Quasidiagonality was introduced by P.R. Halmos for operators and quickly
generalized to C^*-algebras. D. Voiculescu asked whether quasidiagonality
is a topological property. N. Salinas introduced a topology upon the
Kasparov groups KK_*(A,B) and showed that this topology is related to
relative quasidiagonality. This paper has three goals:
1: to show that the Kasparov KK-pairing is continuous with respect
to the Salinas topology (so that a KK-equivalence is a homeomorphism);
2. to identify QD_*(A,B), the quasidiagonal elements in KK_*(A,B),
in terms of K_*(A) and K_*(B);
3. to use these results in various applications.
Here is our central result. Let N denote the bootstrap category of
Rosenberg and Schochet.
THEOREM: Suppose that A is in N and A is quasidiagonal relative to B.
Then there is a natural isomorphism
QD_*(A,B) = PExt_Z^1 ( K_*(A) , K_*(B) ) _{*-1} .
Thus relative quasidiagonality is indeed a topological invariant,
answering D. Voiculescu's question. We also settle a question raised
by L.G. Brown on the relation between relative quasidiagonality and
the kernel of the natural map
Ext_Z^1 ( K_*(A) , K_*(B) ) \to Ext_Z^1 ( tK_*(A) , K_*(B) ) .
where tK_*(A) denotes the torsion subgroup of K_*(A). Finally we
establish a converse to a theorem of Davidson, Herrero, and Salinas,
giving conditions under which the quasidiagonality of A/K implies
the quasidiagonality of A.
Note:
This paper has been supplanted by The Fine Structure of
the Kasparov Groups II: Relative Quasidiagonality, and will not appear.
The dvi files have been removed.