Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation

1. (Conventions)¶ The conventions of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation are in force. Throughout, $d\in\mathbb{N}$; the letter $H$ names the Hilbert space of a tracial W*-probability space $(H,M,\Omega)$, and Hamiltonians are written $\mathcal{H}$.

2. (Combinations and pairing)¶ For $L^{2}$ $d$-tuples $X,Y$ and real $t$, the sum $X+Y$ and real multiple $tX$ and the pairing $\langle X,Y\rangle_{2}$ are those of Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space; differences $X-Y$ and the $L^{2}$ norm $\lVert X\rVert_{2}$ are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples.

3. (Lifts)¶ For $k\in\mathbb{N}$, a function $f:\Sigma^{2}_{k}\to\mathbb{R}$ and a tracial W*-probability space $(H,M,\Omega)$, $f_{M}$ is the lift of $f$ to $(H,M,\Omega)$; in particular $\mathcal{H}_{M}(X,P)=\mathcal{H}(\mathrm{law}(X,P))$ for $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ and $L^{2}$ $d$-tuples $X,P$.

4. (Metrics)¶ For $k\in\mathbb{N}$, $\Sigma^{2}_{k}$ carries the metric $\widehat{W}_{2}$ and $\mathbb{R}$ the metric of The Absolute Value Metric on the Real Line; semicontinuity, continuity and the Lipschitz property of functions $\Sigma^{2}_{k}\to\mathbb{R}$ refer to these metrics.

5. (Background)¶ Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators, Cauchy-Schwarz Inequality in a Complex Inner Product Space and The Induced Norm is a Norm, and Induces a Metric are in force and may be used without restating them.