$C(\mathbb{T})$ embed in $B(\mathcal{H})$ for some Hilbert space $\mathcal{H}$

Question

I was looking at examples of $C^*$ algebras. If $\mathcal{H}$ is a Hilbert space, then $B(\mathcal{H})$ is a standard example. The space of continuous functions on the unit circle $\mathbb{T}$ (denoted $C(\mathbb{T}) $) is also another example.

I know that we can embed $C(\mathbb T)\subset B(\mathcal H)$ for some Hilbert space $\mathcal{H}$. Is there a standard choice for $B(\mathcal{H})$ in this case? If not, is $\mathcal{H}$ necessarily infinite dimensional or can we choose it to be so?

Answer 1

HINT:

I would try $\mathcal{H} = L^2(\mathbb{T})$, and $C(\mathbb{T})\ni f \mapsto L_f\in B(\mathcal{H})$ the multiplication by $f$.