EQUIVARIANT E-THEORY FOR C*-ALGEBRAS 7

For this, it suffices to show that for every m G N there is a neighborhood U of the

identity in G and some n G N such that

sup | |

5

( / ( * ) ) - / ( * ) » - , for all g eU.

tn rn

For each m € N and n € N define a closed subset Wmn of G by

Wmn = {g € G : sup ||5(/(t)) - /(t)|| l/2m} .

Our hypothesis on / amounts to the assertion that for every ra, the union U^=1Wmn

contains a neighborhood of the identity element of G. For every ra, one of the

sets Wmn (n G N) must therefore contain a non-empty open subset of G. This

follows for instance from the Baire category theorem, although it is really the local

compactness of G which is of significance. But if Wmrim contains a non-empty open

set then supt

nm

\\g(f(t)) - f(t)\\ 1/ra for all g G W

m r i m

W^i

m

, and this set

contains a neighborhood of the identity of G. •

1.9.

DEFINITION.

Let 5 b e a G-G*-algebra. We henceforth denote by TJ5 the

G*-algebra of G-continuous, continuous and bounded functions from T = [1, oc) to

B and by %IB the quotient of this G*-algebra by the ideal of continuous functions

from T to B which vanish at infinity.

In view of Lemma 1.8, the new asymptotic algebra %IB is just the G*-subalgebra

of G-continuous elements in the former asymptotic algebra of B. If G is a discrete

group then the definition of %IB is not changed.

We conclude by setting down the obvious notion of equivariant asymptotic

family, and its relation to the notion of asymptotic morphism.

1.10.

DEFINITION.

An asymptotic family {ipt}te[i,oo) '• A — • B is equivariant

if

lim \\M9(o)) ~ g((ft(a))\\ = 0,

t—»oo

for all a G A and g G G.

1.11.

PROPOSITION.

Let A and B be G-C*-algebras. There is a one-to-one

correspondence between equivariant asymptotic morphisms from A to B and equiv-

alence classes of equivariant asymptotic families {Pt}te[i,x) '•