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 $p_{\omega }(x)=\omega (x^{*}x)^{1/2}$ for positive elements $\omega$ of the predual $L_{*}(H)$ that consists of trace class operators. ^[1]^: 68

It was introduced by John von Neumann in 1936. ^[2]

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

References

1 2 3 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", 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
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Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

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Basic concepts	Dual space Dual system Dual topology Duality Operator topologies Polar set Polar topology Topologies on spaces of linear maps
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Main results	Banach–Alaoglu Mackey–Arens
Maps	Transpose of a linear map
Subsets	Saturated family Total set
Other concepts	Biorthogonal system

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Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
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Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

Ultrastrong topology

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References

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