In
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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set ''B''(''H'') of
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
s on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is the
weak-* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
obtained from the
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 In mathematics, specifically functional analysis, a trace-class o ...
''B''
*(''H'') of ''B''(''H''), the
trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
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 ''H''
1 is a separable infinite dimensional Hilbert space
then ''B''(''H'') can be embedded in ''B''(''H''⊗''H''
1) by tensoring with the identity map on ''H''
1. Then the restriction of the weak operator topology on ''B''(''H''⊗''H''
1) is the ultraweak topology of ''B''(''H'').
See also
*
Topologies on the set of operators on a Hilbert space In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space .
Introduction
Let (T_n)_ be a sequence of linear operators on the Banach space ...
*
Ultrastrong topology In functional analysis, the ultrastrong topology, or σ-strong topology, or strongest topology on the set ''B(H)'' of bounded operators on a Hilbert space is the topology defined by the family of seminorms
p_\omega(x) = \omega(x^ x)^
for posit ...
*
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 \langle Tx, y\rangle is ...
References
*
*
*
{{DEFAULTSORT:Ultraweak Topology
Topology of function spaces
Von Neumann algebras