Wouldn't you just use the standard thing where you map elements a and b to the same value in the image and then show that a and b must be the same? That is f(a) = f(b) implies a = b.
Let be the closed ideal consisting of functions
Where denotes the continuous functions vanishing at infinity
from to the unit circle,
Show that the map
is an injective *-homomorphism
Linearity is easy, so is multiplicativity, but how will I show that the adjoint is preserved and that the map is injective?