Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\lambda\in\Sigma_{d}$. $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ is the tracial W*-probability space of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, with conjugation $J_{\lambda}$; its $L^{2}$ $d$-tuples are the $d$-tuples of fixed vectors of $J_{\lambda}$.

1. (Conjugate variables)¶ If $\lambda$ has conjugate variables $\xi_{\lambda}=(\xi_{\lambda,1},\dots,\xi_{\lambda,d})$, then $\xi_{\lambda}$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$.

2. (Wall force)¶ If $R>0$ is real and $\lambda$ has a square-integrable wall force $F^{R}(\lambda)=(F^{R}_{1}(\lambda),\dots,F^{R}_{d}(\lambda))$ of radius $R$, then $F^{R}(\lambda)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$.

3. (Score)¶ If $(\mathcal{D}_{0},\mathcal{E}_{0})$ is a free entropy penalty, $R>0$ is real, and $\lambda$ lies in the score domain $\mathcal{D}_{\Xi}$ of its wall-confined free energy with radius $R$, then the score $\Xi(\lambda)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$.