8 research outputs found
k-Extreme Points in Symmetric Spaces of Measurable Operators
Let be a semifinite von Neumann algebra with a faithful,
normal, semifinite trace and be a strongly symmetric Banach function
space on . We show that an operator in the unit sphere of
is -extreme, , whenever its
singular value function is -extreme and one of the following
conditions hold (i) or (ii)
and , where and
are null and support projections of , respectively. The converse is
true whenever is non-atomic. The global -rotundity property
follows, that is if is non-atomic then is -rotund if and
only if is -rotund. As a consequence of the
noncommutive results we obtain that is a -extreme point of the unit ball
of the strongly symmetric function space if and only if its decreasing
rearrangement is -extreme and . We conclude
with the corollary on orbits and . We get that is a
-extreme point of the orbit , , or
, , , if and only if
and . From this we obtain a
characterization of -extreme points in Marcinkiewicz spaces.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s00020-014-2206-
Completely positive compact operators on non-commutative symmetric spaces
Under natural conditions, it is shown that a completely positive operator between two non-commutative symmetric spaces of ? -measurable operators which is dominated in the sense of complete positivity by a completely positive compact operator is itself compact.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc