Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $m,n\in\mathbb{N}$, let $\gamma\in\Sigma_{m}$, let $a=(a_{1},\dots,a_{n})$ be an $n$-tuple in $\mathcal{P}_{m,\mathrm{sa}}$ with substitution $\sigma_{a}:\mathcal{P}_{n}\to\mathcal{P}_{m}$, and let $\mu=\gamma\circ\sigma_{a}$. For a law $\lambda$ (here $\gamma$ or $\mu$), $\mathcal{H}_{\lambda}$, $\Omega_{\lambda}$ and $\widehat{p}^{\,\lambda}$ are its complex GNS space, vacuum vector and classes; $L_{p}$, $R_{p}$ and $J_{\lambda}$ are the multiplication operators and the conjugation on $\mathcal{H}_{\lambda}$, acting on $\mathcal{H}_{\gamma}$ for polynomials taken in $\mathcal{P}_{m}$ and on $\mathcal{H}_{\mu}$ for polynomials taken in $\mathcal{P}_{n}$; and $\mathcal{M}_{\lambda}$, $\tau_{\lambda}$ are the tracial algebra and its trace.

1. (The marginal law)¶ $\mu\in\Sigma_{n}$; more precisely, $\mu\in\Sigma_{n,C}$ for every real $C>0$ with $\lVert L_{a_{j}}\rVert_{\mathrm{op}}\le C$ for all $j\in[n]$, the operators $L_{a_{j}}$ acting on $\mathcal{H}_{\gamma}$.

2. (The isometry)¶ In this lemma the letter $V$ denotes an operator, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces. There is exactly one $V\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ with $V\widehat{p}^{\,\mu}=\widehat{\sigma_{a}(p)}^{\,\gamma}$ for every $p\in\mathcal{P}_{n}$. It satisfies $V^{*}V=I$, $V\Omega_{\mu}=\Omega_{\gamma}$, $VJ_{\mu}=J_{\gamma}V$, and, for every $p\in\mathcal{P}_{n}$,

3. (The embedding)¶ For every $T\in\mathcal{M}_{\mu}$ there is exactly one $\pi(T)\in\mathcal{M}_{\gamma}$ with $\pi(T)\Omega_{\gamma}=VT\Omega_{\mu}$. It satisfies $\lVert\pi(T)\rVert_{\mathrm{op}}\le\lVert T\rVert_{\mathrm{op}}$ and $\pi(T)V=VT$.

4. (Homomorphism)¶ The map $\pi:\mathcal{M}_{\mu}\to\mathcal{M}_{\gamma}$ is linear and injective, and for all $S,T\in\mathcal{M}_{\mu}$ and $p\in\mathcal{P}_{n}$

5. (Conditional expectation)¶ For $b\in\mathcal{M}_{\gamma}$ let $E(b)=V^{*}bV$. Then $E(b)\in\mathcal{M}_{\mu}$, the map $E:\mathcal{M}_{\gamma}\to\mathcal{M}_{\mu}$ is linear, and for all $b\in\mathcal{M}_{\gamma}$, $S,T\in\mathcal{M}_{\mu}$ and $\zeta,\zeta'\in\mathcal{H}_{\mu}$

In particular $E(I)=I$ and $E(b^{*}b)\ge0$.