nLab operator K-theory

Idea

A variant of algebraic K-theory for operator algebras (Banach algebras and in particular C-star algebras) is operator K-theory: it generalizes topological K-theory via the Serre-Swan theorem from commutative C-star algebras to general $C^\ast$-algebras.

Related concepts

geometric context	universal additive bivariant (preserves split exact sequences)	universal localizing bivariant (preserves all exact sequences in the middle)	universal additive invariant	universal localizing invariant
noncommutative algebraic geometry	noncommutative motives $Mot_{add}$	noncommutative motives $Mot_{loc}$	algebraic K-theory	non-connective algebraic K-theory
noncommutative topology	KK-theory	E-theory	operator K-theory	…

References

A standard textbook account is in section 2 of

• Bruce Blackadar, K-Theory for Operator Algebras (pdf)

Other introductions include

• M. Rørdam, F. Larsen, N. Laustsen, An introduction to K-theory for $C^\ast$-algebras, London Mathematical Society Student Textx