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=\
Previous Page Next Page