About unital *-homomorphisms

Dear **MHF** members,

I have a *functional analysis* problem as follows.

**Problem**. Let $\displaystyle z$ denote the map $\displaystyle z\mapsto z$ on the unit circle $\displaystyle \mathbb{T}$ of the complex plane $\displaystyle \mathbb{C}$.

For $\displaystyle f\in\mathrm{C}(\mathbb{T})$, let $\displaystyle M_{f}\in\mathcal{B}(L^{2}(\mathbb{T}))$ be the mapping $\displaystyle g\mapsto fg$ for $\displaystyle g\in L^{2}(\mathbb{T})$.

- Show that the map $\displaystyle M$ is a unital $\displaystyle \ast$-homomorphism of $\displaystyle \mathrm{C}(\mathbb{T})$ to $\displaystyle \mathcal{B}(L^{2}(\mathbb{T}))$.

Conclude that $\displaystyle M_{z}$ is a unitary. - Show that $\displaystyle \sigma(M_{z})=\mathbb{T}$, while $\displaystyle \sigma_{\mathrm{p}}(M_{z})=\emptyset$.
- Show that $\displaystyle \mathrm{Im}(M_{z}-\mathrm{I})$ is a proper dense subspace of $\displaystyle L^{2}(\mathbb{T})$.

Thanks.

**Notations**.

$\displaystyle \mathrm{C}$ is the set of continuous functions.

$\displaystyle \mathcal{B}$ is the set of bounded operators.

$\displaystyle L^{2}$ is the set of square integrable functions.

$\displaystyle \sigma$ and $\displaystyle \sigma_{\mathrm{p}}$ stand for the spectrum and the point spectrum, respectively.

$\displaystyle \ast$-homomorphism - Star-Homomorphism -- from Wolfram MathWorld

$\displaystyle \ast$-algebra - C*-Algebra -- from Wolfram MathWorld