6 ERIK GUENTNER, NIGEL HIGSON, AND JODY TROU T
Again, if p : A » B is a *-homomorphism, viewed as an asymptotic morphism via
as as above, then this construction gives the usual induced map on i^-theory.
If p is an asymptotic morphism from A to £ , and if ^ is a *-homomorphism
from A\ to A then we can compose ^ with p to obtain an asymptotic morphism from
A\ to B. Similarly, a *-homomorphism from B to B\ induces a *-homomorphism
from %B to 2LE?i (the functoriality of the asymptotic algebra will be considered
more fully in the next chapter) and once again we can compose with p to obtain an
asymptotic morphism from A to B\. It is easy to check that the induced map on
K-theory just defined is compatible with these compositions, in the obvious sense.
The main achievement of the theory of asymptotic morphisms is the construc-
tion of a composition operation for a pair of asymptotic morphisms (not just one
asymptotic morphism and one *-homomorphism, as we have just considered) which
is compatible with the composition of induced maps on if-theory. This is not a
simple matter; for instance the most obvious attempt to compose asymptotic fami-
lies would be to form Lptotpt, but this fails to produce an asymptotic family. In fact
the correct definition of composition is only well defined up to a suitable notion of
homotopy for asymptotic morphisms. The construction will be given in the next
chapter.
We now consider the definition of an equivariant asymptotic morphism between
two G-C*-algebras. Throughout the paper we shall denote by G a locally compact,
second countable, Hausdorff topological group. In the later chapters we will limit
ourselves to consideration of discrete groups, but for the next several chapters this
restriction will not apply. In several places we could assume less of G—for instance
we could drop the hypothesis of second countability—but for simplicity we do not
do so.
1.7.
DEFINITION.
A G-C*-algebra is a C*-algebra A equipped with a continu-
ous action G x A A by *-automorphisms. An equivariant asymptotic morphism
from one G-C*-algebra A to another one B is an equivariant *-homomorphism from
A to the asymptotic C*-algebra 2LB.
It should be noted that while the action of G on B passes in a natural way
to an action by *-automorphisms on the asymptotic algebra 2LB, this action is
seldom continuous; G acts continuously on %QB but not on XJB or 2LB. Never-
theless, an asymptotic morphism from A to B necessarily maps A into the C*-
subalgebra comprised of elements b G %IB which are G-continuous, in the sense
that the map g \- g(b) is continuous from G to B. Indeed it is clear that any
equivariant *-homomorphism between C*-algebras equipped with actions of G by
*-automorphisms must map G-continuous elements to G-continuous elements. This
prompts us to alter the definition of %IB a little. Note first the following not alto-
gether simple fact:
1.8.
LEMMA.
If a bounded continuous function f:T B determines a G-
continuous element of the asymptotic algebra %IB then f is a G-continuous element
of IB.
PROOF .
We must show that for all e 0 there exists a neighborhood U of the
identity in G such that
llfl(/()) " /(Oil £ for all* T and geU.
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