Proof of The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation

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

Notation. For $\gamma\in\Sigma^{2}_{3d}$ put $\Psi(\gamma)=p(\gamma)-h(s(\gamma))-\theta\,s(\gamma)^{2}$, where $s(\gamma)$ and $p(\gamma)$ are the displacement and momentum pairing of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §displacement-pairing; by that clause a realisation of $\gamma$ exists and $s(\gamma)$, $p(\gamma)$ may be computed from any realisation. Put $a_{\pi}=|\pi|_{\mathrm{mom}}$ and $K=a_{\pi}+L+4\theta r\sqrt{d}$, a real number with $K>0$. By The Coupling Test Function of a Square-Integrable Plan §test, $\Psi(\gamma)\le\Phi(\lambda)$ whenever $\lambda\in\Sigma^{2}_{d}$ and $\gamma\in\mathcal{C}(\lambda)$; and since $\Phi(\lambda)$ is a least upper bound in the sense of The Real Numbers: Standing Notation and Background §bounds, for every real $\varepsilon>0$ some $\gamma\in\mathcal{C}(\lambda)$ has $\Psi(\gamma)>\Phi(\lambda)-\varepsilon$. For $L^{2}$ $d$-tuples of one tracial W*-probability space $(H,M,\Omega)$, the pairing is the inner product and $\lVert\cdot\rVert_{2}$ the norm of the complex Hilbert space $H^{d}$, sums and real multiples are those of $H^{d}$, and the pairing is real, by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing. Hence, by the results in force by Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §background, for all $L^{2}$ $d$-tuples $U_{1},U_{2},U_{3}$ of one space and real $t$ we have $|\langle U_{1},U_{2}\rangle_{2}|\le\lVert U_{1}\rVert_{2}\lVert U_{2}\rVert_{2}$, $\lVert U_{1}+U_{2}\rVert_{2}\le\lVert U_{1}\rVert_{2}+\lVert U_{2}\rVert_{2}$, $\lVert tU_{1}\rVert_{2}=|t|\,\lVert U_{1}\rVert_{2}$, $\langle U_{1},U_{2}+tU_{3}\rangle_{2}=\langle U_{1},U_{2}\rangle_{2}+t\langle U_{1},U_{3}\rangle_{2}$, $\lVert U_{1}+U_{2}\rVert_{2}^{2}=\lVert U_{1}\rVert_{2}^{2}+2\langle U_{1},U_{2}\rangle_{2}+\lVert U_{2}\rVert_{2}^{2}$, and $U_{1}=0$ whenever $\lVert U_{1}\rVert_{2}=0$. These rules are used below without further mention.

Step A (the gauge). First, $h'(s)\ge0$ for every real $s\ge0$. Indeed, fix a real $\eta>0$, let $r_{\eta}$ be as in the hypothesis at the point $s$, and put $t=s+r_{\eta}/2$. Since $h$ is nondecreasing in the sense of Monotone Real Function §nondecreasing, $0\le h(t)-h(s)\le h'(s)(t-s)+\eta(t-s)$, and dividing by $t-s>0$ gives $h'(s)\ge-\eta$. As $\eta>0$ was arbitrary, $h'(s)\ge0$.

Second, $|h(t)-h(s)|\le L|t-s|$ for all real $s,t\ge0$. The hypothesis gives, for every real $c\ge0$ and $\eta>0$, a real $r_{\eta}(c)>0$ such that

$$|h(t)-h(c)|\le(|h'(c)|+\eta)|t-c|\le(L+\eta)|t-c|\qquad\text{for every real }t\ge0\text{ with }|t-c|<r_{\eta}(c).$$

Fix reals $0\le a<b$ and $\eta>0$, and let $A$ be the set of real $t$ with $a\le t\le b$ and $h(t)-h(a)\le(L+\eta)(t-a)$. Then $a\in A$ and $b$ is an upper bound of $A$, so $A$ has a least upper bound $c$ with $a\le c\le b$, by The Real Numbers: Standing Notation and Background §bounds. We show $c\in A$. If $c=a$ this is clear. Otherwise, by the least-upper-bound property there is $t\in A$ with $c-r_{\eta}(c)<t\le c$; the displayed estimate at $c$ gives $h(c)-h(t)\le(L+\eta)(c-t)$, and adding $h(t)-h(a)\le(L+\eta)(t-a)$ gives $c\in A$. Next, $c=b$: otherwise $t_{1}=\min(b,c+r_{\eta}(c)/2)$ satisfies $c<t_{1}\le b$, the displayed estimate at $c$ gives $h(t_{1})-h(c)\le(L+\eta)(t_{1}-c)$, and adding $h(c)-h(a)\le(L+\eta)(c-a)$ gives $t_{1}\in A$, contradicting $t_{1}>c$. Hence $b\in A$, that is, $0\le h(b)-h(a)\le(L+\eta)(b-a)$, the first inequality because $h$ is nondecreasing. As $\eta>0$ was arbitrary, $0\le h(b)-h(a)\le L(b-a)$, which gives the claim.

