathcal {A}$, and that $\mathcal {A}$ has the multiplicative Hahn-Banach property if every multiplicative linear functional $\chi$ on $\mathcal {A}$ has a multiplicative linear extension to every commutative extension $\mathcal {B}$ of $\mathcal {A}$. We give characterizations of these properties for semisimple commutative Banach algebras.
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