Weak operator topology

Let $H$ be a Hilbert space. The weak operator topology on the bounded-operator $C^*$ -algebra $B(H)$ is the weakest topology making every matrix coefficient

T\longmapsto \langle T\xi,\eta\rangle

continuous, for $\xi,\eta\in H$ . Equivalently, a net $(T_i)$ converges to $T$ in this topology exactly when $\langle T_i\xi,\eta\rangle\to\langle T\xi,\eta\rangle$ for every pair $\xi,\eta\in H$ . Thus weak operator convergence tests operators on fixed vectors and then tests the resulting vectors weakly. It is weaker than operator-norm and strong-operator convergence.

Locally convex description

The weak operator topology is generated by the seminorms

p_{\xi,\eta}(T)=|\langle T\xi,\eta\rangle|.

Consequently, a basic neighborhood of $T_0$ imposes finitely many bounds $p_{\xi_j,\eta_j}(T-T_0)<\varepsilon$ . The same definition applies to $B(H,K)$ , with $\xi\in H$ and $\eta\in K$ . Nets, rather than sequences, are required to describe this topology in full generality.

Comparison with other weak topologies

The weak operator topology is generally strictly weaker than the Banach-space weak topology $\sigma(B(H),B(H)^*)$ . The ultraweak topology instead uses all functionals in the trace-class predual of $B(H)$ . It is finer than the weak operator topology, but the two induce the same topology on every norm-bounded subset. In particular, every weak-operator convergent net is norm bounded and also converges ultraweakly.

Algebraic continuity

The adjoint operation is weak-operator continuous, and left or right multiplication by a fixed operator is weak-operator continuous. Joint multiplication is not weak-operator continuous, even on the unit ball. By contrast, on uniformly norm-bounded nets, strong convergence of both factors implies strong convergence of their products; this is a property of the strong operator topology. Von Neumann algebras are precisely the unital self-adjoint subalgebras of $B(H)$ that are weak-operator closed.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I, American Mathematical Society, 1997. Publisher record. Relevant: §5.1 on weak and strong operator topologies.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter I, §3 on operator topologies.