  In these notes we develop a link between the Kadison-Singer problem and
questions about certain dynamical systems. We conjecture that whether or not a
given state has a unique extension is related to certain dynamical properties
of the state. We prove that if any state corresponding to a minimal idempotent
point extends uniquely to the von Neumann algebra of the group, then every
state extends uniquely to the von Neumann algebra of the group. We prove that
if any state arising in the Kadsion-Singer problem has a unique extension, then
the injective envelope of a C*-crossed product algebra associated with the
state necessarily contains the full von Neumann algebra of the group. We prove
that this latter property holds for states arising from rare ultrafilters and
$\delta$-stable ultrafilters, independent, of the group action and also for
states corresponding to non-recurrent points in the corona of the group.
