High Quality Content by WIKIPEDIA articles! In mathematics, operator K-theory is a variant of K-theory on the category of Banach algebras (In most applications, these Banach algebras are C*-algebras). Its basic feature that distinguishes it from algebraic K-theory is that it has a Bott periodicity. So there are only two K-groups, namely K0, 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 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, an n-dimensional vector bundle over a topological space X is associated to a projection in Mn(C(X)), where C(X) is the C* algebra of continuous functions over X. Also, it is known that homotopy equivalence of vector bundles translates to Murray-von Neumann equivalence of the associated projection.

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