Relation With The Weak (operator) Topology
The ultraweak topology is similar to the weak operator topology. For example, on any norm-bounded set the weak operator and ultraweak topologies are the same, and in particular the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology.
One problem with the weak operator topology is that the dual of B(H) with the weak operator topology is "too small". The ultraweak topology fixes this problem: the dual is the full predual B*(H) of all trace class operators. In general the ultraweak topology is more useful than the weak operator topology, but it is more complicated to define, and the weak operator topology is often more apparently convenient.
The ultraweak topology can be obtained from the weak operator topology as follows. If H1 is a separable infinite dimensional Hilbert space then B(H) can be embedded in B(H⊗H1) by tensoring with the identity map on H1. Then the restriction of the weak operator topology on B(H⊗H1) is the ultraweak topology of B(H).
Read more about this topic: Ultraweak Topology
Famous quotes containing the words relation and/or weak:
“To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
—Bruno Bettelheim (20th century)
“The opportunities for heroism are limited in this kind of world: the most people can do is sometimes not to be as weak as they’ve been at other times.”
—Angus Wilson (1913–1991)