Ultrastrong operator topology - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space. The ultrastrong operator topology, or $\sigma$ -strong topology, on $\mathcal B(H)$ is the locally convex topology generated by the seminorms

p_{\boldsymbol{\xi}}(T) =\left(\sum_{n=1}^{\infty}\|T\xi_n\|^2\right)^{1/2},

where $\boldsymbol{\xi}=(\xi_n)$ ranges over all square-summable sequences in $H$ . Thus $T_i\to T$ ultrastrongly exactly when $p_{\boldsymbol{\xi}}(T_i-T)\to0$ for every such sequence. A concrete von Neumann algebra $M\subseteq\mathcal B(H)$ carries the topology inherited from this one.

Equivalent description by normal functionals

On a von Neumann algebra $M$ , the $\sigma$ -strong topology is generated by

x\longmapsto \varphi(x^*x)^{1/2},

as $\varphi$ ranges over the positive linear functionals on $M$ that are normal. Such a functional on a concrete $M$ can be expressed using a square-summable family of vector functionals, which recovers the seminorms in the core. This intrinsic formulation shows that the inherited topology does not depend on a chosen faithful normal representation Takesaki, Chapter III, §2.

Comparison with other operator topologies

Ultrastrong convergence implies strong-operator convergence, since a sequence with only one nonzero vector is allowed. Conversely, the two topologies induce the same topology on every norm-bounded subset of $\mathcal B(H)$ Takesaki, Chapter III, §2. They differ on the whole algebra in infinite dimension: a strong neighborhood controls finitely many vectors, whereas one ultrastrong seminorm can impose a square-summed condition on an entire countable family.

The ultrastrong topology is stronger than the ultraweak topology. On norm-bounded sets, ultrastrong convergence still need not agree with ultraweak convergence.

Conventions and scope

Some sources write “strong topology” for the topology generated by the normal-functional seminorms when the ambient algebra is already a von Neumann algebra. Here “strong operator topology” means pointwise norm convergence, while “ultrastrong” means the square-summable refinement above. Applying the definition simultaneously to $x$ and $x^*$ gives the ultrastrong-star topology, which is a different topology.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III, §2 on the $\sigma$ -strong and $\sigma$ -weak topologies.