Step B (bounds on couplings). Let $\nu\in\Sigma_{d,r}$, let $\gamma\in\mathcal{C}(\kappa_{d}(\nu))$ and let $(X,P,X')$ be a realisation of $\gamma$. We show

$$\mathrm{law}(X,P)=\kappa_{2d}(\pi),\qquad\mathrm{law}(X)=\kappa_{d}(\mu),\qquad\lVert P\rVert_{2}=a_{\pi},\qquad\lVert X\rVert_{2}\le r\sqrt{d},\qquad\lVert X'\rVert_{2}\le r\sqrt{d},$$

and consequently $s(\gamma)\le2r\sqrt{d}$ and $p(\gamma)\le a_{\pi}\,s(\gamma)$. Since $B(X,P,X')=(X,P)$ and $C(X,P,X')=X'$, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward and Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings give $\mathrm{law}(X,P)=B_{\#}\gamma=\kappa_{2d}(\pi)$ and $\mathrm{law}(X')=C_{\#}\gamma=\kappa_{d}(\nu)$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling, $\mathrm{law}(X)=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)$ and $\mathrm{law}(P)=\mathrm{pr}^{2}_{\#}\kappa_{2d}(\pi)$. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate, these equal $\kappa_{d}(\pi\circ\iota^{1})$ and $\kappa_{d}(\pi\circ\iota^{2})$ (here $\pi\in\Sigma_{2d}$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, so $\pi\circ\iota^{1},\pi\circ\iota^{2}\in\Sigma_{d}$ by the preamble of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws), and $\pi\circ\iota^{1}=\mu$ because $\pi$ is a bounded plan at $\mu$ in the sense of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments and Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded, $\lVert X\rVert_{2}^{2}=M(\mu)$, $\lVert X'\rVert_{2}^{2}=M(\nu)$ and $\lVert P\rVert_{2}^{2}=M(\pi\circ\iota^{2})$. Since $\mu,\nu\in\Sigma_{d,r}$, each of the real numbers $\mu(x_{j}x_{j})$ and $\nu(x_{j}x_{j})$ has modulus at most $r^{2}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound (words of length $2$), so $M(\mu)\le dr^{2}$ and $M(\nu)\le dr^{2}$. Since $\iota^{2}$ substitutes $x_{d+j}$ for $x_{j}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals and is multiplicative by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism, $M(\pi\circ\iota^{2})=\sum_{j=1}^{d}\pi(x_{d+j}x_{d+j})=a_{\pi}^{2}$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans. Finally $s(\gamma)=\lVert X'-X\rVert_{2}\le\lVert X'\rVert_{2}+\lVert X\rVert_{2}\le2r\sqrt{d}$ and $p(\gamma)=\langle P,X'-X\rangle_{2}\le\lVert P\rVert_{2}\,s(\gamma)=a_{\pi}\,s(\gamma)$.

Step C (law invariance). Let $k,n\in\mathbb{N}$, let $Z$ and $Z_{1}$ be $L^{2}$ $k$-tuples of tracial W*-probability spaces (possibly different) with $\mathrm{law}(Z)=\mathrm{law}(Z_{1})$, and let $T=(E,0)$ be an affine datum from $k$ to $n$ variables. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $\mathrm{law}(TZ)=T_{\#}\mathrm{law}(Z)=T_{\#}\mathrm{law}(Z_{1})=\mathrm{law}(TZ_{1})$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments, $\lVert TZ\rVert_{2}^{2}$ and $\lVert TZ_{1}\rVert_{2}^{2}$ both equal the second moment of this law, so $\lVert TZ\rVert_{2}=\lVert TZ_{1}\rVert_{2}$; and if $n=2d$ and $TZ=(A,A')$, $TZ_{1}=(A_{1},A_{1}')$ with $L^{2}$ $d$-tuples $A,A',A_{1},A_{1}'$, then by the same clause and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing both $\langle A,A'\rangle_{2}$ and $\langle A_{1},A_{1}'\rangle_{2}$ equal $\sum_{i=1}^{d}\mathrm{m}_{i,d+i}(\mathrm{law}(TZ))$, so they are equal. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, every tuple formed below from the coordinates of a tuple $Z$ by selecting or reordering blocks and taking real linear combinations of blocks is $TZ$ for the affine datum $T=(E,0)$ whose coefficients are read off from the formula; pairs and triples are tuples by the same clause. We use this without writing out the matrices $E$.

