Proof of Plan Jets of a Sum along a Common Realisation

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 $\lVert\cdot\rVert_{2}$, 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; so the pairing is bilinear over $\mathbb{R}$, and a tuple of norm $0$ is the zero tuple. Lifts satisfy $f_{M}(Y)=f(\mathrm{law}(Y))$ by Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples §lift; in particular $(\varphi_{1}+\varphi_{2})_{M}(Y)=(\varphi_{1})_{M}(Y)+(\varphi_{2})_{M}(Y)$ for every $L^{2}$ $d$-tuple $Y$.

Step 0 (law invariance). Let $A$ and $A'$ be $L^{2}$ $k$-tuples of two (possibly different) tracial W*-probability spaces with $\mathrm{law}(A)=\mathrm{law}(A')$, and let $T,S$ be affine data from $k$ to $d$ variables with zero constant part. By the entrywise formula of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, the pair $(TA,SA)$ equals $UA$, where $U$ is the affine datum from $k$ to $2d$ variables whose first $d$ rows are those of $T$ and whose last $d$ rows are those of $S$; likewise $(TA',SA')=UA'$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $\mathrm{law}(TA)=T_{\#}\mathrm{law}(A)=T_{\#}\mathrm{law}(A')=\mathrm{law}(TA')$ and similarly $\mathrm{law}(UA)=\mathrm{law}(UA')$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments, applied to $TA$ and to $UA$ and to their primed counterparts,

$$\lVert TA\rVert_{2}^{2}=\widehat{M}(\mathrm{law}(TA))=\lVert TA'\rVert_{2}^{2},\qquad\langle TA,SA\rangle_{2}=\sum_{j=1}^{d}\mathrm{m}_{j,d+j}(\mathrm{law}(UA))=\langle TA',SA'\rangle_{2}.$$

Also $f_{M}(TA)=f(\mathrm{law}(TA))=f(\mathrm{law}(TA'))$ for every $f:\Sigma^{2}_{d}\to\mathbb{R}$, where the second lift is taken in the space of $A'$. Every sub-tuple, sum and difference of blocks of a tuple used below is of the form $TA$ with such a datum $T$, by the same entrywise formula. So norms, pairings and lift values of such combinations depend only on the law of $A$.

Step 1 (the summed law is a plan at $\lambda$). $\mathrm{law}(X,P_{1}+P_{2})\in\Sigma^{2}_{2d}$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, $\mathrm{pr}^{1}_{\#}\mathrm{law}(X,P_{1}+P_{2})=\mathrm{law}(\mathrm{pr}^{1}(X,P_{1}+P_{2}))=\mathrm{law}(X)=\lambda$, since the entries of $\mathrm{pr}^{1}(X,P_{1}+P_{2})$ are those of $X$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate). Hence $\mathrm{law}(X,P_{1}+P_{2})$ is a plan at $\lambda$ by Plans at a Square-Integrable Noncommutative Law §plan.

Step 2 (Claim 1, superjets). Fix a real $\eta>0$. By Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super, applied to $\varphi_{i}$, the plan $\mathrm{law}(X,P_{i})$ and slack $\delta_{i}$ with the positive number $\eta/2$ in place of $\eta$, there are reals $r_{i}>0$ ($i=1,2$), depending only on $\eta$, such that for every tracial W*-probability space $(K,N,\Psi)$ and all $L^{2}$ $d$-tuples $A,B,A'$ of it with $\mathrm{law}(A,B)=\mathrm{law}(X,P_{i})$ and $\lVert A'-A\rVert_{2}<r_{i}$,

$$(\varphi_{i})_{N}(A')\le\varphi_{i}(\lambda)+\langle B,A'-A\rangle_{2}+(\delta_{i}+\eta/2)\lVert A'-A\rVert_{2}.$$

Put $r=\min\{r_{1},r_{2}\}>0$. Now let $(\tilde H,\tilde M,\tilde\Omega)$ be any tracial W*-probability space and $\tilde X,\tilde P,\tilde X'$ any $L^{2}$ $d$-tuples of it with $\mathrm{law}(\tilde X,\tilde P)=\mathrm{law}(X,P_{1}+P_{2})$ and $\lVert\tilde X'-\tilde X\rVert_{2}<r$.

