About unital *-homomorphisms

Dear **MHF** members,

I have a *functional analysis* problem as follows.

**Problem**. Let denote the map on the unit circle of the complex plane .

For , let be the mapping for .

- Show that the map is a unital -homomorphism of to .

Conclude that is a unitary. - Show that , while .
- Show that is a proper dense subspace of .

Thanks.

**Notations**.

is the set of continuous functions.

is the set of bounded operators.

is the set of square integrable functions.

and stand for the spectrum and the point spectrum, respectively.

-homomorphism - Star-Homomorphism -- from Wolfram MathWorld

-algebra - C*-Algebra -- from Wolfram MathWorld