TY - JOUR
TI - Rank one subspaces of bimodules over maximal abelian selfadjoint
algebras
AU - Erdos, JA
AU - Katavolos, A
AU - Shulman, VS
JO - Journal of Functional Analysis
PY - 1998
VL - 157
TODO - 2
SP - 554-587
PB - ACADEMIC PRESS INC ELSEVIER SCIENCE
SN - 0022-1236, 1096-0783
TODO - 10.1006/jfan.1998.3274
TODO - null
TODO - Spaces of operators that are left and right modules over maximal abelian
selfadjoint algebras (masa bimodules For short) are natural
generalizations of algebras with cummutative subspace lattices. This
paper is concerned with density properties of finite rank operators and
of various Classes of compact operators in such modules. It is shown
that every finite rank operator of a norm closed masa bimodule H is in
the trace norm closure of the rank one subspace of H. An important
consequence is that the rank one subspace of a strongly reflexive masa
bimodule (that is, one which is the reflexive hull of its rank one
operators ) is dense in the module in the weak operator topology.
However, in contrast to the situation for algebras, it is shown that
such density need not hold in the ultraweak topology.
A new method of representing masa bimodules is introduced. This uses a
novel concept of all omega-topology. With the appropriate notion of
omega-support, a correspondence is established between reflexive masa
bimodules and their omega-supports. It is shown that, if a l(2)-closed
masa bimodule contains a tract class operator then it must contain rank
one operators. indeed, every such operator is in the l(2)-norm closure
of the rank one subspace of the module. Consequently the weak closure of
any masa bimodule of trace class operators is strongly reflexive.
However. the trace norm closure of the rank one subspace need not
contain all trace class operators of the module. Also, it is shown that
there exists a CSL algebra which contains no trace class operators yet
contains an operator belonging to l(p) for all p > 1. From this it
follows that a transitive bimodule spanned by the rank one operators it
contains need not be dense in l(p) for 1 less than or equal to p <
infinity.
As an application, it is shown that there exists a commutative subspace
lattice L such that L is non-synthetic but every weakly closed algebra
which contains a mesa and has invariant lattice L coincides with Alg L.
(C) 1998 Academic Press.
ER -