Operator K-theory

In mathematics, operator K-theory is a noncommutative analogue of topological K-theory for Banach algebras with most applications used for C*-algebras.

Operator K-theory resembles topological K-theory more than algebraic K-theory. In particular, a Bott periodicity theorem holds. So there are only two K-groups, namely K₀, which is equal to algebraic K₀, and K₁. As a consequence of the periodicity theorem, it satisfies excision. This means that it associates to an extension of C*-algebras to a long exact sequence, which, by Bott periodicity, reduces to an exact cyclic 6-term-sequence.

Operator K-theory is a generalization of topological K-theory, defined by means of vector bundles on locally compact Hausdorff spaces. Here, a vector bundle over a topological space X is associated to a projection in the C* algebra of matrix-valued—that is, ${\displaystyle M_{n}(\mathbb {C} )}$-valued—continuous functions over X. Also, it is known that isomorphism of vector bundles translates to Murray-von Neumann equivalence of the associated projection in K ⊗ C(X), where K is the compact operators on a separable Hilbert space.

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