In functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B(H) of bounded operator

Bounded operator

In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...

s on a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 is the weak-* topology

Weak topology

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...

 obtained from the predual

Predual

In mathematics, the predual of an object D is an object P whose dual space is D.For example, the predual of the space of bounded operators is the space of trace class operators. The predual of the space of differential forms is the space of chainlets....

 B_*(H) of B(H), the trace class

Trace class

In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....

 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 more useful than the weak operator topology, but it is more complicated to define, and the weak operator topology is often more apparently convenient.

The ultraweak topology can be obtained from the weak operator topology as follows.
If H₁ is a separable infinite dimensional Hilbert space
then 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
  Weak operator topology
  In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space.Equivalently, a...