Let $\mathbb{T}:=\lbrace z\in\mathbb{C}:|z|=1\rbrace$ and $A$ be the subalgebra of $C(\mathbb{T})=C(\mathbb{T};\mathbb{C})$ consisting of the Laurent polynomials $\lbrace\sum_{k=-n}^na_kz^k:a_k\in\mathbb{C}\rbrace$.
I have established that $A$ is dense in $C(\mathbb{T})$. I assume from this that $C(\mathbb{T})$ must be the space of convergent Laurent series? How can I prove that the closure of $A$ is the space of convergent Laurent series?
If $f\in C(\mathbb T),$ does there exist a Laurent series $\sum_{-\infty}^{\infty}a_nz^n$ such that $\sum_{-N}^{N}a_nz^n$ converges to $f$ uniformly on $\mathbb T$ as $N\to \infty?$ If this is your question, then the answer is no. If it were true, then the Laurent series would equal the Fourier series of $f.$ But it is well known there are functions in $C(\mathbb T)$ whose Fourier series diverge at "lots of points".