Noncommutative Topology

-algebras

It turns out that commutative -algebras and locally compact Hausdorff spaces are one and the same "thing" (this will be explained further ahead). Every commutative -algebra corresponds to a locally compact Hausdorff space and vice-versa and there is a correspondence between topological properties of spaces and -algebraic properties (see the noncommutative topology dictionary below).

The -algebraic properties and concepts are of course present in noncommutative -algebras too. Thus, although noncommutative -algebras cannot be associated with "standard" topological spaces, all the topological/ concepts are present. For this reason, this area of mathematics was given the name "noncommutative topology".

In this way, noncommutative topology can be seen as "topology, but without spaces".

The Commutative Case

Conversely, given a commutative -algebra , the Gelfand transform provides an isomorphism between and , for a suitable locally compact Hausdorff space .

Furthermore, there is an equivalence between the category of locally compact Hausdorff spaces and the category of commutative -algebras. This is the content of the Gelfand-Naimark theorem.

This equivalence of categories is one of the reasons for saying that locally compact Hausdorff spaces and commutative -algebras are the same thing. The other reason is the correspondence between topological and -algebraic properties, present in the following dictionary.

Noncommutative Topology Dictionary

We only provide a short list of easy-to-state concepts. Some correspondences of properties are technical and could not be easily stated here. Some of them originate new branches of "noncommutative mathematics", such as noncommutative measure theory.

Topological properties and concepts	-algebraic properties and concepts
topological space	-algebra
proper map	*-homomorphism
homeomorphism	*-isomorphism
open subset	ideal
closed subset	quotient
compact space	algebra with unit
compactification	unitization
one-point compactification	minimal unitization
Stone-Cech compactification	maximal unitization
second countable	separable
connected	projectionless
connected components and topological sums	projections
complement of singleton	maximal ideal
Radon measure	positive linear functional