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The spectral extension property and extension of multiplicative linear functionals


Author: Michael J. Meyer
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
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1052578-X
Article copyright: © Copyright 1991 American Mathematical Society

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