Weak-Star Semicontinuity on Bounded Laws of Functions on Square-Integrable Noncommutative Laws

In the setting of Square-Integrable Noncommutative Laws: Standing Notation, let $u:\Sigma^{2}_{d}\to\mathbb{R}$; weak-star convergence is that of Weak-Star Convergence of Noncommutative Laws §weak-star.

1. (Upper)¶ $u$ is weak-star upper semicontinuous on bounded laws if for every real $R>0$, every sequence $(\lambda_{m})_{m\in\mathbb{N}}$ in $\Sigma_{d,R}$ that converges weak-star to some $\lambda\in\Sigma_{d}$, and every real $c>u(\kappa_{d}(\lambda))$, there is $m_{0}\in\mathbb{N}$ with $u(\kappa_{d}(\lambda_{m}))<c$ for every $m\ge m_{0}$.

2. (Lower)¶ $u$ is weak-star lower semicontinuous on bounded laws if for every real $R>0$, every sequence $(\lambda_{m})_{m\in\mathbb{N}}$ in $\Sigma_{d,R}$ that converges weak-star to some $\lambda\in\Sigma_{d}$, and every real $c<u(\kappa_{d}(\lambda))$, there is $m_{0}\in\mathbb{N}$ with $u(\kappa_{d}(\lambda_{m}))>c$ for every $m\ge m_{0}$.

3. (Continuous)¶ $u$ is weak-star continuous on bounded laws if it is weak-star upper and lower semicontinuous on bounded laws.