Noncommutative topology
Open-knowledge reference entry
In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the duality of the category of locally compact Hausdorff spaces and the category of commutative C*-algebras. Noncommutative topology is related to analytic noncommutative geometry.
Examples
The premise behind noncommutative topology is that a noncommutative C*-algebra can be treated like the algebra of complex-valued continuous functions on a 'noncommutative space' which does not exist classically. Several topological properties can be formulated as properties of the C*-algebras without making reference to commutativity or the underlying space, and so have an immediate generalization.
Among these are:
compactness (unital)
σ-compactness (σ-unital)
dimension (real or stable rank)
connectedness (projectionless)
extremally disconnected spaces (AW*-algebras)
Individual elements of a commutative C*-algebra correspond with continuous functions. And so certain types of functions can correspond to certain properties of a C*-algebra. For example, self-adjoint elements of a commutative C*-algebra correspond to real-valued continuous functions. Also, projections (i.e. self-adjoint idempotents) correspond to indicator functions of clopen sets.
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