Continuity of the Kasparov Pairing and Relative Quasidiagonality, by Claude L. Schochet

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.

Claude L. Schochet <claude@math.wayne.edu>