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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.
The strong operator topology, or SOT, on () is the topology of pointwise convergence. Because the inner product is a continuous function, the SOT is stronger than WOT. The following example shows that this inclusion is strict.
For example, the strong operator topology on L(X,Y) is the topology of pointwise convergence. For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x ∈ X: ‖ ‖.
Unfortunately, this topology is also sometimes called the strong operator topology, which may lead to ambiguity; [6] for this reason, this article will avoid referring to this topology by this name. A subset of L ( X ; Y ) {\displaystyle L(X;Y)} is called simply bounded or weakly bounded if it is bounded in L σ ( X ; Y ) {\displaystyle L ...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus. The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the weak operator topology (WOT).
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.
In this definition the weak (operator) topology can be replaced by many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The *-algebras of bounded operators that are closed in the norm topology are C*-algebras , so in particular any von Neumann algebra is a C*-algebra.