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:

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