Gluing Countably Many Noncommutative Couplings with a Common First Marginal in One Tracial W*-Probability Space

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $d\in\mathbb{N}$ and let $R>0$ be real. Self-adjoint $d$-tuples $s$ in $M$ and their laws $\lambda_{s}$ are those of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws, and for self-adjoint $d$-tuples $s,t$ in $M$, $(s,t)$ is the self-adjoint $2d$-tuple of Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling. Let $\mu\in\Sigma_{d,R}$, and for every $k\in\mathbb{N}$ let $\nu_{k}\in\Sigma_{d,R}$ and let $\gamma_{k}\in\Pi(\mu,\nu_{k})$ be a coupling.

¶There are a tracial W*-probability space $(H,M,\Omega)$, a self-adjoint $d$-tuple $s$ in $M$ and, for every $k\in\mathbb{N}$, a self-adjoint $d$-tuple $t^{k}$ in $M$, such that $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ and $\lVert t^{k}_{j}\rVert_{\mathrm{op}}\le R$ for all $j\in[d]$ and $k\in\mathbb{N}$, $\lambda_{s}=\mu$, and $\lambda_{(s,t^{k})}=\gamma_{k}$ and $\lambda_{t^{k}}=\nu_{k}$ for every $k\in\mathbb{N}$.