The following problem is a Exercise 18, section 8, chapter 7 of Conway's A Course in Functional Analysis.
I want to show that Banach algebra $C(\partial D )$ ($\partial D$ the boundary of $D=\{z:|z|\leq1 \}$) does not have a single generator.
definition of generation:If $\mathcal{A}$ is an abelian algebra, say that $a\in \mathcal{A}$ is generator of $ \mathcal{A}$ if $\{P(a): P\; \text{is a polynomial}\}$ is dense in $ \mathcal{A}$.
My Solution
$C[0,1]$ is isomorphic to $C(\partial D )$ and the $\exp(i\pi x)$ is a generator of $C[0,1]$.
but I am not sure how to proceed.