# Operator Algebras and Algebraic K - Theory

#### Abstract

2. Extensions of C*-algebras. Let H be a separable infinite dimensional Hubert space, L(H) the algebra of bounded linear operators on H, K the ideal of compact operators, and A = L(H)/K. In [4] and [5] Ext(X) was defined as the set of equivalence classes of C*-algebra extensions, 0 —• K —• E —* C(X) —• 0, for X a compact metric space and C(X) the algebra of continuous complex functions on X. Ext(A^ was also described as unitary equivalence classes of *isomorphisms r: C(X) —•• A. It was shown that Ext(X) is a group and that it gives rise to a generalized homology theory which is related to ^-theory in roughly the same way as homology is related to cohomology. A Bott periodicity map, Per: Ext(SX) —• Ext(X), was defined and was proved to be injective for all X and surjective for smooth X. Also Ext(Jf) was given the structure of a not necessarily Hausdorff topological group, and the closure of the identity was called PExt(X).

### Cite this paper

@inproceedings{KTHEORY2007OperatorAA, title={Operator Algebras and Algebraic K - Theory}, author={ALGEBRAIC K-THEORY and Lawrence G. Brown}, year={2007} }