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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 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 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 ...
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 ...
Many operators that are studied are operators on Hilbert spaces of holomorphic functions, and the study of the operator is intimately linked to questions in function theory. For example, Beurling's theorem describes the invariant subspaces of the unilateral shift in terms of inner functions, which are bounded holomorphic functions on the unit ...
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).
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
From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general, operator algebras are non-commutative rings. An operator algebra is typically required to be closed in a specified operator topology inside the whole algebra of continuous linear operators. In particular, it is a ...