Ultraweak topology

Remarks

The series defining $\omega_{x,y}(T)$ converges absolutely, by the Cauchy–Schwarz inequality. Equivalently, the ultraweak topology is the weak-$*$ topology of the duality $B(H) \cong \mathcal{L}^1(H)^*$, where $\mathcal{L}^1(H)$ is the space of trace class operators and $T$ acts by $\rho \mapsto \operatorname{tr}(T \rho)$.

Example: for a trace class operator $\rho$, the functional $T \mapsto \operatorname{tr}(T\rho)$ is ultraweakly continuous. Indeed, writing $\rho = \sum_n s_n \langle \cdot, e_n \rangle f_n$ in Schmidt form with $\sum_n s_n < \infty$ gives $\operatorname{tr}(T\rho) = \omega_{x,y}(T)$ with $x_n = \sqrt{s_n} f_n$ and $y_n = \sqrt{s_n} e_n$.

Non-example: let $H$ be infinite-dimensional and let $\varphi$ be a state on $B(H)$ that vanishes on all compact operators. Such states exist because the quotient algebra by the compact operators has states. Then $\varphi$ is norm-continuous but not ultraweakly continuous: every ultraweakly continuous functional has the form $T \mapsto \operatorname{tr}(T\rho)$, vanishing on all rank-one operators forces $\rho = 0$, and yet $\varphi(I) = 1$.

Every weak operator topology open set is ultraweakly open, and the two topologies agree on every norm-bounded subset of $B(H)$. They differ on all of $B(H)$ when $H$ is infinite-dimensional.