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

Each result cited below is universally quantified over the data in its own statement.

Throughout, $d\in\mathbb{N}$ and the real $R>0$ are those of the statement, and $W_{2}$ denotes its restriction to $\Sigma_{d,R}\times\Sigma_{d,R}$, a metric on $\Sigma_{d,R}$ by The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric; convergence, Cauchy sequences, boundedness and interpolation points in $(\Sigma_{d,R},W_{2})$ are those of Convergent Sequence in a Metric Space, Cauchy Sequence in a Metric Space, Bounded Subset of a Metric Space and Metric Space with Interpolation Points §interpolation.

Step 1 (Completeness: a weak-star limit). Let $(\lambda_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence in $(\Sigma_{d,R},W_{2})$. By Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §compact choose a strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ and $\lambda\in\Sigma_{d,R}$ such that $\lambda_{n_{k}}\to\lambda$ weak-star as $k\to\infty$. We show that $(\lambda_{n})$ converges to $\lambda$ in $(\Sigma_{d,R},W_{2})$; since $(\lambda_{n})$ was an arbitrary Cauchy sequence, this proves claim 1 by Complete Metric Space.

Step 2 (Completeness: convergence in $W_{2}$). Let $\varepsilon>0$; then $\varepsilon/2>0$ and $\varepsilon/2<\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. The Cauchy property gives $N\in\mathbb{N}$ with $W_{2}(\lambda_{n},\lambda_{m})<\varepsilon/2$ for all $n,m\ge N$. Fix $n\ge N$, and define two sequences in $\Sigma_{d,R}$ by

The sequence $(\mu_{j})$ is constant, so for every $p\in\mathcal{P}_{d}$ the real sequences $(\operatorname{Re}\mu_{j}(p))_{j}$ and $(\operatorname{Im}\mu_{j}(p))_{j}$ are constant and converge to $\operatorname{Re}\lambda_{n}(p)$ and $\operatorname{Im}\lambda_{n}(p)$ (every term is at distance $0$ from the limit); thus $\mu_{j}\to\lambda_{n}$ weak-star by Weak-Star Convergence of Noncommutative Laws §weak-star. The indices $j\mapsto N+j$ are strictly increasing, so $(\nu_{j})$ is a subsequence, in the sense of Subsequence of a Sequence in a Set, of the sequence $(\lambda_{n_{k}})_{k}$, which converges weak-star to $\lambda$; hence $\nu_{j}\to\lambda$ weak-star by Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §unique. Moreover $N<N+j\le n_{N+j}$ by Strictly Increasing Sequences of Natural Numbers Dominate Their Index, so $W_{2}(\mu_{j},\nu_{j})=W_{2}(\lambda_{n},\lambda_{n_{N+j}})<\varepsilon/2$ for every $j$. Now The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §lsc, applied to these two sequences with $c=\varepsilon/2$, gives $W_{2}(\lambda_{n},\lambda)\le\varepsilon/2<\varepsilon$. As $n\ge N$ was arbitrary, $W_{2}(\lambda_{n},\lambda)<\varepsilon$ for every $n\ge N$, and since $\varepsilon$ was arbitrary, $(\lambda_{n})$ converges to $\lambda$ in $(\Sigma_{d,R},W_{2})$.

Step 3 (Second moments). Let $\kappa\in\Sigma_{d,R}$ and $j\in[d]$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $x_{j}^{2}=x_{j}x_{j}$ is the monomial of the word $(j)(j)$, which has length $2$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid; so $|\kappa(x_{j}^{2})|\le R^{2}=RR$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound and claim 1 of Properties of Natural Number Powers in a Field. Since $x_{j}^{*}=x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $\kappa(x_{j}^{2})=\kappa(x_{j}^{*}x_{j})$ is a nonnegative real by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, hence equals its modulus by claim 8 of Properties of Complex Conjugation and Modulus, and $0\le\kappa(x_{j}^{2})\le R^{2}$. Applying claims 2, 3 and 5 of Properties of Finite Sums to the nonnegative reals $R^{2}-\kappa(x_{j}^{2})$, and using $\sum_{j=1}^{d}R^{2}=d\,R^{2}$ (induction on $d$, from the recursion in claim 1 of Properties of Finite Sums and claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field), we get $M(\kappa)=\sum_{j=1}^{d}\kappa(x_{j}^{2})\le d\,R^{2}$.

Step 4 (Boundedness). Let $\mu,\nu\in\Sigma_{d,R}$. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §moment-bound and Step 3,

the first assertion of claim 2. Put $B=1+4dR^{2}$, a positive real since $4dR^{2}\ge0$. We claim $W_{2}(\mu,\nu)\le B$. Write $s=W_{2}(\mu,\nu)\ge0$. If $s\le1$, then $s\le1\le B$. Otherwise $1<s$, and multiplying by $s>0$ (claim 10 of Elementary Order Arithmetic in an Ordered Field) gives $s<s^{2}\le4dR^{2}<B$. Finally $\Sigma_{d,R}$ is nonempty by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §zero-law; fixing any $\mu_{0}\in\Sigma_{d,R}$, we have $W_{2}(\mu_{0},\nu)\le B$ for every $\nu\in\Sigma_{d,R}$, so $\Sigma_{d,R}$ is bounded in $(\Sigma_{d,R},W_{2})$ by Bounded Subset of a Metric Space. This proves claim 2.

Step 5 (Interpolation points). Let $\mu,\nu\in\Sigma_{d,R}$, let $\gamma\in\Pi(\mu,\nu)$ be optimal, and let $t\in\mathbb{R}$ with $0\le t\le1$. By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §interpolation, $\gamma_{t}\in\Sigma_{d,R}$, and by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §interpolation,

So $\gamma_{t}$ is an interpolation point of $\mu$ and $\nu$ at parameter $t$ in the sense of Metric Space with Interpolation Points §interpolation, which is the second sentence of claim 3. For the first sentence, let $\mu,\nu\in\Sigma_{d,R}$ and $t$ be arbitrary as there; an optimal coupling $\gamma\in\Pi(\mu,\nu)$ exists by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained, and the point $z=\gamma_{t}$ just constructed has the property required by Metric Space with Interpolation Points §interpolation. Hence $(\Sigma_{d,R},W_{2})$ has interpolation points. $\blacksquare$