The Formal Amalgamated Free Product: the Free Vector Space, Positivity of the Nested-Expectation Form, and Linearity and Formal Adjoints of the Actions

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 algebras $N$ and $A_{\varepsilon}$, let $\mathcal{T}$, $\mathcal{F}$, $\delta_{s}$, the nested expectations $X_{j}(s,t)$, the basic pairing $h_{0}$, the form $h$, the vectors $\rho_{\varepsilon,b}(u)$ and the formal actions $\ell_{\varepsilon}(b)$ be those of The Formal Amalgamated Free Product over a Common Marginal: Alternating Tuples, the Free Vector Space, the Nested-Expectation Form and the Formal Actions (labels, free vector space, nested expectations, form, formal actions). The Hilbert space $\mathcal{H}_{\mu}^{M}$ and block entries are those of Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators.

1. (Vector space)¶ With its operations, $\mathcal{F}$ is a complex vector space whose zero vector is the zero map, and $\xi=\sum_{s\in\operatorname{supp}\xi}\xi(s)\,\delta_{s}$ for every $\xi\in\mathcal{F}$ with $\xi\neq0$.

2. (Symmetry of nested expectations)¶ For alternating tuples $s,t$ of the same length $k$ and the same type, and every $j\in[k]$, $X_{j}(t,s)=X_{j}(s,t)^{*}$.

3. (Positive Gram operators)¶ Let $k,M\in\mathbb{N}$ and let $t^{1},\dots,t^{M}$ be alternating tuples of length $k$ and a common type. Then the operator $G\in\mathcal{L}(\mathcal{H}_{\mu}^{M})$ whose block entries are $G_{rq}=X_{k}(t^{r},t^{q})$ for $r,q\in[M]$, which exists and is unique by Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §blocks, satisfies $G\ge0$.

4. (The form)¶ $h$ 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 on $\mathcal{F}$: for all $\xi,\eta,\zeta\in\mathcal{F}$ and $c\in\mathbb{C}$,

and $h(\xi,\xi)$ is a real number with $0\le h(\xi,\xi)$. Moreover $h(\delta_{s},\delta_{t})=h_{0}(s,t)$ for all $s,t\in\mathcal{T}$.

5. (Formal actions)¶ Let $\varepsilon\in\{1,2\}$ and $b\in A_{\varepsilon}$. Then $\ell_{\varepsilon}(b)$ is a linear map with $\ell_{\varepsilon}(b)\delta_{u}=\rho_{\varepsilon,b}(u)$ for every $u\in\mathcal{T}$, and