We consider a paving property for a maximal abelian
-subalgebra
(MASA)
in a von
Neumann algebra ,
that we call
so-paving, involving approximation in the
so-topology,
rather than in norm (as in classical Kadison–Singer paving). If
is the range of a
normal conditional expectation, then
so-paving is equivalent to norm paving in the ultrapower
inclusion
.
We conjecture that any MASA in any von Neumann algebra satisfies
so-paving. We
use work of Marcus, Spielman and Srivastava to check this for all MASAs in
, all
Cartan subalgebras in amenable von Neumann algebras and in group measure space
II factors arising
from profinite actions. By earlier work of Popa, the conjecture also holds true for singular
MASAs in II
factors, and we obtain here an improved paving size
,
which we show to be sharp.
Keywords
Kadison–Singer problem, paving, von Neumann algebra,
maximal abelian subalgebra