Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$. Let ${\cal P}$ be the $C$-algebra spanned by $P$ and the identity $I$. Let $f$ be any *-homomorphism from ${\cal P}$ to the complex numbers. I think its possible to show that:

(1) $f$ is actually a map to the real numbers, because each $p\in P$ Hermitian.

(2) $f$ maps each generator to either $0$ or $1$, since each $p\in P$ is idempotent.

But what other structure does $f$ have? For example can $f$ be the constant function $1$ over the generators? (it can of course be the constant function $0$, but that's not interesting).

Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$. Let ${\cal P}$ be the $C$-algebra spanned by $P$ and the identity $I$. Let $f$ be any *-homomorphism from ${\cal P}$ to the complex numbers. I think its possible to show that:

(1) $f$ maps each generator to the real numbers, because each $p\in P$ Hermitian.

(2) $f$ maps each generator to either $0$ or $1$, since each $p\in P$ is idempotent.

But what other structure does $f$ have? For example can $f$ be the constant function $1$ over the generators? (it can of course be the constant function $0$, but that's not interesting).