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.