Theo's [question][1] made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include:

• A $C^\ast$-[algebra][2] is a noncommutative [topological space][3] (cf. the [Gelfand transform][4]).
• The [multiplier algebra][5] of a nonunital $C^\ast$-algebra is the noncommutative [Stone-Cech compactification][6].
• A [spectral triple][7] is a noncommutative [manifold][8] (add some extra data to the spectral triple to get a noncommutative [Riemannian manifold][9] cf. arXiv:0810.2088).
• A [von Neumann algebra][10] is a noncommutative [measure space][11].

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.

EDIT: still trying to get the hyperlinks to behave. maybe this requires moderator attention? [1]: Which is the correct ring of functions for a topological space? [2]: http://en.wikipedia.org/wiki/C*-algebra [3]: http://en.wikipedia.org/wiki/Topological_space [4]: http://en.wikipedia.org/wiki/Gelfand_transform [5]: http://en.wikipedia.org/wiki/Multiplier_algebra [6]: http://en.wikipedia.org/wiki/Stone–Čech_compactification [7]: http://en.wikipedia.org/wiki/Spectral_triple [8]: http://en.wikipedia.org/wiki/Manifold [9]: http://en.wikipedia.org/wiki/Riemannian_manifold [10]: http://en.wikipedia.org/wiki/Von_Neumann_algebra [11]: http://en.wikipedia.org/wiki/Measure_space