Noncommutative topology in mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 is a term applied to the strictly C*-algebraic part of the noncommutative geometry

Noncommutative geometry

Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

 program. The program has its origins in the Gel'fand duality between the topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 of locally compact spaces and the algebraic structure

Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

 of commutative C*-algebras.

Several topological properties can be formulated as properties for the C*-algebras without making reference to commutativity or the underlying space

Space

Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

, and so have an immediate generalization.

Amongst these are compactness

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 (being unital), dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

 (real or stable rank), connectedness

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 (projectionless algebra) and K-theory

K-theory

In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...

. So we think of a noncommutative C*-algebra as the algebra of functions on a 'noncommutative space' which does not exist classically.

A major tool in the field is a bivariant version of K-theory called KK-theory

KK-theory

In mathematics, KK-theory is a common generalization both of K-homology and K-theory , as an additive bivariant functor on separable C*-algebras...

. It has a composition product



of which the ring structure in ordinary K-theory is a special case. The product gives the structure of a category to KK. It has been related to correspondences

Correspondence (mathematics)

In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings.* In general mathematics, correspondence is an alternative term for a relation between two sets...

of algebraic varieties.