Proof of The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity

Each result cited is universally quantified over the data in its own statement. The results listed in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §background are in force. For $r\in\mathbb{N}$ and $p\in\mathcal{P}_{r}$, $p(\varnothing)$ is the coefficient of $p$ at the empty word, in the sense of The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §polynomials. For $k\in\mathbb{N}$, a $k$-tuple $(e_{1},\dots,e_{k})$ in $\{1,2\}$ is called alternating if $e_{i}\neq e_{i+1}$ whenever $i,i+1\in[k]$; a polynomial of type $1$ is an element of $\mathcal{P}_{m}$ and a polynomial of type $2$ an element of $\mathcal{P}_{n}$. Thus a law $\gamma\in\Sigma_{m+n}$ is free, in the sense of Freeness of Two Groups of Variables under a Noncommutative Law §free, if and only if $\gamma(\iota^{e_{1}}(p_{1})\cdots\iota^{e_{k}}(p_{k}))=0$ whenever $k\in\mathbb{N}$, $(e_{1},\dots,e_{k})$ is alternating, each $p_{i}$ has type $e_{i}$, and $\gamma(\iota^{e_{i}}(p_{i}))=0$ for every $i\in[k]$. Recall that $\iota^{1}=\sigma_{(x_{1},\dots,x_{m})}$ and $\iota^{2}=\sigma_{(x_{m+1},\dots,x_{m+n})}$ are linear, unital and multiplicative by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism.

Step 1: the zero substitution and the zero law. Let $r\in\mathbb{N}$ and let $o$ be the $1$-tuple $(0)$ in $\mathcal{P}_{r}$; it lies in $\mathcal{P}_{r,\mathrm{sa}}$ since $0$ is self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. We claim that $\sigma_{o}(p)=p(\varnothing)1$ for every $p\in\mathcal{P}_{1}$. Indeed, $x_{\varnothing}=1$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, and $\sigma_{o}(1)=1$; if $w$ is a word of length at least $1$ in the single letter $1$, write $w=(1)w'$ with $w'\in W_{1}$; then $\sigma_{o}(x_{w})=o_{w}=o_{(1)}o_{w'}=0\cdot o_{w'}=0$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra. The map $p\mapsto p(\varnothing)1$ from $\mathcal{P}_{1}$ to $\mathcal{P}_{r}$ is linear, because the linear operations act coefficientwise, and it takes the same values on the monomials, since $x_{w}(\varnothing)$ equals $1$ for $w=\varnothing$ and $0$ otherwise by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials. By the uniqueness in part (b) of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension the two linear maps coincide.

Let $\delta\in\Sigma_{1}$ be the law $\delta(p)=p(\varnothing)$ of Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §zero-law for $d=1$; it lies in $\Sigma_{1,R}$ for every real $R>0$. Consequently, for every $r\in\mathbb{N}$, every law $\lambda\in\Sigma_{r}$ and the $1$-tuple $o=(0)$ in $\mathcal{P}_{r,\mathrm{sa}}$ we have $(\lambda\circ\sigma_{o})(p)=p(\varnothing)\lambda(1)=p(\varnothing)$, since $\lambda$ is linear with $\lambda(1)=1$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state; that is, $\lambda\circ\sigma_{o}=\delta$.

