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) '•
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