The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State

In the setting of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product, let $\gamma_{1}\in\Sigma_{m_{1}}$ and $\gamma_{2}\in\Sigma_{m_{2}}$ be laws with the common marginal $\mu\in\Sigma_{n}$, with tracial algebras $N=\mathcal{M}_{\mu}$ and $A_{\varepsilon}=\mathcal{M}_{\gamma_{\varepsilon}}$, traces $\tau_{\mu}$ and $\tau_{\gamma_{\varepsilon}}$, and embeddings $\pi_{\varepsilon}$. Let $\mathcal{H}$ be the amalgamated free product space, with vectors $\Xi_{N}(x)$ and $\Xi(t)$, vacuum vector $\Omega$ and vacuum state $\varphi$, and let $\Lambda_{\varepsilon}(b)\in\mathcal{L}(\mathcal{H})$ be the actions of $b\in A_{\varepsilon}$.

1. (Representations)¶ For each $\varepsilon\in\{1,2\}$ the map $\Lambda_{\varepsilon}:A_{\varepsilon}\to\mathcal{L}(\mathcal{H})$ is linear, and for all $b,b'\in A_{\varepsilon}$

2. (Amalgamation)¶ $\Lambda_{1}(\pi_{1}(x))=\Lambda_{2}(\pi_{2}(x))$ for every $x\in N$.

3. (Freeness)¶ $\Lambda_{\varepsilon}(\pi_{\varepsilon}(x))\Omega=\Xi_{N}(x)$ for every $\varepsilon\in\{1,2\}$ and $x\in N$. For every alternating tuple $(a_{1},\dots,a_{k})$ of type $(e_{1},\dots,e_{k})$,

4. (Marginals)¶ $\lVert\Omega\rVert_{\mathcal{H}}=1$, and $\varphi(\Lambda_{\varepsilon}(b))=\tau_{\gamma_{\varepsilon}}(b)$ for every $\varepsilon\in\{1,2\}$ and $b\in A_{\varepsilon}$.

5. (Traciality)¶ Let $r\in\mathbb{N}$ with $r\ge2$, let $f_{1},\dots,f_{r}\in\{1,2\}$ (not necessarily alternating) and let $c_{j}\in A_{f_{j}}$ for $j\in[r]$. Then