## The spectral extension property and extension of multiplicative linear functionals

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- by Michael J. Meyer
- Proc. Amer. Math. Soc.
**112**(1991), 855-861 - DOI: https://doi.org/10.1090/S0002-9939-1991-1052578-X
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## 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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## Bibliographic Information

- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**112**(1991), 855-861 - MSC: Primary 46J05; Secondary 46H05
- DOI: https://doi.org/10.1090/S0002-9939-1991-1052578-X
- MathSciNet review: 1052578