Moments of Noncommutative Laws are Lipschitz in the Wasserstein Distance

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d\in\mathbb{N}$ and let $R>0$ be real. By The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric the restriction of $W_{2}$ to $\Sigma_{d,R}\times\Sigma_{d,R}$ is a metric on $\Sigma_{d,R}$, and openness of subsets of $\Sigma_{d,R}$ refers to the metric space $(\Sigma_{d,R},W_{2})$ as in Open Subset of a Metric Space.

1. (Lipschitz moments)¶ For every $p\in\mathcal{P}_{d}$ there is a nonnegative real $C$ such that

2. (Wasserstein convergence implies weak-star convergence)¶ Let $(\mu_{m})_{m\in\mathbb{N}}$ be a sequence in $\Sigma_{d,R}$ and $\mu\in\Sigma_{d,R}$ such that the real sequence $(W_{2}(\mu_{m},\mu))_{m\in\mathbb{N}}$ converges to $0$. Then $\mu_{m}\to\mu$ weak-star.

3. (Closed sets)¶ Let $K\subseteq\Sigma_{d,R}$ be such that $\lambda\in K$ whenever $\lambda\in\Sigma_{d,R}$ and some sequence in $K$ converges weak-star to $\lambda$. Then $\Sigma_{d,R}\setminus K$ is open in $(\Sigma_{d,R},W_{2})$.