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=\