Editing Noncommutative topology (section)

== Examples ==

The premise behind noncommutative topology is that a noncommutative C*-algebra can be treated like the algebra of complex-valued [[continuous function]]s on a 'noncommutative space' which does not exist classically. Several topological properties can be formulated as properties for the C*-algebras without making reference to commutativity or the underlying space, and so have an immediate generalization.
Among these are:
* [[compact space|compactness]] ([[unital algebra|unital]])
* [[σ-compact space|σ-compactness]] ([[σ-unital algebra|σ-unital]])
* [[Lebesgue covering dimension|dimension]] ([[real rank (C*-algebras)|real]] or [[topological stable rank|stable rank]])
* [[connected space|connectedness]] ([[projectionless C*-algebra|projectionless]])
* [[extremally disconnected space]]s ([[AW*-algebra]]s)

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, [[projection (linear algebra)#Orthogonal projections|projections]] (i.e. self-adjoint [[idempotent]]s) correspond to [[indicator function]]s of [[clopen set]]s.

Categorical constructions lead to some examples. For example, the [[coproduct]] of spaces is the [[Disjoint union (topology)|disjoint union]] and thus corresponds to the [[direct sum of algebras]], which is the [[product (category theory)|product]] of C*-algebras. Similarly, [[product topology]] corresponds to the coproduct of C*-algebras, the [[tensor product of algebras]]. In a more specialized setting,
compactifications of topologies correspond to unitizations of algebras. So the [[Alexandroff extension|one-point compactification]] corresponds to the minimal unitization of C*-algebras, the [[Stone–Čech compactification]] corresponds to the [[multiplier algebra]], and [[corona set]]s correspond with [[corona algebra]]s.

There are certain examples of properties where multiple generalizations are possible and it is not clear which is preferable. For example, [[probability measure]]s can correspond either to [[state (functional analysis)|states]] or tracial states. Since all states are vacuously 
tracial states in the commutative case, it is not clear whether the tracial condition is necessary to be a useful generalization.