Ultraweak topology

Ultraweak topology

In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set "B"("H") of bounded operators on a Hilbert space is the weak-* topology obtained from the predual "B"*("H") of "B"("H"), the trace class 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 spacethen "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
*ultrastrong topology
*weak operator topology


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Operator topology — In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B(H) of bounded linear operators on a Hilbert space H. Contents 1 Introduction 2 List of topologies on B(H) 3 …   Wikipedia

  • Ultrastrong topology — In functional analysis, the ultrastrong topology, or sigma; 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 pw(x) for positive elements w of the… …   Wikipedia

  • Von Neumann algebra — In mathematics, a von Neumann algebra or W* algebra is a * algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann,… …   Wikipedia

  • Abelian von Neumann algebra — In functional analysis, an Abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebra is the algebra L^infty(X,mu) for μ a σ… …   Wikipedia

  • List of functional analysis topics — This is a list of functional analysis topics, by Wikipedia page. Contents 1 Hilbert space 2 Functional analysis, classic results 3 Operator theory 4 Banach space examples …   Wikipedia

  • Théorème du bicommutant de von Neumann — Le théorème du bicommutant de von Neumann est un théorème d analyse fonctionnelle qui établit un lien entre l adhérence d un ensemble d opérateurs linéaires bornés sur un espace de Hilbert dans certaines topologies et le bicommutant de cet… …   Wikipédia en Français

  • List of mathematics articles (U) — NOTOC U U duality U quadratic distribution U statistic UCT Mathematics Competition Ugly duckling theorem Ulam numbers Ulam spiral Ultraconnected space Ultrafilter Ultrafinitism Ultrahyperbolic wave equation Ultralimit Ultrametric space… …   Wikipedia

  • Expectation value (quantum mechanics) — In quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. It is a fundamental concept in all areas of quantum physics. Operational definition Quantum physics shows an inherent statistical behaviour:… …   Wikipedia

  • Von Neumann bicommutant theorem — In mathematics, the von Neumann bicommutant theorem in functional analysis relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”