Next, let $\mathcal{H}_{\delta}$ be the complex GNS space of $\delta$ with vacuum vector $\Omega_{\delta}$, and let $p\in\mathcal{P}_{1}$ and $q=p-p(\varnothing)1$, so that $q(\varnothing)=0$. With $o=(0)$ in $\mathcal{P}_{1}$, the first claim of this step and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism give $\delta(q^{*}q)1=\sigma_{o}(q^{*}q)=\sigma_{o}(q^{*})\sigma_{o}(q)=\sigma_{o}(q^{*})\,(0\cdot1)=0$, hence $\delta(q^{*}q)=0$ (evaluate the coefficient at $\varnothing$). By Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §algebra, applied to $\delta\in\Sigma_{1,1}$, $\widehat{q}=L_{q}\Omega_{\delta}=L_{p}\Omega_{\delta}-p(\varnothing)\Omega_{\delta}=\widehat{p}-p(\varnothing)\Omega_{\delta}$ and $\lVert\widehat{q}\rVert^{2}=\delta(q^{*}q)=0$. Hence $\widehat{p}=p(\varnothing)\Omega_{\delta}$ for every $p\in\mathcal{P}_{1}$, and $\lVert\Omega_{\delta}\rVert=1$ by the same vacuum claim. Now let $\xi\in\mathcal{H}_{\delta}$, put $c=\langle\Omega_{\delta},\xi\rangle$ and $\eta=\xi-c\Omega_{\delta}$. Then $\langle\Omega_{\delta},\eta\rangle=0$, so $\langle\widehat{p},\eta\rangle=0$ for every $p\in\mathcal{P}_{1}$. The classes are dense in $\mathcal{H}_{\delta}$ by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense (they are the image of the canonical map by The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes), so for every real $\epsilon>0$ there is $p$ with $\lVert\eta-\widehat{p}\rVert<\epsilon$, and by the Cauchy-Schwarz inequality $\lVert\eta\rVert^{2}=|\langle\eta-\widehat{p},\eta\rangle|\le\epsilon\lVert\eta\rVert$; thus $\lVert\eta\rVert\le\epsilon$ for every $\epsilon>0$, so $\eta=0$ and $\xi=\langle\Omega_{\delta},\xi\rangle\Omega_{\delta}$. Consequently every $S\in\mathcal{L}(\mathcal{H}_{\delta})$ is a scalar multiple of the identity: with $c_{S}=\langle\Omega_{\delta},S\Omega_{\delta}\rangle$ we have $S\Omega_{\delta}=c_{S}\Omega_{\delta}$ and hence $S\xi=\langle\Omega_{\delta},\xi\rangle S\Omega_{\delta}=c_{S}\xi$ for every $\xi$, i.e. $S=c_{S}I$.

Step 2: existence of a free law with given marginals and norm bound. Let $R>0$ be real, $\alpha\in\Sigma_{m,R}$ and $\beta\in\Sigma_{n,R}$. We construct a free law $\gamma\in\Sigma_{m+n,R}$ with $\gamma\circ\iota^{1}=\alpha$ and $\gamma\circ\iota^{2}=\beta$.

We use the amalgamation setting with $m_{1}=m$, $m_{2}=n$, the common-marginal dimension (the letter $n$ of that clause) equal to $1$, $\gamma_{1}=\alpha$, $\gamma_{2}=\beta$, and the $1$-tuples $a^{1}=(0)$ in $\mathcal{P}_{m,\mathrm{sa}}$ and $a^{2}=(0)$ in $\mathcal{P}_{n,\mathrm{sa}}$. By Step 1, $\alpha\circ\sigma_{a^{1}}=\delta=\beta\circ\sigma_{a^{2}}$, so the common marginal is $\mu=\delta$. As in Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §algebras and Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §embeddings, $N=\mathcal{M}_{\delta}\subseteq\mathcal{L}(\mathcal{H}_{\delta})$, $A_{1}=\mathcal{M}_{\alpha}$, $A_{2}=\mathcal{M}_{\beta}$, and $\pi_{\varepsilon}$, $E_{\varepsilon}$ are the embedding and the expectation of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation for $\gamma_{\varepsilon}$ and $a^{\varepsilon}$; let $V_{\varepsilon}$ be the isometry of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry for the same data. Let $\varepsilon\in\{1,2\}$ and $b\in A_{\varepsilon}$. By Step 1, $E_{\varepsilon}(b)\in N$ is a scalar multiple of $I$, with scalar

$$\langle\Omega_{\delta},E_{\varepsilon}(b)\Omega_{\delta}\rangle=\langle V_{\varepsilon}\Omega_{\delta},bV_{\varepsilon}\Omega_{\delta}\rangle=\langle\Omega_{\gamma_{\varepsilon}},b\,\Omega_{\gamma_{\varepsilon}}\rangle=\tau_{\gamma_{\varepsilon}}(b),$$

by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §expectation, the identity $V_{\varepsilon}\Omega_{\delta}=\Omega_{\gamma_{\varepsilon}}$ of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, and The Tracial Algebra of a Noncommutative Law and Its Trace §trace. So $E_{\varepsilon}(b)=\tau_{\gamma_{\varepsilon}}(b)I$, and since $I\neq0$ (because $\lVert\Omega_{\delta}\rVert=1$) the centred elements are $A_{\varepsilon}^{\circ}=\{b\in A_{\varepsilon}:\tau_{\gamma_{\varepsilon}}(b)=0\}$.