Step D (gluing). Let $k,m,n\in\mathbb{N}$, let $Z,W$ be $L^{2}$ tuples of lengths $k,m$ of one tracial W*-probability space and $Z_{1},V$ $L^{2}$ tuples of lengths $k,n$ of another, with $\mathrm{law}(Z)=\mathrm{law}(Z_{1})$. With $F^{m},F^{n}$ as in Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal, $F^{m}(Z,W)=Z$ and $F^{n}(Z_{1},V)=Z_{1}$, so $F^{m}_{\#}\mathrm{law}(Z,W)=\mathrm{law}(Z)=\mathrm{law}(Z_{1})=F^{n}_{\#}\mathrm{law}(Z_{1},V)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward. By Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue there are a tracial W*-probability space and $L^{2}$ tuples $\hat{Z},\hat{W},\hat{V}$ of it, of lengths $k,m,n$, with $\mathrm{law}(\hat{Z},\hat{W})=\mathrm{law}(Z,W)$ and $\mathrm{law}(\hat{Z},\hat{V})=\mathrm{law}(Z_{1},V)$.

Step E (transfer). Let $\nu,\nu'\in\Sigma_{d,r}$, put $w=W_{2}(\nu,\nu')$, and let $\gamma\in\mathcal{C}(\kappa_{d}(\nu))$. We show that there is $\gamma''\in\mathcal{C}(\kappa_{d}(\nu'))$ with $|s(\gamma'')-s(\gamma)|\le w$ and $\Psi(\gamma'')\ge\Psi(\gamma)-Kw$. Let $(X,P,X')$ be a realisation of $\gamma$. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained (with $R=r$) there is an optimal coupling $\varrho\in\Pi(\nu,\nu')$, so $\varrho\in\Sigma_{2d}$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §bounded-couplings and $I(\varrho)=w^{2}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal; by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded, $\kappa_{2d}(\varrho)\in\Pi^{2}(\kappa_{d}(\nu),\kappa_{d}(\nu'))$ and $\mathcal{I}(\kappa_{2d}(\varrho))=w^{2}$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling and Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §couplings there are $L^{2}$ $d$-tuples $Y,Y''$ of a tracial W*-probability space with $\mathrm{law}(Y)=\kappa_{d}(\nu)$, $\mathrm{law}(Y'')=\kappa_{d}(\nu')$ and $\lVert Y-Y''\rVert_{2}=w$. Since $\mathrm{law}(X')=\kappa_{d}(\nu)$ by Step B, Step D with $Z=X'$, $W=(X,P)$, $Z_{1}=Y$ and $V=Y''$ gives $L^{2}$ $d$-tuples $\hat{X}',\hat{X},\hat{P},\hat{Y}''$ of one tracial W*-probability space with $\mathrm{law}(\hat{X}',(\hat{X},\hat{P}))=\mathrm{law}(X',(X,P))$ and $\mathrm{law}(\hat{X}',\hat{Y}'')=\mathrm{law}(Y,Y'')$. By Step C, $\mathrm{law}(\hat{X},\hat{P},\hat{X}')=\mathrm{law}(X,P,X')=\gamma$, $\mathrm{law}(\hat{Y}'')=\mathrm{law}(Y'')=\kappa_{d}(\nu')$ and $\lVert\hat{Y}''-\hat{X}'\rVert_{2}=\lVert Y''-Y\rVert_{2}=w$. Put $\gamma''=\mathrm{law}(\hat{X},\hat{P},\hat{Y}'')$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $B_{\#}\gamma''=\mathrm{law}(\hat{X},\hat{P})=B_{\#}\gamma=\kappa_{2d}(\pi)$ and $C_{\#}\gamma''=\mathrm{law}(\hat{Y}'')=\kappa_{d}(\nu')$, so $\gamma''\in\mathcal{C}(\kappa_{d}(\nu'))$ by Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings.

