Ultraweak topology

In functional analysis, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B(H) of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B_*(H) of B(H), the trace class operators on H. In other words it is the weakest topology such that all elements of the predual are continuous (when considered as functions on B(H)).

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 better than the weak operator topology, but is more complicated to define so people usually use the weak operator topology if they can get away with it.

The ultraweak topology can be obtained from the weak operator topology as follows. If H₁ is a separable infinite dimensional Hilbert spacethen 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).

See also

*Topologies on the set of operators on a Hilbert space
*ultrastrong topology
*weak operator topology