Operator K-theory

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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 K_0, equal to algebraic K_0, and K_1. 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, an n-dimensional vector bundle over a topological space X is associated to a projection in the C* algebra of matrix-valued—that is, 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 \otimes C(X), where K is the compact operators on a separable Hilbert space.

Hence, the K_0 group of a (not necessarily commutative) C* algebra A is defined as Grothendieck group generated by the Murray-von Neumann equivalence classes of projections in K \otimes C(X). K_0 is a functor from the category of C* Algebras and *-homomorphisms, to the category of abelian groups and group homomorphisms. The higher K-functors are defined via a C*-version of the suspension:

K_n(A) = K_0(S^n(A)) where

SA = C_0(0,1) \otimes A.

However, by Bott periodicity, it turns out that K_{n+2}(A) and K_n(A) are isomorphic for each n, and thus the only groups produced by this construction are K_0 and K_1.

The key reason for the introduction of K-theoretic methods into the study of C*-algebras was the Fredholm index: Given a bounded linear operator on a Hilbert space that has finite-dimensional kernel and co-kernel, one can associate to it an integer, which, as it turns out, reflects the 'defect' on the operator - i.e. the extent to which it is not invertible. The Fredholm index map appears in the 6-term exact sequence given by the Calkin algebra. In analysis on manifolds, this index and its generalizations played a crucial role in the index theory of Atiyah and Singer, where the topological index of the manifold can be expressed via the index of elliptic operators on it. Later on, Brown, Douglas and Fillmore observed that the Fredholm index was the missing ingredient in classifying essentially normal operators up to certain natural equivalence. These ideas, together with Elliott's classification of AF C*-algebras via K-theory led to a great deal of interest in adapting methods such as K-theory from algebraic topology into the study of operator algebras.

This, in turn, led to K-homology, Kasparov's bivariant KK-Theory, and, more recently, Connes and Higson's E-theory.