Is the dual of the category of unital C*-algebras concretizable?

Question

The dual to the category of commutative unital C*-algebras is equivalent to the category of compact Hausdorff spaces, a concrete category. Can the dual to the category of unital C*-algebras also be concretized, yielding a concrete notion of noncommutative topological space? If so, is there a known description of this concrete category in terms of sets with some structure and maps that preserve that structure (like how a compact Hausdorff space is a set with a collection of its subsets satisfying certain properties)?

Answer 1

The opposite of any concretizable category is concretizable: if $U : C \to \text{Set}$ is a faithful functor, then so is the composite

$$C^{op} \xrightarrow{U^{op}} \text{Set}^{op} \xrightarrow{\text{Hom}(-, 2)} \text{Set}.$$

Really we can pick any concretization of $\text{Set}^{op}$ we want; the one above corresponds to thinking of $\text{Set}^{op}$ as a particular category of Boolean algebras (e.g. the profinite ones, but also referred to as the complete atomic Boolean algebras).

This isn't particularly satisfying. A natural strengthening of the notion of concretizability is to ask for $U$ to be representable; then a category is concretizable in this sense (the nLab calls it representably concrete) iff it has a generator. Unfortunately the opposite of the category of C*-algebras does not admit a generator; equivalently, C*-algebras do not admit a cogenerator. This is because there are simple C*-algebras of arbitrarily large cardinality. This same argument prevents e.g. $\text{Grp}^{op}$ and $\text{Ring}^{op}$ from being representably concrete.