Let $\mathcal{H}$ be the amalgamated free product space of these data, with vacuum vector $\Omega$, vectors $\Xi(t)$ for alternating tuples $t$, and vacuum state $\varphi(X)=\langle\Omega,X\Omega\rangle$, and let $\Lambda_{\varepsilon}(b)$, for $b\in A_{\varepsilon}$, be the actions. For $j\in[m]$ the operator $L_{x_{j}}$ on $\mathcal{H}_{\alpha}$, and for $j\in[n]$ the operator $L_{x_{j}}$ on $\mathcal{H}_{\beta}$, lies in $A_{1}$, respectively $A_{2}$, 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, is self-adjoint by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint (the variables being self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint), and has operator norm at most $R$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication. Let $T$ be the $(m+n)$-tuple in $\mathcal{L}(\mathcal{H})$ with $T_{j}=\Lambda_{1}(L_{x_{j}})$ for $j\in[m]$ and $T_{m+j}=\Lambda_{2}(L_{x_{j}})$ for $j\in[n]$. By The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation every $T_{j}$ is self-adjoint with $\lVert T_{j}\rVert_{\mathrm{op}}\le R$, and $\lVert\Omega\rVert=1$ by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §marginals.

Traciality. Let $u,v\in W_{m+n}$. If $u=\varnothing$ or $v=\varnothing$, then $T_{\varnothing}=I$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products and $T_{u}T_{v}=T_{v}T_{u}$. Otherwise let $s,t\in\mathbb{N}$ be the lengths of $u$ and $v$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values and Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products, $T_{u}T_{v}=T_{u_{1}}\cdots T_{u_{s}}T_{v_{1}}\cdots T_{v_{t}}$, a product of $r=s+t\ge2$ factors each of the form $\Lambda_{f}(c)$ with $f\in\{1,2\}$ and $c\in A_{f}$. For $i\in\{0,\dots,s\}$ let $X_{i}=T_{u_{i+1}}\cdots T_{u_{s}}T_{v_{1}}\cdots T_{v_{t}}T_{u_{1}}\cdots T_{u_{i}}$ (the first $i$ factors moved to the end). Then $X_{0}=T_{u}T_{v}$ and $X_{s}=T_{v}T_{u}$, and for $i<s$ the product $X_{i+1}$ arises from the $r$ factors of $X_{i}$ by moving the first one to the end, so $\varphi(X_{i})=\varphi(X_{i+1})$ by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §trace. Hence $\langle\Omega,T_{u}T_{v}\Omega\rangle=\langle\Omega,T_{v}T_{u}\Omega\rangle$ for all $u,v$, and by The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law the map $\gamma:\mathcal{P}_{m+n}\to\mathbb{C}$, $\gamma(p)=\langle\Omega,p(T)\Omega\rangle$, belongs to $\Sigma_{m+n,R}$, hence to $\Sigma_{m+n}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

Marginals. Let $p\in\mathcal{P}_{m}$ and let $a=(x_{1},\dots,x_{m})$, an $m$-tuple in $\mathcal{P}_{m+n}$, so that $\iota^{1}=\sigma_{a}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution, $(\iota^{1}(p))(T)=p(a(T))$, and $a(T)=(T_{1},\dots,T_{m})$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values. Apply Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport to the $m$-tuple $(L_{x_{1}},\dots,L_{x_{m}})$ in $\mathcal{L}(\mathcal{H}_{\alpha})$, the set $A_{1}$ (which contains $I$ and the $L_{x_{j}}$ and is closed under sums, multiplication by complex numbers and composition 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 the map $\Lambda_{1}:A_{1}\to\mathcal{L}(\mathcal{H})$ (additive, homogeneous, multiplicative and unital by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation): it gives $\Lambda_{1}(p(L_{x_{1}},\dots,L_{x_{m}}))=p(T_{1},\dots,T_{m})$, and $p(L_{x_{1}},\dots,L_{x_{m}})=L_{p}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns. Hence $(\iota^{1}(p))(T)=\Lambda_{1}(L_{p})$, and in the same way, with the $n$-tuple $(x_{m+1},\dots,x_{m+n})$, $(\iota^{2}(q))(T)=\Lambda_{2}(L_{q})$ for $q\in\mathcal{P}_{n}$, where now $L_{q}$ acts on $\mathcal{H}_{\beta}$. By The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §marginals and 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 §trace,

$$\gamma(\iota^{1}(p))=\varphi(\Lambda_{1}(L_{p}))=\tau_{\alpha}(L_{p})=\alpha(p),\qquad\gamma(\iota^{2}(q))=\varphi(\Lambda_{2}(L_{q}))=\tau_{\beta}(L_{q})=\beta(q).$$

