CHAPTER 1
Asymptotic Morphisms
Let B be a C*-algebra. Denote by %B the C*-algebra of continuous, bounded
functions from the locally compact space T [l,oo) into B. Denote by loB the
ideal in %B comprised of continuous functions from T to B which vanish in norm
at infinity.
1.1.
DEFINITION.
Let A and B be C*-algebras. The asymptotic algebra of B
is the quotient C*-algebra
2UB = T £ / T
0
£ .
An asymptotic morphism from A to B is a *-homomorphism from A into the C*-
algebra 2LB.
1.2.
REMARK.
In a moment, when we start to consider equivariant asymptotic
morphisms, we shall modify the definition of %IB very slightly (see Definition 1.9
below).
One can extract from a *-homomorphism (p: A %IB a family of functions
{Pt}te[i,oo) ' A- B
by composing p with any set-theoretic section from the quotient algebra %B to
TJB, then composing with the *-homomorphisms from %B to B given by evaluation
at t G [1, oc). The family {pt} so obtained has the following properties:
(i) for every a G A the map t i— pt(a), from [l,oo) into # , is continuous and
bounded; and
(ii) for every a,
a1
G A and A G C,
lim
Pt(a)* -Pt{a*)
pt(a) + \ipt{a') -ipt{a + \a')
pt(a)(pt(af) -(pt(aa/)
= 0 .
Conversely, a family of functions {Pt}te[i,oo) ' A -^ B satisfying these conditions
determines an asymptotic morphism from Ato B. Indeed if a G A then the function
t i— (pt(a) belongs to %B and by associating to a the class of this function in the
quotient %IB = %B/%QB we obtain a *-homomorphism from A into %B.
1.3.
DEFINITION.
Let A and B be C*-algebras. An asymptotic family mapping
A to B is a family of functions
{Pt}te[i,oo) '• A^B
satisfying the conditions (i) and (ii) above. Two asymptotic families {/?*} {^t} :
A—+B are equivalent if \imt^oc((pt(a) ^t(a)) = 0, for all a e A.
The following result is clear from the above discussion:
4
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