17.4. Topology on Ext 4391 7.4. Topology on Ext
Finally, we would like to show that noninvertible elements in Ext are often
abundant. This will follow from topological considerations and a result of
Dadarlat (whom we thank for some helpful conversations regarding his work
cited below).
First, though, we define a quotient of Ext by relaxing the equivalence
relation on Busby invariants. Namely, we say two unital -monomorphisms
cp, 'ljJ: A ----+ Q(1-i) are weakly equivalent if there exists a unitary u E Q(1-i)
such that ucp(a)u = 'lj;(a) for all a E A. Defining Extw(A) to be weak
equivalence classes of unital *-monomorphisms into the Calkin algebra, it
is obvious that Extw(A) is a quotient of Ext(A) and one can check that
invertibility in Extw(A) is still characterized by the existence of a u.c.p.
splitting (as in Proposition 17.1.8).
There is a natural topology on Extw: the quotient of the pointwise-
convergence topology on the set of unital *-monomorphisms from A into
Q(1-i). This topology is not Hausdorff, but it is pseudometrizable since A is
separable. To be more precise, we fix a dense sequence x1, x2, ... in the unit
ball of A and define a pseudometric d on Extw (A) by
00 1
d( [ cp], ['lj;]) = inf"°' u 62i -:-II ucp( Xi)u* - 'l/J (xi) II,
i=lwhere u E Q(H) is a unitary. The topology on Extw(A) defined by this
pseudometric is called the (Larry) Brown-Salinas topology. It is independent
of the choice of dense sequence, of course. We note that the group Extw(A)-^1
of invertible elements is closed in this topology, by Lemma C.2. It is not very
difficult to show that this topology makes Extw(A) a topological semigroup:
Addition and inversion are continuous operations. A theorem of Dadarlat
(which depends on KK-theory; hence we won't prove it - see [52]) states
that Extw(A)-^1 is separable. In this section, we'll show that the set of
quasidiagonal extensions is nonseparable in many cases; hence noninvertible
elements abound.
Definition 17.4.1. A unital *-monomorphism cp: A----+ Q(1-i) is said to be
quasidiagonal if 7f-^1 (cp(A)) c IIB(1-i) is a quasidiagonal set of operators. We
say [cp] E Extw(A) is quasidiagonal if cp is.^7
Lemma 17.4.2. If cp: A ----+ Q(1-i) is quasidiagonal and liftable, then [cp]
belongs to the closure of the neutral element in Extw(A).
7It isn't entirely obvious, but this is well-defined, as one should check (or see [171, Corollary
2.6]).