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The order topology for a von Neumann algebra
The order topology (resp. the sequential order topology
) on a poset is the topology that has as its closed sets
those that contain the order limits of all their order convergent nets (resp.
sequences). For a von Neumann algebra we consider the following three
posets: the self-adjoint part , the self-adjoint part of the unit ball
, and the projection lattice . We study the order topology (and
the corresponding sequential variant) on these posets, compare the order
topology to the other standard locally convex topologies on , and relate the
properties of the order topology to the underlying operator-algebraic structure
of
Vector lattices with a Hausdorff uo-Lebesgue topology
We investigate the construction of a Hausdorff uo-Lebesgue topology on a
vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal,
and what the properties of the topologies thus obtained are. When the vector
lattice has an order dense ideal with a separating order continuous dual, it is
always possible to supply it with such a topology in this fashion, and the
restriction of this topology to a regular sublattice is then also a Hausdorff
uo-Lebesgue topology. A regular vector sublattice of
for a semi-finite measure falls into this
category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is
then the convergence in measure on subsets of finite measure. When a vector
lattice not only has an order dense ideal with a separating order continuous
dual, but also has the countable sup property, we show that every net in a
regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology
always contains a sequence that is uo-convergent to the same limit. This
enables us to give satisfactory answers to various topological questions about
uo-convergence in this context.Comment: 37 pages. Minor changes; a few references added. Final version, to
appear in J. Math. Anal. App
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