Gluing. Put $\pi_{1}=\mathrm{law}((X,P_{1}+P_{2}),(P_{1},P_{2}))\in\Sigma^{2}_{4d}$ and $\pi_{2}=\mathrm{law}(\tilde X,\tilde P,\tilde X')\in\Sigma^{2}_{3d}$. With $k=2d$ and the data $F^{2d},F^{d}$ of Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal, which select the first $2d$ entries, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward gives $F^{2d}_{\#}\pi_{1}=\mathrm{law}(X,P_{1}+P_{2})=\mathrm{law}(\tilde X,\tilde P)=F^{d}_{\#}\pi_{2}$. By Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue (with $k=2d$, $m=2d$, $n=d$ and these $\pi_{1},\pi_{2}$) there are a tracial W*-probability space $(\hat H,\hat M,\hat\Omega)$, an $L^{2}$ $2d$-tuple $(\hat X,\hat P)$ (written by its two blocks of $d$ entries), an $L^{2}$ $2d$-tuple $(Y_{1},Y_{2})$ and an $L^{2}$ $d$-tuple $\hat X'$ of it with

$$\mathrm{law}((\hat X,\hat P),(Y_{1},Y_{2}))=\mathrm{law}((X,P_{1}+P_{2}),(P_{1},P_{2})),\qquad\mathrm{law}(\hat X,\hat P,\hat X')=\mathrm{law}(\tilde X,\tilde P,\tilde X').$$

Consequences of law invariance. By Step 0 applied to the first equality of laws: $\lVert\hat P-Y_{1}-Y_{2}\rVert_{2}=\lVert(P_{1}+P_{2})-P_{1}-P_{2}\rVert_{2}=0$, so $\hat P=Y_{1}+Y_{2}$; and $\mathrm{law}(\hat X,Y_{i})=\mathrm{law}(X,P_{i})$ for $i=1,2$ (push-forward under the datum selecting the corresponding blocks, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward). By Step 0 applied to the second equality of laws: $\lVert\hat X'-\hat X\rVert_{2}=\lVert\tilde X'-\tilde X\rVert_{2}<r$, $\langle\hat P,\hat X'-\hat X\rangle_{2}=\langle\tilde P,\tilde X'-\tilde X\rangle_{2}$, and $(\varphi_{1}+\varphi_{2})_{\hat M}(\hat X')=(\varphi_{1}+\varphi_{2})_{\tilde M}(\tilde X')$.

Adding the inequalities. The triples $(\hat X,Y_{i},\hat X')$ in $(\hat H,\hat M,\hat\Omega)$ are admissible in the displayed inequality for $i=1,2$, since $\mathrm{law}(\hat X,Y_{i})=\mathrm{law}(X,P_{i})$ and $\lVert\hat X'-\hat X\rVert_{2}<r\le r_{i}$. Adding the two resulting inequalities, and using the additivity of lifts, bilinearity of the pairing and $Y_{1}+Y_{2}=\hat P$,

$$(\varphi_{1}+\varphi_{2})_{\hat M}(\hat X')\le(\varphi_{1}+\varphi_{2})(\lambda)+\langle\hat P,\hat X'-\hat X\rangle_{2}+(\delta_{1}+\delta_{2}+\eta)\lVert\hat X'-\hat X\rVert_{2}.$$

By the consequences of law invariance, each term equals its counterpart for $(\tilde X,\tilde P,\tilde X')$, so

$$(\varphi_{1}+\varphi_{2})_{\tilde M}(\tilde X')\le(\varphi_{1}+\varphi_{2})(\lambda)+\langle\tilde P,\tilde X'-\tilde X\rangle_{2}+(\delta_{1}+\delta_{2}+\eta)\lVert\tilde X'-\tilde X\rVert_{2}.$$

As $\eta>0$ was arbitrary and $r$ depends only on $\eta$, the plan $\mathrm{law}(X,P_{1}+P_{2})$ at $\lambda$ (Step 1) is a plan superdifferential of $\varphi_{1}+\varphi_{2}$ at $\lambda$ with slack $\delta_{1}+\delta_{2}$ in the sense of Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super, that is, it lies in $J^{+}_{\delta_{1}+\delta_{2}}(\varphi_{1}+\varphi_{2})(\lambda)$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet.

Step 3 (Claim 2, subjets). The argument of Step 2 applies verbatim with Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet in place of 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 §superjet: the inequalities are reversed and the slack terms $(\delta_{i}+\eta/2)\lVert A'-A\rVert_{2}$ enter with a minus sign, so adding them gives

$$(\varphi_{1}+\varphi_{2})_{\tilde M}(\tilde X')\ge(\varphi_{1}+\varphi_{2})(\lambda)+\langle\tilde P,\tilde X'-\tilde X\rangle_{2}-(\delta_{1}+\delta_{2}+\eta)\lVert\tilde X'-\tilde X\rVert_{2}$$

for every admissible realisation $(\tilde X,\tilde P,\tilde X')$ with $\lVert\tilde X'-\tilde X\rVert_{2}<r$. Hence $\mathrm{law}(X,P_{1}+P_{2})\in J^{-}_{\delta_{1}+\delta_{2}}(\varphi_{1}+\varphi_{2})(\lambda)$.