Does the category of $C^*$-algebras embed into the category of frames?

Question

The category of commutative $C^*$-algebras is dual to the category of LCH topological spaces. My understanding is that authors in operator algebras often understand a 'noncommutative LCH space' to be an object in the dual of the category of (general) $C^*$-algebras. This way of defining spaces resembles the approach of pointless topology, where the category of locales is defined to be the dual of the category of frames. This leads me to ask the title question, which can also be phrased as "are noncommutative spaces (in the $C^*$-algebra sense) locales?"

Answer 1

No, the opposite category of $C^*$-algebras is a generalization of topological spaces in a very different direction than locales are. For instance, there are infinitely many different embeddings of $C^*$-algebras $\mathbb{C}^2\to M_2(\mathbb{C})$. If $M_2(\mathbb{C})$ corresponded to some locale $X$, then these embeddings should give infinitely many different clopen subsets of $X$ (i.e., complemented elements of the corresponding frame, which form a sublattice which is a Boolean algebra). In particular, then, if $X$ has infinitely many different clopen subsets, it should have a surjective map to a 3-point discrete space (just partition it into three nonempty clopen subsets). But $\mathbb{C}^3$ does not embed into $M_2(\mathbb{C})$.