Put $\hat{a}=\hat{X}'-\hat{X}$ and $v=\hat{Y}''-\hat{X}'$, so that $\hat{Y}''-\hat{X}=\hat{a}+v$ and $\lVert v\rVert_{2}=w$. Computing from the realisations $(\hat{X},\hat{P},\hat{X}')$ of $\gamma$ and $(\hat{X},\hat{P},\hat{Y}'')$ of $\gamma''$, we get $s(\gamma)=\lVert\hat{a}\rVert_{2}$, $s(\gamma'')=\lVert\hat{a}+v\rVert_{2}$, hence $|s(\gamma'')-s(\gamma)|\le w$, and $p(\gamma'')-p(\gamma)=\langle\hat{P},v\rangle_{2}\ge-\lVert\hat{P}\rVert_{2}\,w=-a_{\pi}w$ by Step B. By Step A, $h(s(\gamma''))-h(s(\gamma))\le Lw$. By Step B, $s(\gamma)+s(\gamma'')\le4r\sqrt{d}$, so $\theta s(\gamma'')^{2}-\theta s(\gamma)^{2}=\theta(s(\gamma'')-s(\gamma))(s(\gamma'')+s(\gamma))\le4\theta r\sqrt{d}\,w$. Adding, $\Psi(\gamma'')\ge\Psi(\gamma)-(a_{\pi}+L+4\theta r\sqrt{d})w=\Psi(\gamma)-Kw$.

Proof of clause 1. Fix $\nu\in\Sigma_{d,r}$. Since $\pi\in\Sigma_{2d}$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans, there is a real $R_{\pi}>0$ with $\pi\in\Sigma_{2d,R_{\pi}}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law; put $R'=\max(R_{\pi},r)$.

(a) Every $\gamma\in\mathcal{C}(\kappa_{d}(\nu))$ equals $\kappa_{3d}(\lambda)$ for some $\lambda\in\Sigma_{3d,R'}$. Indeed, let $(X,P,X')$ be a realisation of $\gamma$ in $(H,M,\Omega)$. By Step B, $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$ and $\mathrm{law}(X')=\kappa_{d}(\nu)$, so by Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §operator and Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §bound (for $2d$ and $d$ variables) there are a self-adjoint $2d$-tuple $u$ and a self-adjoint $d$-tuple $u'$ in $M$ with $u\Omega=(X,P)$, $u'\Omega=X'$, and all operator norms of their entries at most $R'$. The concatenation $(u,u')$ is a self-adjoint $3d$-tuple in $M$ in the sense of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §tuple, and its vacuum tuple is $(X,P,X')$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, $\lambda=\lambda_{(u,u')}\in\Sigma_{3d,R'}$, and by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, $\gamma=\mathrm{law}(X,P,X')=\kappa_{3d}(\lambda)$.

(b) Let $\sigma_{B}$, $\sigma_{C}$ and $\sigma_{U}$ be the affine substitutions of the affine data $B$, $C$ and $U$ of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing. For $\lambda\in\Sigma_{3d}$ we have $\lambda\circ\sigma_{B}\in\Sigma_{2d}$ and $\lambda\circ\sigma_{C}\in\Sigma_{d}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint, and $\kappa_{3d}(\lambda)\in\mathcal{C}(\kappa_{d}(\nu))$ if and only if $\lambda\circ\sigma_{B}=\pi$ and $\lambda\circ\sigma_{C}=\nu$. Indeed $B_{\#}\kappa_{3d}(\lambda)=\kappa_{2d}(\lambda\circ\sigma_{B})$ and $C_{\#}\kappa_{3d}(\lambda)=\kappa_{d}(\lambda\circ\sigma_{C})$ by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded, and the canonical maps $\kappa_{2d},\kappa_{d}$ of the completions of Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws are injective by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry; now apply Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings.

