View source for Ultrastrong topology

In [[functional analysis]], the '''ultrastrong topology''',  or '''&sigma;-strong topology''', or '''strongest topology''' on the set ''B(H)'' of [[bounded operator]]s on a [[Hilbert space]] is the topology defined by the family of seminorms
<math display=block>p_\omega(x) = \omega(x^{*} x)^{1/2}</math>
for positive elements <math>\omega</math> of the [[predual]] <math>L_{*}(H)</math> that consists of [[trace class]] operators.
<ref name="TakesakiI">{{Cite book
|last=Takesaki
|first=Masamichi
|title=Theory of operator algebras. I.
|publisher=Springer-Verlag|location=[[Berlin]]
|year=2002
|isbn=3-540-42248-X
}}</ref>{{rp|68}}

It was introduced by [[John von Neumann]] in 1936.
<ref>{{Citation
 | last=von Neumann
 | first=John
 | title=On a Certain Topology for Rings of Operators
 | journal=Annals of Mathematics |series=Second Series
 | volume=37
 | issue=1
 | year=1936
 | pages=111–115
 | jstor=1968692
 | doi=10.2307/1968692
 }}</ref>

==Relation with the strong (operator) topology==

The ultrastrong topology is similar to the strong (operator) topology. 
For example, on any norm-bounded set the strong operator and ultrastrong topologies are the same. The ultrastrong topology is stronger than the strong operator topology.

One problem with the strong operator topology is that the dual of ''B(H)'' with the strong operator topology is "too small". The ultrastrong topology fixes this problem: the dual is the full predual ''B<sub>*</sub>(H)'' of all trace class operators. In general the ultrastrong topology is better than the strong operator topology, but is more complicated to define so people usually use the strong operator topology if they can get away with it.

The ultrastrong topology can be obtained from the strong operator topology as follows. 
If ''H''<sub>1</sub> is a separable infinite dimensional Hilbert space then ''B(H)'' can be embedded in ''B''(''H''&otimes;''H''<sub>1</sub>) by [[Tensor product|tensor]]ing with the identity map on ''H''<sub>1</sub>. Then the restriction of the strong operator topology on 
''B''(''H''&otimes;''H''<sub>1</sub>) is the ultrastrong topology of ''B(H)''.
Equivalently, it is given by the family of seminorms
<math display=block>x \mapsto \left(\sum_{n=1}^\infty \|x\xi_n\|^2\right)^{1/2},</math>
where <math>\sum_{n=1}^\infty \|\xi_n\|^2 < \infty.</math><ref name=TakesakiI/>{{rp|68}}

The adjoint map is not continuous in the ultrastrong topology. There is another topology called the ultrastrong*  topology, which is the weakest topology stronger than the ultrastrong topology such that the adjoint map is continuous.<ref name=TakesakiI/>{{rp|68}}

==See also==

* {{annotated link|Strong operator topology}}
* {{annotated link|Topological tensor product}}
* {{annotated link|Topologies on the set of operators on a Hilbert space}}
* {{annotated link|Ultraweak topology}}

==References==

{{reflist}}
{{reflist|group=note}}

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->

{{Banach spaces}}
{{Hilbert space}}
{{Duality and spaces of linear maps}}
{{Topological vector spaces}}

[[Category:Topology of function spaces]]
[[Category:Von Neumann algebras]]