KK-theory - Definition

Definition

The following definition is quite close to the one originally given by Kasparov. This is the form in which most KK-elements arise in applications.

Let A and B be separable C*-algebras, where B is also assumed to be σ-unital. The set of cycles is the set of triples (H, ρ, F), where H is a countably generated graded Hilbert module over B, ρ is a *-representation of A on H as even bounded operators which commute with B, and F is a bounded operator on H of degree 1 which again commutes with B. They are required to fulfill the condition that

for a in A are all B-compact operators. A cycle is said to be degenerate if all three expressions are 0 for all a.

Two cycles are said to be homologous, or homotopic, if there is a cycle between A and IB, where IB denotes the C*-algebra of continuous functions from to B, such that there is an even unitary operator from the 0-end of the homotopy to the first cycle, and a unitary operator from the 1-end of the homotopy to the second cycle.

The KK-group KK(A, B) between A and B is then defined to be the set of cycles modulo homotopy. It becomes an abelian group under the direct sum operation of bimodules as the addition, and the class of the degenerate modules as its neutral element.

There are various, but equivalent definitions of the KK-theory, notably the one due to Joachim Cuntz which eliminates bimodule and 'Fredholm' operator F from the picture and puts the accent entirely on the homomorphism ρ. More precisely it can be defined as the set of homotopy classes

,

of *-homomorphisms from the classifying algebra qA of quasi-homomorphisms to the C*-algebra of compact operators of an infinite dimensional separable Hilbert space tensored with B. Here, qA is defined as the kernel of the map from the C*-algebraic free product A*A of A with itself to A defined by the identity on both factors.