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
\( p_{\omega }(x)=\omega (x^{{*}}x)^{{1/2}} \)
for positive elements \( \omega \) of the predual L\( L_{{*}}(H) \) that consists of trace class operators. [1]:68
It was introduced by John von Neumann in 1936. [2]
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*(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 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 strong operator topology on B(H⊗H1) is the ultrastrong topology of B(H). Equivalently, it is given by the family of seminorms
\( x\mapsto \left(\sum _{{n=1}}^{\infty }||x\xi _{n}||^{2}\right)^{{1/2}}, \)
where \( \sum _{{n=1}}^{\infty }||\xi _{n}||^{2}<\infty \).[1]: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.[1]:68
See also
Topologies on the set of operators on a Hilbert space
ultraweak topology
strong operator topology
References
Takesaki, Masamichi (2002). Theory of operator algebras. I. Berlin: Springer-Verlag. ISBN 3-540-42248-X.
von Neumann, John (1936), "On a Certain Topology for Rings of Operators", The Annals of Mathematics, Second Series, 37 (1): 111–115, doi:10.2307/1968692, JSTOR 1968692
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Functional analysis (topics – glossary)
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Hilbert space Banach space Fréchet space topological vector space
Theorems
Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
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