Spectrum of a direct product of commutative $C^*$-algebras.

Question

Let $\{A_i: i \in I\}$ be a collection of commutative $C^*$-algebras. Given a commutative $C^*$-algebra $A$, denote its spectrum (consisting of non-zero algebra morphisms) by $\Omega(A)$. It is true that we have a homeomorphism $$\Omega\left(\bigoplus_{i \in I}^{c_0} A_i\right)\cong \bigsqcup_{i\in I}\Omega(A_i)$$ where the right is the disjoint union of topological spaces. This makes a fun exercise (for those who make it: where does the proof go wrong if we would replace the $c_0$-direct sum with the $\ell^\infty$-direct sum, i.e. the direct product?)

Is it possible to say something about $$\Omega\left(\prod_{i\in I}A_i\right)\cong \quad ?$$

I'm guessing not? For example, let $X$ be a discrete topological space. Then $$\beta X = \Omega(\ell^\infty(X))= \Omega\left(\prod_{x\in X} \mathbb{C}\right)$$ which suggests that an explicit description (in terms of the spectra of the $A_i$) is probably out of reach, but I just wanted to make sure I'm not missing something.

Answer 1

When the $A_i$ are unital, there is a simple description (at least, if you consider Stone-Cech compactifications simple): the spectrum of $\prod A_i$ is $\beta(\coprod \Omega(A_i))$, the Stone-Cech compactification of the disjoint union of their spectra. This follows immediately from Gelfand duality: since $\prod A_i$ is the categorical product of the $A_i$ (in the category of commutative unital $C^*$-algbras), its spectrum is the coproduct of their spectra in the category of compact Hausdorff spaces, which is just the Stone-Cech compactification of their disjoint union.

In the non-unital case things are a bit more awkward. You can deduce a somewhat complicated description by passing to the unitalizations. Namely, note that $U(\prod A_i)$ can be identified with the subalgebra of $\prod U(A_i)$ consisting of elements that have the same "unit part" on each coordinate, where $U$ denotes unitalization. So, $\Omega(U(\prod A_i))$ is the quotient of $\Omega(\prod U(A_i))=\beta(\coprod(\Omega(A_i)\cup\{\infty\}))$ by a corresponding equivalence relation, which is just the equivalence relation that collapses the closure of the set of all the points at $\infty$ to a single point. This can equivalently be described as $\beta(\bigvee(\Omega(A_i)\cup\{\infty\}))$, where $\bigvee$ denotes a wedge sum with $\infty$ as the basepoint. Thus, $\Omega(\prod A_i)$ can be constructed in the following way: take the one-point compactifications of the $\Omega(A_i)$, glue them together at their points at infinity, take the Stone-Cech compactification, and then remove the point at infinity.