Freeness. Let $k\in\mathbb{N}$, let $(e_{1},\dots,e_{k})$ be alternating, and let $p_{i}$ have type $e_{i}$ with $\gamma(\iota^{e_{i}}(p_{i}))=0$ for every $i\in[k]$. Put $b_{i}=L_{p_{i}}\in A_{e_{i}}$ (on $\mathcal{H}_{\alpha}$ if $e_{i}=1$, on $\mathcal{H}_{\beta}$ if $e_{i}=2$). By the previous paragraph $\tau_{\gamma_{e_{i}}}(b_{i})=\gamma(\iota^{e_{i}}(p_{i}))=0$, so $b_{i}\in A_{e_{i}}^{\circ}$ and $(b_{1},\dots,b_{k})$ is an alternating tuple of type $(e_{1},\dots,e_{k})$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, applied $k-1$ times, $(\iota^{e_{1}}(p_{1})\cdots\iota^{e_{k}}(p_{k}))(T)=\Lambda_{e_{1}}(b_{1})\cdots\Lambda_{e_{k}}(b_{k})$, so by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §freeness

$$\gamma\bigl(\iota^{e_{1}}(p_{1})\cdots\iota^{e_{k}}(p_{k})\bigr)=\langle\Omega,\Lambda_{e_{1}}(b_{1})\cdots\Lambda_{e_{k}}(b_{k})\Omega\rangle=\langle\Omega,\Xi(b_{1},\dots,b_{k})\rangle=0.$$

Thus $\gamma$ is free.

Step 3: uniqueness. Let $\gamma,\gamma'\in\Sigma_{m+n}$ be free with $\gamma\circ\iota^{1}=\gamma'\circ\iota^{1}$ and $\gamma\circ\iota^{2}=\gamma'\circ\iota^{2}$. For $k\in\mathbb{N}$ let $\mathsf{A}(k)$ be the statement: for every alternating $k$-tuple $(e_{1},\dots,e_{k})$ and all polynomials $p_{1},\dots,p_{k}$ with $p_{i}$ of type $e_{i}$,

$$\gamma\bigl(\iota^{e_{1}}(p_{1})\cdots\iota^{e_{k}}(p_{k})\bigr)=\gamma'\bigl(\iota^{e_{1}}(p_{1})\cdots\iota^{e_{k}}(p_{k})\bigr).$$

We prove $\mathsf{A}(k)$ for every $k\in\mathbb{N}$ by strong induction on $k$: the inductive set is $K=\{k\in\mathbb{N}:\mathsf{A}(l)\text{ holds for every }l\in[k]\}$, and it suffices to prove $\mathsf{A}(k)$ assuming $\mathsf{A}(l)$ for every $l\in\mathbb{N}$ with $l<k$. Fix such $k$, an alternating $(e_{1},\dots,e_{k})$ and $p_{1},\dots,p_{k}$ as in $\mathsf{A}(k)$. For $i\in[k]$ let $c_{i}=\gamma(\iota^{e_{i}}(p_{i}))$, which equals $\gamma'(\iota^{e_{i}}(p_{i}))$ because the marginals agree, and let $y_{i}=\iota^{e_{i}}(p_{i}-c_{i}1)=\iota^{e_{i}}(p_{i})-c_{i}1$; since $\gamma(1)=\gamma'(1)=1$, we get $\gamma(y_{i})=\gamma'(y_{i})=0$. For $S\subseteq[k]$ let $y_{S}$ be the product of the $y_{i}$ with $i\in S$ in increasing order of $i$, with $y_{\varnothing}=1$. By the distributive and scalar rules of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and $1p=p1=p$ from Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, by induction on the number of factors,

$$\iota^{e_{1}}(p_{1})\cdots\iota^{e_{k}}(p_{k})=\prod_{i=1}^{k}(y_{i}+c_{i}1)=\sum_{S\subseteq[k]}\Bigl(\prod_{i\in[k]\setminus S}c_{i}\Bigr)\,y_{S},$$

