432 17. K-Homologyfundamental facts, but it is really a bare minimum treatment of the subject
(cf. [86]). We will only deal with unital C-algebras and unital maps.
If A is a C -algebra, then we define an essential extension of lK by A to
be a short exact sequence
O ---+ lK -.!'..+ & ~ A ---+ 0,
where & is a unital C -algebra, both i arid 1l' are -homomorphisms and
i(JK) <l E is an essential ideal (Definition 8.4.1). It is often more convenient
to work with the Busby invariant of an extension which, by definition, is
a -homomorphism^2 into the Calkin algebra. It is clear that any injective
-homomorphism into the Calkin algebra gives rise to an essential extension
- identify A with its image and let & be the pullback in JB(H) - but the
converse is also true. Indeed, suppose that
0---+JK-!.+&~A-+O
is an essential extension and lK is given as the compact operators on some
concrete Hilbert space Ji. The map i-^1 : i(JK) ---+ lK c JB(H) extends to a
-homomorphism & ---+ JB(Ji) and this map is faithful, since we assumed that
i(JK) is essential (if there were a kernel, it would be orthogonal to i(JK)).
Hence we get an induced -homomorphism a: A ---+ Q(Ji) by composing
the representation & ---+ JB(H) with the quotient map .IB(H) ---+ Q(Ji). Note,
however, that this procedure does not produce a unique map into the Calkin
algebra, since there are infinitely many ways to identify an abstract copy of
lK with the (concrete) compact operators on a Hilbert space Ji. Hence, at
this point, it is an abuse of terminology to refer to "the" Busby invariant
associated to an extension. Luckily, we can mod out by a natural equivalence
relation and eliminate this ambiguity.
Definition 17.1.1. Let U: 1i---+ K be a unitary operator. We will let
Adu: Q(H) ---+ Q(K)denote the isomorphism induced by the isomorphism JB(Ji) ---+ JB(K), TH
UTU*.
Definition 17.1.2. Two.unital injective *-homomorphisms cp: A---+ Q(Ji)
and 'ljJ : A ---+ Q (K) · are called equivalent if there exists a unitary operator
U: 1i ---+ K such that
Adu o 'P = 'lj;.
This gives the appropriate equivalence relation on maps to the Calkin
algebra and we use the term Busby invariant to refer to the equivalence class
of a map. Now we define an equivalence relation on the essential extensions.
(^2) Actually, an equivalence class - see Definition 17.1.2.