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It is stronger than the weak operator topology. The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h 1), h 2)| for h 1, h 2 ∈ H. (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)
The weak topology on a JW algebra M is define by the seminorms |f(a)| where f is a normal state; the strong topology is defined by the seminorms |f(a 2)| 1/2. The quadratic representation and Jordan product operators L(a)b = a ∘ b are continuous operators on M for both the weak and strong topology. An idempotent p in a JBW algebra M is called ...
The predual of B(H) is the trace class operators C 1 (H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H). A net {T α} ⊂ B(H) converges to T in WOT if and only Tr(T α F) converges to Tr(TF) for all finite-rank ...
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators , and consideration may be given to nonlinear operators .
The SOT is stronger than the weak operator topology and weaker than the norm topology. The SOT lacks some of the nicer properties that the weak operator topology has, but being stronger, things are sometimes easier to prove in this topology. It can be viewed as more natural, too, since it is simply the topology of pointwise convergence.
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.
The ultrastrong topology is stronger than the strong operator topology. One problem with the strong operator topology is that the dual of B(H) with the strong operator topology is "too small". The ultrastrong topology fixes this problem: the dual is the full predual B * (H) of all trace class operators. In general the ultrastrong topology is ...
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