The Formal Amalgamated Free Product over a Common Marginal: Alternating Tuples, the Free Vector Space, the Nested-Expectation Form and the Formal Actions

We work 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 $A_{\varepsilon}$, embeddings $\pi_{\varepsilon}$, expectations $E_{\varepsilon}$.

1. (Centred parts)¶ For $b\in A_{\varepsilon}$, the centred part of $b$ is $b^{\circ}=b-\pi_{\varepsilon}(E_{\varepsilon}(b))$, and $A_{\varepsilon}^{\circ}=\{b\in A_{\varepsilon}:E_{\varepsilon}(b)=0\}$. For $b\in A_{\varepsilon}$, $a\in A_{\varepsilon}^{\circ}$ and $x\in N$, the elements $b^{\circ}$, $\pi_{\varepsilon}(x)\,a$ and $a\,\pi_{\varepsilon}(x)$ belong to $A_{\varepsilon}^{\circ}$: they lie in $A_{\varepsilon}$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra, and $E_{\varepsilon}$ annihilates them because $E_{\varepsilon}$ is linear with $E_{\varepsilon}(\pi_{\varepsilon}(T))=T$ and $E_{\varepsilon}(\pi_{\varepsilon}(S)\,c\,\pi_{\varepsilon}(T))=S\,E_{\varepsilon}(c)\,T$ for $S,T\in N$ and $c\in A_{\varepsilon}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §expectation, and $\pi_{\varepsilon}(I)=I$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism.

2. (Alternating tuples)¶ Let $k\in\mathbb{N}$. An alternating tuple of length $k$ is a $k$-tuple $t=((e_{1},a_{1}),\dots,(e_{k},a_{k}))$ of pairs with $e_{j}\in\{1,2\}$ and $a_{j}\in A_{e_{j}}^{\circ}$ for every $j\in[k]$, and $e_{j}\neq e_{j+1}$ for every $j\in[k]$ with $j<k$. Its type is $(e_{1},\dots,e_{k})$; when the type is clear, $t$ is also written $(a_{1},\dots,a_{k})$.

3. (Labels)¶ $\mathcal{T}$ is the set of the pairs $(0,x)$ with $x\in N$ together with the pairs $(k,t)$ with $k\in\mathbb{N}$ and $t$ an alternating tuple of length $k$; here $0$ is the real number zero, which is not a natural number.

4. (Free vector space)¶ $\mathcal{F}$ is the set of maps $\xi:\mathcal{T}\to\mathbb{C}$ whose support $\operatorname{supp}\xi=\{s\in\mathcal{T}:\xi(s)\neq0\}$ is finite. For $\xi,\eta\in\mathcal{F}$ and $c\in\mathbb{C}$, $\xi+\eta$ and $c\xi$ are the maps $s\mapsto\xi(s)+\eta(s)$ and $s\mapsto c\,\xi(s)$; their supports lie in $\operatorname{supp}\xi\cup\operatorname{supp}\eta$ and in $\operatorname{supp}\xi$, so they belong to $\mathcal{F}$ by claim 3 of Peeling an Element off a Finite Set, and Unions of Finite Sets and claim 3 of Basic Properties of Finite Sets. The zero map is written $0$. For $s\in\mathcal{T}$, $\delta_{s}$ is the map with $\delta_{s}(s)=1$ and $\delta_{s}(s')=0$ for $s'\neq s$, which belongs to $\mathcal{F}$ by claim 2 of Basic Properties of Finite Sets; we write $\delta_{x}^{N}=\delta_{(0,x)}$ for $x\in N$ and $\delta_{t}=\delta_{(k,t)}$ for an alternating tuple $t$ of length $k$. For a nonempty finite set $F$ and maps $c:F\to\mathbb{C}$ and $\rho:F\to\mathcal{F}$, $\sum_{u\in F}c(u)\rho(u)$ is the map $s\mapsto\sum_{u\in F}c(u)\,\rho(u)(s)$ (a sum over a finite index set); it belongs to $\mathcal{F}$, because by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing its support lies in $\bigcup_{u\in F}\operatorname{supp}\rho(u)$, which is finite by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §finite-union.

5. (Nested expectations)¶ Let $s=(a_{1},\dots,a_{k})$ and $t=(b_{1},\dots,b_{k})$ be alternating tuples of the same length $k$ and the same type $(e_{1},\dots,e_{k})$. Their nested expectations $X_{1}(s,t),\dots,X_{k}(s,t)\in N$ are

the sequence given by Definition of Sequences by Recursion on the Natural Numbers §recursion with first term $E_{e_{1}}(a_{1}^{*}b_{1})$ and step map $(j,Y)\mapsto E_{e_{j+1}}(a_{j+1}^{*}\pi_{e_{j+1}}(Y)b_{j+1})$ for $j<k$ and $(j,Y)\mapsto Y$ for $j\ge k$, on the set $N$; the arguments of $E_{e_{j}}$ lie in $A_{e_{j}}$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra.

6. (Form)¶ The basic pairing $h_{0}:\mathcal{T}\times\mathcal{T}\to\mathbb{C}$ is given by $h_{0}((0,x),(0,y))=\tau_{\mu}(x^{*}y)$ for $x,y\in N$, where $x^{*}y\in N$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra; by $h_{0}((k,s),(k,t))=\tau_{\mu}(X_{k}(s,t))$ for alternating tuples $s,t$ of the same length $k$ and the same type; and by $h_{0}(u,v)=0$ for all other pairs $(u,v)$. The nested-expectation form $h:\mathcal{F}\times\mathcal{F}\to\mathbb{C}$ is

if $\xi\neq0$ and $\eta\neq0$, and $h(\xi,\eta)=0$ otherwise.

7. (Formal actions)¶ Let $\varepsilon\in\{1,2\}$ and $b\in A_{\varepsilon}$. For $u\in\mathcal{T}$ the vector $\rho_{\varepsilon,b}(u)\in\mathcal{F}$ is:

(a) for $u=(0,x)$ with $x\in N$: $\rho_{\varepsilon,b}(u)=\delta_{((\varepsilon,(b\pi_{\varepsilon}(x))^{\circ}))}+\delta^{N}_{E_{\varepsilon}(b\pi_{\varepsilon}(x))}$;

(b) for $u=(k,t)$ with $t=((e_{1},a_{1}),\dots,(e_{k},a_{k}))$ and $e_{1}=\bar{\varepsilon}$:

(c) for $u=(k,t)$ with $t$ as in (b) but $e_{1}=\varepsilon$: if $k\ge2$,

and if $k=1$, $\rho_{\varepsilon,b}(u)=\delta_{((\varepsilon,(ba_{1})^{\circ}))}+\delta^{N}_{E_{\varepsilon}(ba_{1})}$.

The tuples displayed are alternating by the membership facts of clause 1, since $e_{2}=\bar{\varepsilon}$ in (c). The formal action of $b$ is the map $\ell_{\varepsilon}(b):\mathcal{F}\to\mathcal{F}$ with $\ell_{\varepsilon}(b)0=0$ and $\ell_{\varepsilon}(b)\xi=\sum_{u\in\operatorname{supp}\xi}\xi(u)\,\rho_{\varepsilon,b}(u)$ for $\xi\neq0$.