1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

17.1. BDF preliminaries 433

Definition 17.1.3. Two essential extensions, 0 -----+ lK ~ £ 1 ~ A -----+ 0 and

0 -----+ lK ~ £2 ~ A -----+ 0, are equivalent if there exists a *-isomorphism

g?: £1 -----+ £2 such that g?(i1(K)) = i2(lK) and the induced isomorphism of

quotients is the identity on A - i.e., there exists a commutative diagram

O ---+ lK ---+ £1 ---+ A ---+ O

O ---+ lK ---+ £2 ---+ A ---+ 0.
If we identify equivalent extensions, then there is a one-to-one corre-
spondence between Busby invariants and essential extensions. This is not
immediate, though not too difficult, and we give some hints in Exercises
17.1.1and17.1.2. For our purposes, the Busby invariant will be more use-
ful, so the reader can safely concentrate on maps to the Calkin algebra.

Having dispensed with the technicalities, here's the BDF extension semi-

group.

Definition 17.1.4. Let A be a unital separable C*-algebra. Then Ext(A)

denotes the set of (equivalence classes of) essential extensions of A by lK -

i.e., Ext(A) is just the set of Busby invariants. If cp: A -----+ Q(H) is a unital

*-monomorphism, then [cp] E Ext(A) will denote its equivalence class.

As mentioned above, Ext(A) is always a semigroup: If cp: A-----+ Q(H)
and 'lj;: A-----+ Q(JC) are unital -monomorphisms, there is a natural diagonal
embedding of Q(H) E9 Q(K) into Q(H E9 K) and hence we get a unital -
monomorphism cp E9 'lj;: A-----+ Q(H E9 JC) via the embedding Q(H) E9 Q(JC) <-t
Q(H E9 K); this allows us to define addition in Ext(A) as
[cp] +['I/;] = [cp E9 'I/;]
(you'll have the opportunity in Exercise 17.1.3 to check that this is well-
defined and turns Ext (A) into an abelian semigroup).
Though it is far from obvious, it turns out that Ext(A) always has a
neutral element, given by a trivial extension.
Definition 17.1.5. A unital -monomorphism cp: A-----+ Q(H) is called triv-
ial if it has a unital -homomorphic lifting - i.e., there exists a unital repre-
sentation O": A -----+ ll:B(H) such that 7r?-l o O" = cp, where 7r?-l: ll:B(H) -----+ Q(H) is
the quotient map.

Note that, since cp is injective, O": A -----+ ll:B(H) is necessarily faithful and

essential. Hence Voiculescu's Theorem (Theorem 1.7.3) implies that all triv-

ial extensions give rise to the same element in Ext(A). In fact, more is true.

Theorem 17.1.6 (Voiculescu). If 0 E Ext( A) denotes the class of a trivial

extension, then 0 is an additive identity for Ext(A).