EQUIVARIANT E-THEORY FOR C*-ALGEBRAS 5 1.4. PROPOSITION. There is a one-to-one correspondence between asymptotic morphisms from A to B and equivalence classes of asymptotic families {Pt}te[i,oo) '• A-B. D 1.5. REMARK. Our definition of asymptotic family is virtually the same as the original definition of asymptotic morphism ([11, Section 2] or [10]). We have added to the original the requirement that (ft {a) be a bounded function of t. In fact boundedness follows from the other parts of the definition of asymptotic family, but since the proof is not altogether simple (the one suggested in [11] is incomplete) it seems simpler to incorporate boundedness into our definition. Every *-homomorphism from A to B determines an asymptotic morphism from A to B by means of the following device: 1.6. DEFINITION. If B is a C*-algebra then denote by aB'.B 2LB the *- homomorphism which associates to b G B the class in %B of the constant function t\—be%B. Thus if ip: A B is a *-homomorphism then composing with OLB we ob- tain an asymptotic morphism from A to B. Of course, all we are doing here is constructing from if the constant asymptotic family {(pt ip}te[i,oo): A » B. This idea is slightly generalized by the observation that a continuous family of *- homomorphisms {(pt}te[i,oc): A B defines an asymptotic family, and hence an asymptotic morphism from A to B. The most important feature of asymptotic morphisms is that they induce ho- momorphisms of C*-algebra if-theory groups. From an asymptotic morphism (p: A $IB we obtain a homomorphism of abelian groups (p*:K*(A)^K*{B), in such a way that if / ? is actually a *-homomorphism from A to B then ip* is the usual induced map on if-theory groups. To see how this comes about, let p be an asymptotic morphism from A to B and for simplicity consider a class in KQ(A) represented by a projection p G A. Let {ift}te[i,oc) : A B be an asymptotic family corresponding to ip and consider the norm-continuous family of elements ft (ftip) m B. It has the property that lim{ / t 2 _ / t l=0. By an easy application of the functional calculus for C*-algebras, there is a norm- continuous family of actual projections et £ B such that \iirit^00(et ft) = 0. The projections et define a common class [e] G KQ(B) and we define K0(A) 3\p}^ [e] G K0(B). To give a fuller description of the induced map on if-theory, applicable to all classes in KQ(A) as well as all classes in K\(A), we note that the ideal %QB \ %B is contractible that is, it is homotopy equivalent in the C*-algebra sense to the zero C*-algebra (see the next chapter for a quick review of homotopy for C*-algebras). It follows [6,Chapter 4] that the projection map %B 2U3 induces an isomorphism in if-theory, and we define £* : K*(A) » K*(B) to be the composition K*(A) ^ K^B) ^ K*(ZB) Evaluation K*(B). at t=\
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