Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $u:\Sigma^{2}_{d}\to\mathbb{R}$, let $\mu\in\Sigma^{2}_{d}$, let $\delta\ge0$ be real, and let $\pi$ be a plan at $\mu$. The lifts $u_{M}$ are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §lifts, differences and the $L^{2}$ norm of $L^{2}$ $d$-tuples are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples, and the pairing $\langle\cdot,\cdot\rangle_{2}$ is that of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing.

1. (Superdifferentials)¶ $\pi$ is a plan superdifferential of $u$ at $\mu$ with slack $\delta$ if for every real $\eta>0$ there is a real $r>0$ such that, for every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P,X'$ of $(H,M,\Omega)$ with $\mathrm{law}(X,P)=\pi$ and $\lVert X'-X\rVert_{2}<r$,

2. (Subdifferentials)¶ $\pi$ is a plan subdifferential of $u$ at $\mu$ with slack $\delta$ if for every real $\eta>0$ there is a real $r>0$ such that, for every tracial W*-probability space $(H,M,\Omega)$ and all $L^{2}$ $d$-tuples $X,P,X'$ of $(H,M,\Omega)$ with $\mathrm{law}(X,P)=\pi$ and $\lVert X'-X\rVert_{2}<r$,

3. (Superjet)¶ The plan superjet $J^{+}_{\delta}u(\mu)$ of $u$ at $\mu$ with slack $\delta$ is the set of plans at $\mu$ that are plan superdifferentials of $u$ at $\mu$ with slack $\delta$; $J^{+}u(\mu)=J^{+}_{0}u(\mu)$.

4. (Subjet)¶ The plan subjet $J^{-}_{\delta}u(\mu)$ of $u$ at $\mu$ with slack $\delta$ is the set of plans at $\mu$ that are plan subdifferentials of $u$ at $\mu$ with slack $\delta$; $J^{-}u(\mu)=J^{-}_{0}u(\mu)$.