The Amalgamated Free Product Space: Multilinearity of the Tuple Vectors and the Bounded Left Actions of the Two Tracial Algebras

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 $\mu$, tracial algebras $N$ and $A_{\varepsilon}$ and embeddings $\pi_{\varepsilon}$, let $A_{\varepsilon}^{\circ}$ be the centred elements, $\mathcal{F}$ the free vector space and $\ell_{\varepsilon}(b)$ the formal actions, and let $\mathcal{H}$, $J$, $\Xi_{N}$ and $\Xi$ be the amalgamated free product space, its canonical map and its vectors. Tuples $(a_{1},\dots,a_{k})$ below are alternating tuples of a fixed type $(e_{1},\dots,e_{k})$, and $j\in[k]$.

1. (Multilinearity)¶ For all $x,y\in N$ and $c\in\mathbb{C}$: $\Xi_{N}(x+y)=\Xi_{N}(x)+\Xi_{N}(y)$ and $\Xi_{N}(cx)=c\,\Xi_{N}(x)$. For all $a_{j}'\in A_{e_{j}}^{\circ}$ and $c\in\mathbb{C}$,

where only the $j$-th entry is changed. In particular $\Xi(a_{1},\dots,a_{k})=0$ if $a_{j}=0$ for some $j$.

2. (Balance over $N$)¶ If $j<k$ and $x\in N$, then

3. (Relations of the formal actions)¶ For all $\varepsilon\in\{1,2\}$, $b,b'\in A_{\varepsilon}$, $c\in\mathbb{C}$, $x\in N$ and $\xi\in\mathcal{F}$:

and $J\ell_{1}(\pi_{1}(x))\xi=J\ell_{2}(\pi_{2}(x))\xi$.

4. (Bound)¶ For all $\varepsilon\in\{1,2\}$, $b\in A_{\varepsilon}$ and $\xi\in\mathcal{F}$, $\lVert J\ell_{\varepsilon}(b)\xi\rVert_{\mathcal{H}}\le\lVert b\rVert_{\mathrm{op}}\lVert J\xi\rVert_{\mathcal{H}}$.

5. (Actions)¶ For every $\varepsilon\in\{1,2\}$ and $b\in A_{\varepsilon}$ there is exactly one $\Lambda_{\varepsilon}(b)\in\mathcal{L}(\mathcal{H})$ with $\Lambda_{\varepsilon}(b)J\xi=J\ell_{\varepsilon}(b)\xi$ for every $\xi\in\mathcal{F}$, and $\lVert\Lambda_{\varepsilon}(b)\rVert_{\mathrm{op}}\le\lVert b\rVert_{\mathrm{op}}$.