Structure of Homomorphisms of commutative C^* algebra

Question

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).

Answer 1

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.

Answer 2

The algebra $\mathcal P$ is isomorphic to continuous functions on its spectrum, which will be a compact Hausdorff totally disconnected space. Your projections will be characteristic functions of clopen subsets and as Andreas said, you homomorphisms will basically be evaluation at a point. Alternatively, the projections of $\mathcal P$ form a boolean algebra, the spectrum of $\mathcal P$ is the Stone space of ultrafilters on this boolean algebra. Given an ultrafilter, the corresponding homomorphism sends a projection to 1, if the ultrafilter contains that projection.