(c) For $\lambda\in\Sigma_{3d}$ and $i,j\in[2d]$ put $c_{ij}(\lambda)=\lambda(\sigma_{U}(x_{i}x_{j}))$. Since $\lambda\circ\sigma_{U}$ is a tracial state by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint, $c_{ij}(\lambda)=\mathrm{m}_{ij}(\lambda\circ\sigma_{U})$ is real by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §moments, and $\mathrm{m}_{ij}(U_{\#}\kappa_{3d}(\lambda))=\mathrm{m}_{ij}(\kappa_{2d}(\lambda\circ\sigma_{U}))=c_{ij}(\lambda)$ by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded. Hence, by the formulas of Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §displacement-pairing,

$$s(\kappa_{3d}(\lambda))^{2}=\sum_{i=1}^{d}c_{d+i,d+i}(\lambda),\qquad p(\kappa_{3d}(\lambda))=\sum_{i=1}^{d}c_{i,d+i}(\lambda).$$

Now choose, for every $n\in\mathbb{N}$, some $\gamma_{n}\in\mathcal{C}(\kappa_{d}(\nu))$ with $\Psi(\gamma_{n})>\Phi(\kappa_{d}(\nu))-2^{-n}$ (Notation), and by (a) some $\lambda_{n}\in\Sigma_{3d,R'}$ with $\gamma_{n}=\kappa_{3d}(\lambda_{n})$. By Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §compact there are a subsequence $(\lambda_{n_{k}})_{k\in\mathbb{N}}$ in the sense of Subsequence of a Sequence in a Set and $\lambda\in\Sigma_{3d,R'}$ with $\lambda_{n_{k}}\to\lambda$ weak-star; by Weak-Star Convergence of Noncommutative Laws §weak-star, for every $q\in\mathcal{P}_{3d}$ the real and imaginary parts of $\lambda_{n_{k}}(q)$ converge to those of $\lambda(q)$. For $q'\in\mathcal{P}_{2d}$, (b) gives $\lambda_{n_{k}}(\sigma_{B}(q'))=\pi(q')$ for every $k$, so $\lambda(\sigma_{B}(q'))=\pi(q')$; thus $\lambda\circ\sigma_{B}=\pi$, and likewise $\lambda\circ\sigma_{C}=\nu$. By (b), $\gamma^{*}=\kappa_{3d}(\lambda)\in\mathcal{C}(\kappa_{d}(\nu))$. The real numbers $c_{ij}(\lambda_{n_{k}})$ converge to $c_{ij}(\lambda)$, so by (c) $s(\gamma_{n_{k}})^{2}\to s(\gamma^{*})^{2}$ and $p(\gamma_{n_{k}})\to p(\gamma^{*})$. Since $|x-y|^{2}\le|x-y|(x+y)=|x^{2}-y^{2}|$ for reals $x,y\ge0$, also $s(\gamma_{n_{k}})\to s(\gamma^{*})$, and by Step A $|h(s(\gamma_{n_{k}}))-h(s(\gamma^{*}))|\le L|s(\gamma_{n_{k}})-s(\gamma^{*})|\to0$. Hence $\Psi(\gamma_{n_{k}})\to\Psi(\gamma^{*})$ by the limit rules of The Real Numbers: Standing Notation and Background §sequences. Since $n_{k}\ge k$, we have $\Phi(\kappa_{d}(\nu))-2^{-k}<\Psi(\gamma_{n_{k}})\le\Phi(\kappa_{d}(\nu))$ for every $k$, so $\Psi(\gamma^{*})=\Phi(\kappa_{d}(\nu))$ and $\gamma^{*}$ is a maximising coupling for $\nu$.

Proof of clause 2. We show $|\Phi(\kappa_{d}(\nu))-\Phi(\kappa_{d}(\nu'))|\le K\,W_{2}(\nu,\nu')$ for all $\nu,\nu'\in\Sigma_{d,r}$. Let $\gamma$ be a maximising coupling for $\nu$, which exists by clause 1. Step E gives $\gamma''\in\mathcal{C}(\kappa_{d}(\nu'))$ with $\Phi(\kappa_{d}(\nu'))\ge\Psi(\gamma'')\ge\Psi(\gamma)-K\,W_{2}(\nu,\nu')=\Phi(\kappa_{d}(\nu))-K\,W_{2}(\nu,\nu')$. Exchanging $\nu$ and $\nu'$ and using $W_{2}(\nu',\nu)=W_{2}(\nu,\nu')$ from The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry gives the claim. Consequently, for $\nu\in\Sigma_{d,r}$ and real $\varepsilon>0$, the choice $\delta=\varepsilon/K$ gives $|\Phi(\kappa_{d}(\nu))-\Phi(\kappa_{d}(\nu'))|<\varepsilon$ whenever $\nu'\in\Sigma_{d,r}$ and $W_{2}(\nu,\nu')<\delta$, which is continuity in the sense of Continuous Map Between Metric Spaces for the metric space of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points and the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics.

