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).



Structure of Homomorphisms of commutative C^* algebra
This depends on the structure of your set of projections. For example, if they are pairwise orthogonal, then $f$ can send at most one of them to 1, because the sum of any two is again a projection. But if one of their ranges is non-trivial and is included in the ranges of the others, then $f$ can map them all to 1. In general the projections (like any elements in a commutative $C^\*$-algebra) can (and probably should) be regarded as continuous functions on the spectrum of the algebra; any homomorphism to $\mathbb C$ is then just evaluation at a point of the spectrum.


