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 criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
—Henry James (1843–1916)
“The weak are the most treacherous of us all. They come to the strong and drain them. They are bottomless. They are insatiable. They are always parched and always bitter. They are everyone’s concern and like vampires they suck our life’s blood.”
—Bette Davis (1908–1989)