an empty product of numbers being $1$. Since $\gamma$ and $\gamma'$ are linear, it suffices to show $\gamma(y_{S})=\gamma'(y_{S})$ for every $S\subseteq[k]$. For $S=[k]$ both values are $0$ by freeness of $\gamma$ and of $\gamma'$, since $y_{i}=\iota^{e_{i}}(p_{i}-c_{i}1)$ with $p_{i}-c_{i}1$ of type $e_{i}$. For $S=\varnothing$ both values are $1$. If $\varnothing\neq S\neq[k]$, list $S$ as $i_{1}<\dots<i_{s}$ with $1\le s<k$ and cut this list into maximal runs of consecutive entries with equal $e_{i}$; by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism the product of the $y_{i}$ along one run of common value $f$ is $\iota^{f}(q)$, where $q$ is the product of the corresponding $p_{i}-c_{i}1$ and has type $f$, and neighbouring runs have different values. Thus $y_{S}=\iota^{f_{1}}(q_{1})\cdots\iota^{f_{l}}(q_{l})$ with $(f_{1},\dots,f_{l})$ alternating, $q_{j}$ of type $f_{j}$ and $1\le l\le s<k$, and $\gamma(y_{S})=\gamma'(y_{S})$ by $\mathsf{A}(l)$. This proves $\mathsf{A}(k)$ (for $k=1$ only $S=\varnothing$ and $S=[1]$ occur), and hence $\mathsf{A}(k)$ for every $k\in\mathbb{N}$.

Now let $w\in W_{m+n}$ be nonempty. Cutting $w$ at every position where the letter passes from $[m]$ to $\{m+1,\dots,m+n\}$ or back (formally, by induction on the length of $w$: appending a letter either extends the last piece or starts a new one), we write $w=w^{1}\cdots w^{k}$ with $k\in\mathbb{N}$ and nonempty words $w^{i}$ whose letters all lie in $[m]$ (put $e_{i}=1$) or all in $\{m+1,\dots,m+n\}$ (put $e_{i}=2$), with $(e_{1},\dots,e_{k})$ alternating. Let $u^{i}=w^{i}\in W_{m}$ if $e_{i}=1$, and let $u^{i}\in W_{n}$ be $w^{i}$ with $m$ subtracted from every letter if $e_{i}=2$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\iota^{e_{i}}(x_{u^{i}})$ is the product of the variables $x_{j}$ of $\mathcal{P}_{m+n}$ along the letters $j$ of $w^{i}$, which is $x_{w^{i}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials; by the same claim $x_{w}=x_{w^{1}}\cdots x_{w^{k}}=\iota^{e_{1}}(x_{u^{1}})\cdots\iota^{e_{k}}(x_{u^{k}})$, so $\gamma(x_{w})=\gamma'(x_{w})$ by $\mathsf{A}(k)$. Also $\gamma(x_{\varnothing})=\gamma(1)=1=\gamma'(x_{\varnothing})$. By the uniqueness in part (a) of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, $\gamma=\gamma'$.

Claims 1 and 2. Let $\alpha\in\Sigma_{m}$ and $\beta\in\Sigma_{n}$. By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are real $r_{1},r_{2}>0$ with $\alpha\in\Sigma_{m,r_{1}}$ and $\beta\in\Sigma_{n,r_{2}}$; with $R=\max(r_{1},r_{2})$ we have $\alpha\in\Sigma_{m,R}$ and $\beta\in\Sigma_{n,R}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone. Step 2 yields a free law in $\Sigma_{m+n}$ with marginals $\alpha$ and $\beta$, and Step 3 shows that there is at most one; this is claim 1. For claim 2, let $R>0$ be any real number with $\alpha\in\Sigma_{m,R}$ and $\beta\in\Sigma_{n,R}$: Step 2 with this $R$ yields a free law in $\Sigma_{m+n,R}$ with marginals $\alpha,\beta$, which equals $\alpha\star\beta$ by Step 3.

Since $m$ and $n$ were arbitrary, Steps 2 and 3 and claim 1 hold for every split of a number of variables into two groups; this is used below for the splits of $m'+n'$ and of $n+m$ variables.