Proof of clause 3. Put $c=h'(s)/s+2\theta$ if $s>0$ and $c=0$ if $s=0$, so that $S=P-ca$ in both cases (for $s=0$, $S=P$), and $S$ is an $L^{2}$ $d$-tuple by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations. Since $(X,P,X')$ realises $\gamma$, $s=s(\gamma)$ and $p(\gamma)=\langle P,a\rangle_{2}$. By Step B, $\mathrm{law}(X')=\kappa_{d}(\nu)$, and by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $\mathrm{pr}^{1}_{\#}\mathrm{law}(X',S)=\mathrm{law}(X')$, so $\mathrm{law}(X',S)$ is a plan at $\kappa_{d}(\nu)$ in the sense of Plans at a Square-Integrable Noncommutative Law §plan. By Step A, $h'(s)\ge0$, so $c\ge0$; if $s>0$, $\lVert S-P\rVert_{2}=c\,s=h'(s)+2\theta s$, and if $s=0$, $\lVert S-P\rVert_{2}=0=h'(0)+2\theta\cdot0$ because $h'(0)=0$. This proves the norm bound.

Fix a real $\eta>0$. The radius is chosen as follows, in this order. If $s>0$, let $\rho>0$ be the radius $r_{\eta/2}$ of the hypothesis on $h$ at the point $s$ with $\eta/2$ in place of $\eta$, and put $r_{*}=\min\bigl(\rho,\ \eta/(2(L/(2s)+\theta))\bigr)$. If $s=0$, let $\rho>0$ be the radius $r_{\eta/2}$ of the hypothesis on $h$ at the point $0$, and put $r_{*}=\min(\rho,\ \eta/(2\theta))$. We verify the inequality of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub with slack $0$ and radius $r_{*}$. Let $(H_{1},M_{1},\Omega_{1})$ be a tracial W*-probability space and $Y,\tilde{S},Y''$ $L^{2}$ $d$-tuples of it with $\mathrm{law}(Y,\tilde{S})=\mathrm{law}(X',S)$ and $\lVert Y''-Y\rVert_{2}<r_{*}$.

Step D with $Z=(X',S)$, $W=(X,P)$, $Z_{1}=(Y,\tilde{S})$ and $V=Y''$ gives $L^{2}$ $d$-tuples $\hat{X}',\hat{S},\hat{X},\hat{P},\hat{Y}''$ of one tracial W*-probability space with $\mathrm{law}((\hat{X}',\hat{S}),(\hat{X},\hat{P}))=\mathrm{law}((X',S),(X,P))$ and $\mathrm{law}((\hat{X}',\hat{S}),\hat{Y}'')=\mathrm{law}((Y,\tilde{S}),Y'')$. By Step C: $\mathrm{law}(\hat{X},\hat{P},\hat{X}')=\mathrm{law}(X,P,X')=\gamma$; $\lVert\hat{S}-\hat{P}+c(\hat{X}'-\hat{X})\rVert_{2}=\lVert S-P+ca\rVert_{2}=0$, so $\hat{S}=\hat{P}-c\hat{a}$ with $\hat{a}=\hat{X}'-\hat{X}$; and, with $v=\hat{Y}''-\hat{X}'$, $\mathrm{law}(\hat{Y}'')=\mathrm{law}(Y'')$, $\lVert v\rVert_{2}=\lVert Y''-Y\rVert_{2}$ and $\langle\hat{S},v\rangle_{2}=\langle\tilde{S},Y''-Y\rangle_{2}$. Computing from the realisation $(\hat{X},\hat{P},\hat{X}')$ of $\gamma$, $s=\lVert\hat{a}\rVert_{2}$ and $p(\gamma)=\langle\hat{P},\hat{a}\rangle_{2}$. Put $\gamma''=\mathrm{law}(\hat{X},\hat{P},\hat{Y}'')$. As in Step E, $B_{\#}\gamma''=\mathrm{law}(\hat{X},\hat{P})=B_{\#}\gamma=\kappa_{2d}(\pi)$ and $C_{\#}\gamma''=\mathrm{law}(\hat{Y}'')$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, so $\gamma''\in\mathcal{C}(\mathrm{law}(\hat{Y}''))$ by Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings, and $s(\gamma'')=t$ with $t=\lVert\hat{a}+v\rVert_{2}$ and $p(\gamma'')=\langle\hat{P},\hat{a}+v\rangle_{2}$. We have $|t-s|\le\lVert v\rVert_{2}<r_{*}\le\rho$ and $t^{2}-s^{2}=2\langle\hat{a},v\rangle_{2}+\lVert v\rVert_{2}^{2}$.

