Operator K-theory

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 \( M_n(C(X)) \), 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 in K⊗C(X), where K is 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⊗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. 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.
References

Rordam, M.; Larsen, Finn; Laustsen, N. (2000), An introduction to K-theory for C∗-algebras, London Mathematical Society Student Texts, 49, Cambridge University Press, ISBN 978-0-521-78334-7; 978-0-521-78944-8

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