Claim 3 (substitutions). Let $m',n',a,b,c$ be as in the statement and let $\alpha\in\Sigma_{m}$, $\beta\in\Sigma_{n}$ and $\gamma=\alpha\star\beta$. Let $\kappa^{1}:\mathcal{P}_{m'}\to\mathcal{P}_{m'+n'}$ and $\kappa^{2}:\mathcal{P}_{n'}\to\mathcal{P}_{m'+n'}$ be the substitutions of $(x_{1},\dots,x_{m'})$ and $(x_{m'+1},\dots,x_{m'+n'})$, i.e. the maps $\iota^{1},\iota^{2}$ of the split of $m'+n'$ variables. The entries of $c$ are self-adjoint by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint, the variables being self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint; hence $\gamma\circ\sigma_{c}\in\Sigma_{m'+n'}$, $\alpha\circ\sigma_{a}\in\Sigma_{m'}$ and $\beta\circ\sigma_{b}\in\Sigma_{n'}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law. For $q\in\mathcal{P}_{m'}$, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition gives $\sigma_{c}(\kappa^{1}(q))=\sigma_{c'}(q)$, where $c'_{j}=\sigma_{c}(x_{j})=c_{j}=\iota^{1}(a_{j})$ for $j\in[m']$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, and it also gives $\iota^{1}(\sigma_{a}(q))=\sigma_{c'}(q)$, the tuple being again $(\iota^{1}(a_{1}),\dots,\iota^{1}(a_{m'}))$. Hence $\sigma_{c}\circ\kappa^{1}=\iota^{1}\circ\sigma_{a}$, and in the same way, since $\sigma_{c}(x_{m'+j})=c_{m'+j}=\iota^{2}(b_{j})$ for $j\in[n']$, $\sigma_{c}\circ\kappa^{2}=\iota^{2}\circ\sigma_{b}$. Therefore $(\gamma\circ\sigma_{c})\circ\kappa^{1}=\alpha\circ\sigma_{a}$ and $(\gamma\circ\sigma_{c})\circ\kappa^{2}=\beta\circ\sigma_{b}$. Freeness: let $k\in\mathbb{N}$, let $(e_{1},\dots,e_{k})$ be alternating, and let $q_{i}\in\mathcal{P}_{m'}$ if $e_{i}=1$ and $q_{i}\in\mathcal{P}_{n'}$ if $e_{i}=2$, with $(\gamma\circ\sigma_{c})(\kappa^{e_{i}}(q_{i}))=0$ for every $i$. Put $s_{i}=\sigma_{a}(q_{i})\in\mathcal{P}_{m}$ if $e_{i}=1$ and $s_{i}=\sigma_{b}(q_{i})\in\mathcal{P}_{n}$ if $e_{i}=2$. Then $\sigma_{c}(\kappa^{e_{i}}(q_{i}))=\iota^{e_{i}}(s_{i})$, so $\gamma(\iota^{e_{i}}(s_{i}))=0$, and by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and freeness of $\gamma$,

$$(\gamma\circ\sigma_{c})\bigl(\kappa^{e_{1}}(q_{1})\cdots\kappa^{e_{k}}(q_{k})\bigr)=\gamma\bigl(\iota^{e_{1}}(s_{1})\cdots\iota^{e_{k}}(s_{k})\bigr)=0.$$

So $\gamma\circ\sigma_{c}$ is free for the split of $m'+n'$ variables with marginals $\alpha\circ\sigma_{a}$ and $\beta\circ\sigma_{b}$, and it equals $(\alpha\circ\sigma_{a})\star(\beta\circ\sigma_{b})$ by claim 1 for that split.

