The spectral extension property and extension of multiplicative linear functionals

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.