Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include:
- A $C^\ast$-algebra is a noncommutative topological space (cf. the Gelfand transform).
- The multiplier algebra of a nonunital $C^\ast$-algebra is the noncommutative Stone-Cech compactification.
- A spectral triple is a noncommutative manifold (add some extra data to the spectral triple to get a noncommutative Riemannian manifold cf. arXiv:0810.2088).
- A von Neumann algebra is a noncommutative measure space.
Are there any other good examples? If you know more in operator algebras, that's great too.
EDIT: these algebras should be considered as various functions spaces for noncommutative spaces as per @Yemon's answer. I'm going to leave the above text as is unless there are requests for another edit.
Best Answer
Quantales are another sort of noncommutative space.
A general remark: It was said in two other answers that "a good category of spaces should be self dual". If by "self dual" you mean "equivalent to its opposite category", then this should rather not be the case for a category of spaces. Dual cats to cats of spaces are rather of algebraic character, a resonable definition for being "of algebraic character" is being "locally presentable". But a theorem of Gabriel/Ulmer states that if the opposite of a locally presentable cat is locally presentable again, then the cat is preorder (i.e. we are in a trivial case) - this can be seen as a mathematical statement reflecting the duality between algebra and geometry