Claim 4 (symmetry). Let $\alpha\in\Sigma_{m}$, $\beta\in\Sigma_{n}$, $\gamma=\alpha\star\beta$, and let $\kappa^{1}:\mathcal{P}_{n}\to\mathcal{P}_{n+m}$ and $\kappa^{2}:\mathcal{P}_{m}\to\mathcal{P}_{n+m}$ be the substitutions of $(x_{1},\dots,x_{n})$ and $(x_{n+1},\dots,x_{n+m})$. The entries of $s$ are variables, hence self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint, so $\gamma\circ\sigma_{s}\in\Sigma_{n+m}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\sigma_{s}\circ\kappa^{1}$ is the substitution of $(\sigma_{s}(x_{1}),\dots,\sigma_{s}(x_{n}))=(x_{m+1},\dots,x_{m+n})$, which is $\iota^{2}$, and $\sigma_{s}\circ\kappa^{2}$ is the substitution of $(\sigma_{s}(x_{n+1}),\dots,\sigma_{s}(x_{n+m}))=(x_{1},\dots,x_{m})$, which is $\iota^{1}$. Hence $(\gamma\circ\sigma_{s})\circ\kappa^{1}=\beta$ and $(\gamma\circ\sigma_{s})\circ\kappa^{2}=\alpha$. Freeness: let $(e_{1},\dots,e_{k})$ be alternating and let $q_{i}\in\mathcal{P}_{n}$ if $e_{i}=1$ and $q_{i}\in\mathcal{P}_{m}$ if $e_{i}=2$, with $(\gamma\circ\sigma_{s})(\kappa^{e_{i}}(q_{i}))=0$. With $f_{i}=3-e_{i}$, the tuple $(f_{1},\dots,f_{k})$ is alternating, $q_{i}$ has type $f_{i}$ for the split of $m+n$ variables, $\sigma_{s}(\kappa^{e_{i}}(q_{i}))=\iota^{f_{i}}(q_{i})$ and $\gamma(\iota^{f_{i}}(q_{i}))=0$; so by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and freeness of $\gamma$, $(\gamma\circ\sigma_{s})(\kappa^{e_{1}}(q_{1})\cdots\kappa^{e_{k}}(q_{k}))=\gamma(\iota^{f_{1}}(q_{1})\cdots\iota^{f_{k}}(q_{k}))=0$. By claim 1 for the split of $n+m$ variables, $\gamma\circ\sigma_{s}=\beta\star\alpha$.

Claim 5 (weak-star continuity). We say that a sequence $(z_{l})_{l\in\mathbb{N}}$ of complex numbers converges to $z\in\mathbb{C}$ if $(\operatorname{Re}z_{l})_{l}$ and $(\operatorname{Im}z_{l})_{l}$ converge to $\operatorname{Re}z$ and $\operatorname{Im}z$ in the sense of Limit of a Sequence of Real Numbers. Constant sequences converge to their value by Constant Sequences and Index-Shifted Sequences of Real Numbers §constant; limits are unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, applied to real and imaginary parts, which determine a complex number; and if $z_{l}\to z$ and $z'_{l}\to z'$, then $z_{l}+z'_{l}\to z+z'$ and $z_{l}z'_{l}\to zz'$, by claims 1, 2 and 3 of Arithmetic of Limits of Real Sequences together with $\operatorname{Re}(zz')=\operatorname{Re}z\operatorname{Re}z'-\operatorname{Im}z\operatorname{Im}z'$ and $\operatorname{Im}(zz')=\operatorname{Re}z\operatorname{Im}z'+\operatorname{Im}z\operatorname{Re}z'$. By induction on the number of terms, finite sums and finite products of convergent sequences converge to the sum and the product of the limits.

Let $R$, $(\alpha_{k})$, $(\beta_{k})$, $\alpha$ and $\beta$ be as in claim 5, and let $\gamma_{k}=\alpha_{k}\star\beta_{k}$, which lies in $\Sigma_{m+n,R}$ by claim 2.

(a) Limits of subsequences. Let $(k_{j})_{j\in\mathbb{N}}$ be a strictly increasing sequence in $\mathbb{N}$. By Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §compact, applied to the sequence $(\gamma_{k_{j}})_{j}$ in $\Sigma_{m+n,R}$, we first choose a strictly increasing sequence $(j_{l})_{l\in\mathbb{N}}$ in $\mathbb{N}$ and then a law $\gamma\in\Sigma_{m+n,R}$ with $\gamma_{\nu_{l}}\to\gamma$ weak-star, where $\nu_{l}=k_{j_{l}}$. By claim 3 of A Subsequence of a Subsequence is a Subsequence, $(\alpha_{\nu_{l}})_{l}$ and $(\beta_{\nu_{l}})_{l}$ are subsequences of $(\alpha_{k})$ and $(\beta_{k})$, so they converge weak-star to $\alpha$ and $\beta$ by Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §unique. We show $\gamma=\alpha\star\beta$.

