The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points

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 $\Sigma_{d,R}$; we write $(\Sigma_{d,R},W_{2})$ for this metric space.

1. (Completeness)¶ The metric space $(\Sigma_{d,R},W_{2})$ is complete.

2. (Boundedness)¶ $W_{2}(\mu,\nu)^{2}\le4dR^{2}$ for all $\mu,\nu\in\Sigma_{d,R}$, where $d$ is read as a real number; in particular $\Sigma_{d,R}$ is bounded in $(\Sigma_{d,R},W_{2})$.

3. (Interpolation points)¶ The metric space $(\Sigma_{d,R},W_{2})$ has interpolation points. More precisely, for $\mu,\nu\in\Sigma_{d,R}$, an optimal coupling $\gamma\in\Pi(\mu,\nu)$ and a real $t$ with $0\le t\le1$, the displacement interpolant $\gamma_{t}$ is an interpolation point of $\mu$ and $\nu$ at parameter $t$.