The Fine Structure of the Kasparov Groups II Relative Quasidiagonality, by C.L. Schochet

Quasidiagonality was introduced by P.R. Halmos for operators and quickly generalized to C^*-algebras. D. Voiculescu asked how quasidiagonality (in its various forms) was related to topological invariants. N. Salinas systematically studied the topology of the Kasparov groups KK_*(A,B) and showed that this topology is related to relative quasidiagonality.

In this paper we identify QD_*(A,B), the quasidiagonal elements in KK_*(A,B), in terms of K_*(A) and K_*(B), and we use these results in various applications. Here is our central result. Let N denote the bootstrap category.

Theorem: Suppose that A \in N and A is quasidiagonal relative to B. Then there is a natural isomorphism

QD_*(A,B) \cong Pext_{\Bbb Z}^1( {K_*(A)},{K_*(B)}) _{*-1}

Thus for A \in N relative quasidiagonality is indeed a topological invariant. We also settle a question raised by L.G. Brown on the relation between relative quasidiagonality and the kernel of the natural map

Ext_{\Bbb Z}^1 ( K_*(A), K_*(B) ) \to Ext_{\Bbb Z}^1 ( K_*(A)_t, K_*(B) )

where K_*(A)_t 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, and to a small extent "Fine Structure I" replace and very substantially extend the preliminary preprint Continuity of the Kasparov pairing and relative quasidiagonality which will not appear.


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