Characterization of small Banach subalgebras

Question

Let $A$ be a unital Banach algebra and $x \in A$ nonzero. We can consider the subalgebra $B$ of $A$ generated by $\{1,x\}$. This is the norm closure of the subspace of polynomials in $x$. So for any $y \in B$ there exists a sequence of polynomials $p_n(z) \in \mathbb C[z]$ such that

$$\lim_{n\rightarrow \infty} ||p_n(x) - y||=0. $$ In particular we notice that any power series with radius of convergence less than $||x||$ evaluated at $x$ is in $B$. My question is whether or not every element of $B$ can be written as a power series in $x$. It seems like this might follow easily from the classical proof that a normed vector space is complete if and only if every absolutely convergent series converges, but I haven't been able to work through the details.

Answer 1

Not necessarily. If $A=C[0,1]$, the space of continuous functions on $[0,1]$ ith real values, and $x$ is the map $t\mapsto t$, then $B=A$. But not all element of $A$ can be written as a power series (just take a map which is not $C^1$).

Answer 2

Another example:

Let $A$ be the algebra of functions on the closed unit disk, continuous on the closed disk and analytic on the interior, with pointwise operations and sup norm, and let $x$ be the map $z\mapsto z$. Then $A=B$. In this case every element of $A$ has a power series converging pointwise on the interior of the disk, but they are not all norm convergent (the convergence may be nonuniform even for a series that converges everywhere on the closed disk). Thus $A$ gives another counterexample.

However, the sequence of Cesàro means of the power series of each $f\in A$ converges in norm to $f$.