Proof of Plan Jets at a Common Realisation: Superdifferential and Subdifferential Momenta are Close, and the Plan Jets of a Constant

Each result cited is universally quantified over the data in its own statement. For $L^{2}$ $d$-tuples of one tracial W*-probability space, sums and real multiples are those of $H^{d}$ and $\langle\cdot,\cdot\rangle_{2}$ is the inner product of $H^{d}$ with norm $\lVert\cdot\rVert_{2}$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations, Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing); so bilinearity over $\mathbb{R}$ and the Cauchy--Schwarz inequality $|\langle Y,Z\rangle_{2}|\le\lVert Y\rVert_{2}\lVert Z\rVert_{2}$ hold (Cauchy-Schwarz Inequality in a Complex Inner Product Space). Lifts satisfy $u_{M}(Y)=u(\mathrm{law}(Y))$ (Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift).

Claim 1. Put $D=G-P$ and fix $\eta>0$. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub there are $r_{1},r_{2}>0$ such that, for every $L^{2}$ $d$-tuple $X'$ of $(H,M,\Omega)$ with $\lVert X'-X\rVert_{2}<\min\{r_{1},r_{2}\}$ (the realisations $(X,P,X')$ and $(X,G,X')$ being admissible, since $\mathrm{law}(X,P)$ and $\mathrm{law}(X,G)$ are the given plans),

If $D=0$ there is nothing to prove. Otherwise take $X'=X+tD$ with real $t>0$ and $t\lVert D\rVert_{2}<\min\{r_{1},r_{2}\}$. Subtracting the two inequalities gives $t\langle G-P,D\rangle_{2}\le(\delta+\delta'+2\eta)t\lVert D\rVert_{2}$, i.e. $\lVert D\rVert_{2}^{2}\le(\delta+\delta'+2\eta)\lVert D\rVert_{2}$, hence $\lVert D\rVert_{2}\le\delta+\delta'+2\eta$. As $\eta>0$ was arbitrary, $\lVert P-G\rVert_{2}\le\delta+\delta'$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above.

Claim 2. Let $\pi$ be a plan at $\mu$. For every realisation $(X,P)$ of $\pi$ in any tracial W*-probability space, $\lVert P\rVert_{2}^{2}=\widehat{M}(\mathrm{law}(P))=\widehat{M}(\mathrm{pr}^{2}_{\#}\pi)$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward (the entries of $\mathrm{pr}^{2}(X,P)$ being those of $P$, Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations). So the third condition says $\lVert P\rVert_{2}\le\delta$ for every realisation (Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field).

Third $\Rightarrow$ first and second. If $\lVert P\rVert_{2}\le\delta$, then for every $X'$, by Cauchy--Schwarz, $u_{M}(X')=c=u(\mu)$ lies between $c+\langle P,X'-X\rangle_{2}-\delta\lVert X'-X\rVert_{2}$ and $c+\langle P,X'-X\rangle_{2}+\delta\lVert X'-X\rVert_{2}$; so $\pi$ satisfies both defining inequalities with any $\eta>0$ and any $r>0$.

First $\Rightarrow$ third. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling there is a realisation $(X,P)$ of $\pi$. Fix $\eta>0$, let $r$ be as in Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super, and suppose $P\ne0$. With $X'=X-tP$, $0<t\lVert P\rVert_{2}<r$, the superdifferential inequality reads $c\le c-t\lVert P\rVert_{2}^{2}+(\delta+\eta)t\lVert P\rVert_{2}$, so $\lVert P\rVert_{2}\le\delta+\eta$. Hence $\lVert P\rVert_{2}\le\delta$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above; this also holds if $P=0$.

Second $\Rightarrow$ third. The same argument with $X'=X+tP$ and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub gives $c\ge c+t\lVert P\rVert_{2}^{2}-(\delta+\eta)t\lVert P\rVert_{2}$, hence again $\lVert P\rVert_{2}\le\delta$ (trivially so if $P=0$).