Operator Topologies - List of Topologies On B(H)

List of Topologies On B(H)

There are many topologies that can be defined on B(H) besides the ones used above. These topologies are all locally convex, which implies that they are defined by a family of seminorms.

In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak. (In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.) The diagram on the right is a summary of the relations, with the arrows pointing from strong to weak.

The Banach space B(H) has a (unique) predual B(H)_*, consisting of the trace class operators, whose dual is B(H). The seminorm p_w(x) for w positive in the predual is defined to be (w, x*x)1/2.

If B is a vector space of linear maps on the vector space A, then σ(A, B) is defined to be the weakest topology on A such that all elements of B are continuous.

• The norm topology or uniform topology or uniform operator topology is defined by the usual norm ||x|| on B(H). It is stronger than all the other topologies below.
• The weak (Banach space) topology is σ(B(H), B(H)*), in other words the weakest topology such that all elements of the dual B(H)* are continuous. It is the weak topology on the Banach space B(H). It is stronger than the ultraweak and weak operator topologies. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
• The Mackey topology or Arens-Mackey topology is the strongest locally convex topology on B(H) such that the dual is B(H)_*, and is also the uniform convergence topology on σ(B(H)_*, B(H)-compact convex subsets of B(H)_*. It is stronger than all topologies below.
• The σ-strong* topology or ultrastrong* topology is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous. It is defined by the family of seminorms p_w(x) and p_w(x*) for positive elements w of B(H)_*. It is stronger than all topologies below.
• The σ-strong topology or ultrastrong topology or strongest topology or strongest operator topology is defined by the family of seminorms p_w(x) for positive elements w of B(H)_*. It is stronger than all the topologies below other than the strong* topology. Warning: in spite of the name "strongest topology", it is weaker than the norm topology.)
• The σ-weak topology or ultraweak topology or weak* operator topology or weak * topology or weak topology or σ(B(H), B(H)_*) topology is defined by the family of seminorms |(w, x)| for elements w of B(H)_*. It is stronger than the weak operator topology. (Warning: the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different.)
• The strong* operator topology or strong* topology is defined by the seminorms ||x(h)|| and ||x*(h)|| for h in H. It is stronger than the strong and weak operator topologies.
• The strong operator topology (SOT) or strong topology is defined by the seminorms ||x(h)|| for h in H. It is stronger than the weak operator topology.
• The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h₁), h₂)| for h₁ and h₂ in 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.)