Marginals: for $q\in\mathcal{P}_{m}$ we have $\gamma_{\nu_{l}}(\iota^{1}(q))=\alpha_{\nu_{l}}(q)$ for every $l$; by Weak-Star Convergence of Noncommutative Laws §weak-star the left side converges to $\gamma(\iota^{1}(q))$ and the right side to $\alpha(q)$, so $\gamma(\iota^{1}(q))=\alpha(q)$ by uniqueness of limits. In the same way $\gamma\circ\iota^{2}=\beta$.

Freeness: let $r\in\mathbb{N}$, let $(e_{1},\dots,e_{r})$ be alternating, and let $p_{i}$ have type $e_{i}$ with $\gamma(\iota^{e_{i}}(p_{i}))=0$ for $i\in[r]$. Put $y_{i}=\iota^{e_{i}}(p_{i})$, and for $S\subseteq[r]$ let $y_{S}$ be the product of the $y_{i}$ with $i\in S$ in increasing order of $i$, with $y_{\varnothing}=1$. For $l\in\mathbb{N}$ and $i\in[r]$ let $c^{(l)}_{i}=\gamma_{\nu_{l}}(y_{i})$. Then $\gamma_{\nu_{l}}(\iota^{e_{i}}(p_{i}-c^{(l)}_{i}1))=0$, so freeness of $\gamma_{\nu_{l}}$ and the expansion of Step 3 (with $-c^{(l)}_{i}$ in place of $c_{i}$) give

$$0=\gamma_{\nu_{l}}\Bigl(\prod_{i=1}^{r}\bigl(y_{i}-c^{(l)}_{i}1\bigr)\Bigr)=\sum_{S\subseteq[r]}\Bigl(\prod_{i\in[r]\setminus S}\bigl(-c^{(l)}_{i}\bigr)\Bigr)\gamma_{\nu_{l}}(y_{S}), \qquad\text{so}\qquad \gamma_{\nu_{l}}(y_{[r]})=-\sum_{S\subsetneq[r]}\Bigl(\prod_{i\in[r]\setminus S}\bigl(-c^{(l)}_{i}\bigr)\Bigr)\gamma_{\nu_{l}}(y_{S}).$$

As $l\to\infty$, $c^{(l)}_{i}\to\gamma(y_{i})=0$ and $\gamma_{\nu_{l}}(y_{S})\to\gamma(y_{S})$ for every $S$, by Weak-Star Convergence of Noncommutative Laws §weak-star. For $S\subsetneq[r]$ the set $[r]\setminus S$ is nonempty, so the corresponding summand converges to a product with a factor $0$, that is, to $0$; hence the right side converges to $0$, while the left side converges to $\gamma(y_{[r]})$. By uniqueness of limits $\gamma(y_{[r]})=0$. Thus $\gamma$ is free with marginals $\alpha$ and $\beta$, and $\gamma=\alpha\star\beta$ by claim 1.

(b) Convergence of the whole sequence. Fix $p\in\mathcal{P}_{m+n}$. Consider the real line $(\mathbb{R},d_{\mathbb{R}})$ with $d_{\mathbb{R}}(s,t)=|s-t|$, a metric space by The Absolute Value Metric on the Real Line; convergence of a real sequence in this metric space in the sense of Convergent Sequence in a Metric Space is the same condition as convergence in the sense of Limit of a Sequence of Real Numbers, the two definitions reading identically for $d_{\mathbb{R}}$. Let $x_{k}=\operatorname{Re}\gamma_{k}(p)$. For every strictly increasing sequence $(k_{j})_{j}$ in $\mathbb{N}$, part (a) provides a strictly increasing $(j_{l})_{l}$ with $\gamma_{k_{j_{l}}}\to\alpha\star\beta$ weak-star, so $(x_{k_{j_{l}}})_{l}$ converges to $\operatorname{Re}(\alpha\star\beta)(p)$. By The Subsequence Criterion for Convergence in a Metric Space §criterion, $(x_{k})$ converges to $\operatorname{Re}(\alpha\star\beta)(p)$. The same argument with imaginary parts shows that $(\operatorname{Im}\gamma_{k}(p))$ converges to $\operatorname{Im}(\alpha\star\beta)(p)$. As $p$ was arbitrary, $\alpha_{k}\star\beta_{k}\to\alpha\star\beta$ weak-star by Weak-Star Convergence of Noncommutative Laws §weak-star.