Case $s>0$. Since $0\le(t-s)^{2}$ gives $2s(t-s)\le t^{2}-s^{2}$, we get $t-s\le\langle\hat{a},v\rangle_{2}/s+\lVert v\rVert_{2}^{2}/(2s)$. By the choice of $\rho$, $h(t)-h(s)\le h'(s)(t-s)+\tfrac{\eta}{2}|t-s|$, and, since $0\le h'(s)\le L$ and $|t-s|\le\lVert v\rVert_{2}$,

$$h(t)-h(s)\le\frac{h'(s)}{s}\langle\hat{a},v\rangle_{2}+\frac{L}{2s}\lVert v\rVert_{2}^{2}+\frac{\eta}{2}\lVert v\rVert_{2}.$$

Therefore

$$\Psi(\gamma'')-\Psi(\gamma)=\langle\hat{P},v\rangle_{2}-\bigl(h(t)-h(s)\bigr)-\theta\bigl(2\langle\hat{a},v\rangle_{2}+\lVert v\rVert_{2}^{2}\bigr)\ge\langle\hat{P}-c\hat{a},v\rangle_{2}-\Bigl(\frac{\eta}{2}+\Bigl(\frac{L}{2s}+\theta\Bigr)\lVert v\rVert_{2}\Bigr)\lVert v\rVert_{2},$$

and $(L/(2s)+\theta)\lVert v\rVert_{2}<\eta/2$ by the choice of $r_{*}$. Hence $\Psi(\gamma'')\ge\Psi(\gamma)+\langle\hat{S},v\rangle_{2}-\eta\lVert v\rVert_{2}$.

Case $s=0$. Then $\hat{a}=0$, $\hat{S}=\hat{P}$, $t=\lVert v\rVert_{2}$, $p(\gamma)=0$ and $\Psi(\gamma)=-h(0)=0$. By the choice of $\rho$ and $h(0)=h'(0)=0$, $h(t)\le\tfrac{\eta}{2}t$; and $\theta t^{2}\le\tfrac{\eta}{2}t$ since $t<\eta/(2\theta)$. Hence $\Psi(\gamma'')=\langle\hat{P},v\rangle_{2}-h(t)-\theta t^{2}\ge\Psi(\gamma)+\langle\hat{S},v\rangle_{2}-\eta\lVert v\rVert_{2}$.

In both cases, since $\gamma''\in\mathcal{C}(\mathrm{law}(\hat{Y}''))$, $\mathrm{law}(\hat{Y}'')=\mathrm{law}(Y'')$ and $\Psi(\gamma)=\Phi(\kappa_{d}(\nu))$, the Notation and the identities from Step C give

$$\Phi(\mathrm{law}(Y''))\ge\Psi(\gamma'')\ge\Phi(\kappa_{d}(\nu))+\langle\tilde{S},Y''-Y\rangle_{2}-(0+\eta)\lVert Y''-Y\rVert_{2}.$$

The left side is $\Phi_{M_{1}}(Y'')$ by Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts and Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift. So $\mathrm{law}(X',S)$ is a plan subdifferential of $\Phi$ at $\kappa_{d}(\nu)$ with slack $0$ in the sense of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub, that is, $\mathrm{law}(X',S)\in J^{-}\Phi(\kappa_{d}(\nu))$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet.

Proof of clause 4. First, $\Psi(\gamma)\le-\tfrac{\theta}{2}s(\gamma)^{2}$ for every $\gamma\in\mathcal{C}(\kappa_{d}(\mu))$. Write $s=s(\gamma)$. If $s<r_{0}$, the hypothesis gives $p(\gamma)\le\tfrac12h(s)$, so $\Psi(\gamma)\le-\tfrac12h(s)-\theta s^{2}\le-\tfrac{\theta}{2}s^{2}$ because $h\ge0$. If $s\ge r_{0}$, Step B (with $\nu=\mu\in\Sigma_{d,r}$) gives $p(\gamma)\le a_{\pi}s\le\tfrac{\theta r_{0}}{2}s\le\tfrac{\theta}{2}s^{2}$, so $\Psi(\gamma)\le\tfrac{\theta}{2}s^{2}-h(s)-\theta s^{2}\le-\tfrac{\theta}{2}s^{2}$. Hence $\Phi(\kappa_{d}(\mu))\le0$.

