Ultraweak operator topology - Imported Knowlpedia Content

Definition

Let $\mathcal H$ be a Hilbert space. The ultraweak operator topology, or $\sigma$ -weak topology, on the bounded-operator algebra $\mathcal B(\mathcal H)$ is the weak-star topology $\sigma(\mathcal B(\mathcal H),\mathcal C_1(\mathcal H))$ coming from the trace-class predual. Explicitly, a net $T_i$ converges ultraweakly to $T$ exactly when

\operatorname{Tr}(T_iS)\longrightarrow\operatorname{Tr}(TS)

for every trace-class operator $S$ . Equivalently, it is the weakest locally convex topology making all such trace-pairing functionals continuous. This realizes $\mathcal B(\mathcal H)$ as a dual Banach space.

Equivalent seminorm description

The topology is generated by seminorms

T\longmapsto\left|\sum_{n=1}^{\infty}\langle T\xi_n,\eta_n\rangle\right|,

where $(\xi_n)$ and $(\eta_n)$ are square-summable sequences in $\mathcal H$ . The series converges absolutely by Cauchy–Schwarz and represents pairing with a trace-class operator. This description avoids choosing a particular trace-class factorization.

Comparison with other operator topologies

Ultraweak convergence implies weak-operator convergence because vector functionals correspond to rank-one trace-class operators. On norm-bounded subsets of $\mathcal B(\mathcal H)$ , the two topologies agree, but they differ globally. The ultraweak topology is generally much weaker than the operator-norm topology and is not the Banach-space weak topology $\sigma(\mathcal B(\mathcal H),\mathcal B(\mathcal H)^*)$ .

Von Neumann algebras

For a von Neumann algebra $M$ , “ultraweak” means the weak-star topology induced by its canonical predual $M_*$ . On a concrete $M\subseteq\mathcal B(\mathcal H)$ , this agrees with the topology inherited from $\mathcal B(\mathcal H)$ . Ultraweakly continuous linear functionals on $M$ are precisely the elements of $M_*$ Takesaki, Chapter III, §2.

References

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