Printed: Jun 2013
In this paper we analyze states on C*-algebras and
their relationship to filter-like structures of projections and
positive elements in the unit ball. After developing the basic theory
we use this to investigate the Kadison-Singer conjecture, proving its
equivalence to an apparently quite weak paving conjecture and the
existence of unique maximal centred extensions of projections coming
from ultrafilters on the natural numbers. We then prove that Reid's
positive answer to this for q-points in fact also holds for rapid
p-points, and that maximal centred filters are obtained in this case.
We then show that consistently such maximal centred filters do not
exist at all meaning that, for every pure state on the Calkin algebra,
there exists a pair of projections on which the state is 1, even
though the state is bounded strictly below 1 for projections below
this pair. Lastly we investigate towers, using cardinal invariant
equalities to construct towers on the natural numbers that do and do
not remain towers when canonically embedded into the Calkin algebra.
Finally we show that consistently all towers on the natural numbers
remain towers under this embedding.
C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towers
46L03 - unknown classification 46L03
03E35 - Consistency and independence results