Noncommutative topology

In

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 for the C*-algebras without making reference to commutativity or the underlying space, and so have an immediate generalization. Among these are:

• unital
  )
• 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,

There are certain examples of properties where multiple generalizations are possible and it is not clear which is preferable. For example, probability measures can correspond either to 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.

K-theory

One of the major examples of this idea is the generalization of topological K-theory to noncommutative C*-algebras in the form of operator K-theory.

A further development in this is a bivariant version of K-theory called KK-theory, which has a composition product

${\displaystyle KK(A,B)\times KK(B,C)\rightarrow KK(A,C)}$

of which the

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