Second, by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling there are $L^{2}$ $d$-tuples $X,P$ of a tracial W*-probability space with $\mathrm{law}(X,P)=\kappa_{2d}(\pi)$, and then $\mathrm{law}(X)=\mathrm{pr}^{1}_{\#}\kappa_{2d}(\pi)=\kappa_{d}(\mu)$ as computed in Step B. Put $\gamma_{0}=\mathrm{law}(X,P,X)$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $B_{\#}\gamma_{0}=\mathrm{law}(X,P)=\kappa_{2d}(\pi)$ and $C_{\#}\gamma_{0}=\mathrm{law}(X)=\kappa_{d}(\mu)$, so $\gamma_{0}\in\mathcal{C}(\kappa_{d}(\mu))$ by Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing §couplings. From this realisation, $s(\gamma_{0})=\lVert X-X\rVert_{2}=0$ and $p(\gamma_{0})=\langle P,X-X\rangle_{2}=0$, so $\Psi(\gamma_{0})=-h(0)=0$ and $\Phi(\kappa_{d}(\mu))\ge0$. Therefore $\Phi(\kappa_{d}(\mu))=0$.

Proof of clause 5. Put $w_{n}=W_{2}(\nu_{n},\mu)$ and $s_{n}=s(\gamma_{n})$. The hypotheses of clause 4 hold, so $\Phi(\kappa_{d}(\mu))=0$, and the estimate proved for clause 2 gives $|\Phi(\kappa_{d}(\nu_{n}))|\le Kw_{n}$ for every $n$; hence $(\Phi(\kappa_{d}(\nu_{n})))_{n}$ converges to $0$. For every $n$, Step E (with $\nu=\nu_{n}$, $\nu'=\mu$, $\gamma=\gamma_{n}$) gives $\gamma''_{n}\in\mathcal{C}(\kappa_{d}(\mu))$ with $|s(\gamma''_{n})-s_{n}|\le w_{n}$ and $\Psi(\gamma''_{n})\ge\Psi(\gamma_{n})-Kw_{n}=\Phi(\kappa_{d}(\nu_{n}))-Kw_{n}\ge-2Kw_{n}$. By the first part of the proof of clause 4, $\Psi(\gamma''_{n})\le-\tfrac{\theta}{2}s(\gamma''_{n})^{2}$, so $s(\gamma''_{n})^{2}\le4Kw_{n}/\theta$. Given a real $\varepsilon>0$, choose $N\in\mathbb{N}$ such that $w_{n}<\min\bigl(\varepsilon/2,\ \theta\varepsilon^{2}/(16K)\bigr)$ for all $n\ge N$, by the convergence of $(w_{n})_{n}$ to $0$ in the sense of The Real Numbers: Standing Notation and Background §sequences; then for $n\ge N$, $s(\gamma''_{n})^{2}<\varepsilon^{2}/4$, so $s(\gamma''_{n})<\varepsilon/2$, and $0\le s_{n}\le s(\gamma''_{n})+w_{n}<\varepsilon$. Hence $(s_{n})_{n}$ converges to $0$.

For the last assertion, let $(X_{n},P_{n},X'_{n})$ and $S_{n}$ be as stated. Since $\gamma_{n}$ is a maximising coupling for $\nu_{n}$, clause 3 gives $\lVert S_{n}-P_{n}\rVert_{2}\le h'(s_{n})+2\theta s_{n}$, and $h'(s_{n})\ge0$ by Step A. Given a real $\varepsilon>0$, let $r_{h}>0$ be the radius of the hypothesis on $h'$ for $\varepsilon/2$ in place of $\eta$, and then choose $N\in\mathbb{N}$ with $s_{n}<\min\bigl(r_{h},\ \varepsilon/(4\theta)\bigr)$ for $n\ge N$. Then $0\le\lVert S_{n}-P_{n}\rVert_{2}<\varepsilon/2+\varepsilon/2=\varepsilon$ for $n\ge N$, so $(\lVert S_{n}-P_{n}\rVert_{2})_{n}$ converges to $0$.