  We introduce the notion of proper Kasparov cycles for Kasparov's
G-equivariant KK-theory for a general locally compact, second countable
topological group G. We show that for any proper Kasparov cycle, its induced
map on K-theory factors through the left-hand side of the Baum-Connes
conjecture. This allows us to upgrade the direct splitting method, a recent new
approach to the Baum-Connes conjecture which, in contrast to the standard gamma
element method (the Dirac dual-Dirac method), avoids the need of constructing
proper algebras and the Dirac and the dual-Dirac elements. We introduce the
notion of Kasparov cycles with Property (gamma) removing the G-compact
assumption on the universal space EG in the previous paper "Direct Splitting
Method for the Baum-Connes Conjecture". We show that the existence of a cycle
with Property (gamma) implies the split-injectivity of the Baum-Connes assembly
map for all coefficients. We also obtain results concerning the surjectivity of
the assembly map.
