The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State

In the setting of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product, with laws $\gamma_{1},\gamma_{2}$ of common marginal $\mu$ and tracial algebra $N$ of $\mu$, let $\mathcal{F}$ be the free vector space, with the vectors $\delta^{N}_{x}$ ($x\in N$) and $\delta_{t}$ ($t$ an alternating tuple), and let $h$ be the nested-expectation form; $\mathcal{F}$ is a complex vector space by The Formal Amalgamated Free Product: the Free Vector Space, Positivity of the Nested-Expectation Form, and Linearity and Formal Adjoints of the Actions §vector-space, and $h$ satisfies the hypotheses of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §real-structure by The Formal Amalgamated Free Product: the Free Vector Space, Positivity of the Nested-Expectation Form, and Linearity and Formal Adjoints of the Actions §form.

1. (Space)¶ The amalgamated free product space of $\gamma_{1}$ and $\gamma_{2}$ over $\mu$ is the complex Hilbert completion $\mathcal{H}$ of $(\mathcal{F},h)$, which is a complex Hilbert space by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert. Its inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ is the pairing $\langle\cdot,\cdot\rangle_{h}$ of that definition, $\lVert\cdot\rVert_{\mathcal{H}}$ is the induced norm, and $J:\mathcal{F}\to\mathcal{H}$ is the canonical map $J_{h}$.

2. (Vectors)¶ For $x\in N$ let $\Xi_{N}(x)=J\delta^{N}_{x}$; the vacuum vector is $\Omega=\Xi_{N}(I)$. For an alternating tuple $t=(a_{1},\dots,a_{k})$ let $\Xi(t)=J\delta_{t}$, also written $\Xi(a_{1},\dots,a_{k})$.

3. (Vacuum state)¶ The vacuum state is the map $\varphi:\mathcal{L}(\mathcal{H})\to\mathbb{C}$, $\varphi(T)=\langle\Omega,T\Omega\rangle_{\mathcal{H}}$.