Sequence of complex numbers converging to a value in the spectrum of an element in a commutative Banach algebra

Question

Let $A$ be a commutative unital Banach algebra over $\mathbb{C},$ $x \in A$ and $\mu \in \sigma(x),$ where $\sigma(x)$ denotes the spectrum of $x.$

Is is true that if $(x_n)_{n=1}^\infty$ is a sequence in $A$ converging to $x,$ then there exists a sequence $(\mu_n)_{n=1}^\infty$ in $\mathbb{C}$ converging to $\mu$ such that $\mu_n \in \sigma(x_n)$ for all $n \in \mathbb{N}?$

Any help will be appreciated.

Answer 1

Let $\hat A$ denote the space of characters (homomorphisms from $A$ to $\Bbb C$). Note that for any $x \in A$, $\sigma(x) = \{\gamma(x) : \gamma \in \hat A\}$.

We can select $\gamma \in \hat A$ such that $\gamma(x) = \mu$. Given a sequence $x_n \to x$, the sequence $\mu_n = \gamma(x_n)$ is such that $\mu_n \in \sigma(x_n)$ and $\mu_n \to \mu$.