Proof of Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family

Each result cited below is universally quantified over the data in its own statement.

Letters. We use the following consequences of Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters. For $j\in[d]$ the letter $j$ has $g(j)=j$, $\varepsilon(j)=1$ and $j^{-1}=d+j$, and the letter $d+j$ (which lies in $[2d]$ and exceeds $d$, by Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters) has $g(d+j)=j$, $\varepsilon(d+j)=-1$ and $(d+j)^{-1}=j$, because $j$ is the only natural number $m$ with $d+j=d+m$. Consequently, for every letter $l\in[2d]$: $g(l^{-1})=g(l)$, $\varepsilon(l^{-1})=-\varepsilon(l)$ and $(l^{-1})^{-1}=l$; and $l=g(l)$ if $\varepsilon(l)=1$, $l=d+g(l)$ if $\varepsilon(l)=-1$; and these two cases are exhaustive, since by Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §letters the sign $\varepsilon(l)$ takes only the values $1$ and $-1$. For $j\in[d]$ and $s\in\{1,-1\}$ we write $\ell(j,s)$ for the letter $j$ if $s=1$ and $d+j$ if $s=-1$; thus $g(\ell(j,s))=j$, $\varepsilon(\ell(j,s))=s$, and $\ell(g(l),\varepsilon(l))=l$ for every letter $l$. For a word $w$ of length $k$ and a letter $l$, the word $w\,l$ has length $k+1$ and adjoint $(w\,l)^{*}=l^{-1}\,w^{*}$: by Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words §adjoint and reversal, $((wl)^{*})_{1}=((wl)_{k+1})^{-1}=l^{-1}$ and $((wl)^{*})_{i}=(w_{k+2-i})^{-1}=(w^{*})_{i-1}$ for $2\le i\le k+1$; also $l^{*}=l^{-1}$ and $\varnothing^{*}=\varnothing$. Every nonempty word is of the form $w\,l$ or is a letter, by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter.

Fact U (unitaries). Let $H$ be a complex Hilbert space and $T:H\to H$ linear. Then $T$ is unitary if and only if $T\in\mathcal{L}(H)$ and $T^{*}T=TT^{*}=I$. Indeed, let $T$ be unitary. By claim 3 of Properties of Unitary Operators and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound, $T\in\mathcal{L}(H)$, so $T$ has an adjoint $T^{*}\in\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint, and $T$ is the adjoint of $T^{*}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus; by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint this gives $\langle u,T^{*}Tv\rangle=\langle Tu,Tv\rangle=\langle u,v\rangle$ for all $u,v\in H$, the second equality being the defining property of $T$. Taking $u=T^{*}Tv-v$ gives $\lVert T^{*}Tv-v\rVert^{2}=0$, so $T^{*}Tv=v$. Given $v\in H$, write $v=Tu$ by surjectivity; then $TT^{*}v=T(T^{*}Tu)=Tu=v$. Conversely, if $T\in\mathcal{L}(H)$ and $T^{*}T=TT^{*}=I$, then every $v$ equals $T(T^{*}v)$, and $\langle Tu,Tv\rangle=\langle u,T^{*}Tv\rangle=\langle u,v\rangle$ as above; so $T$ is unitary.

Proof of clause 1 (law). Fix $(H,M,\Omega)$ and $V$ as in clause 1, and write $\tau=\tau_{M}$; by Tracial W*-Probability Spaces §trace and Cyclic Tracial Operator Algebras and Their Traces §trace, $\tau(T)=\langle\Omega,T\Omega\rangle$ for $T\in M$. Each $V_{j}$ is unitary, so $V_{j}\in\mathcal{L}(H)$ by claim 3 of Properties of Unitary Operators and Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound, and hence $V_{j}$ has an adjoint $V_{j}^{*}\in\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint, as in Fact U.

(i) $V^{(uv)}=V^{(u)}V^{(v)}$ for all $u,v\in W_{2d}$. If $u=\varnothing$ or $v=\varnothing$ this holds because $V^{(\varnothing)}=I$ and $\varnothing v=v$, $u\varnothing=u$ (Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation); otherwise the letters of $uv$ are those of $u$ followed by those of $v$ (Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation), and composition of maps is associative.

(ii) $(V^{(l)})^{*}=V^{(l^{-1})}$ for every letter $l$. If $\varepsilon(l)=1$, then $V^{(l)}=V_{g(l)}$ and, as $\varepsilon(l^{-1})=-1$ and $g(l^{-1})=g(l)$, $V^{(l^{-1})}=V_{g(l)}^{*}$. If $\varepsilon(l)=-1$, then $V^{(l)}=V_{g(l)}^{*}$, whose adjoint is $V_{g(l)}=V^{(l^{-1})}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus.

(iii) $(V^{(w)})^{*}=V^{(w^{*})}$ for every $w\in W_{2d}$. For $w=\varnothing$ both sides are $I$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus); for a letter this is (ii); and if it holds for $w$ of length $k$, then for a letter $l$, by (i), Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus and (ii), $(V^{(wl)})^{*}=(V^{(w)}V^{(l)})^{*}=V^{(l^{-1})}V^{(w^{*})}=V^{(l^{-1}w^{*})}=V^{((wl)^{*})}$. Induction on the length gives (iii).

(iv) $V^{(l)}V^{(l^{-1})}=I$ for every letter $l$: by Fact U, $V_{g(l)}V_{g(l)}^{*}=I$ and $V_{g(l)}^{*}V_{g(l)}=I$, and by (ii) the product $V^{(l)}V^{(l^{-1})}$ is the first of these if $\varepsilon(l)=1$ and the second if $\varepsilon(l)=-1$.

Now we check the four properties of Laws of d-Tuples of Unitaries §law. Normalised: $\lambda_{V}(\varnothing)=\tau(I)=\langle\Omega,\Omega\rangle=\lVert\Omega\rVert^{2}=1$ by Cyclic Tracial Operator Algebras and Their Traces §cyclic. Cancellation: by (i) and (iv), $V^{(u\,l\,l^{-1}\,v)}=V^{(u)}V^{(l)}V^{(l^{-1})}V^{(v)}=V^{(u)}V^{(v)}=V^{(uv)}$, so $\lambda_{V}(u\,l\,l^{-1}\,v)=\lambda_{V}(uv)$. Cyclic: by (i) and Cyclic Tracial Operator Algebras and Their Traces §tracial, $\lambda_{V}(uv)=\tau(V^{(u)}V^{(v)})=\tau(V^{(v)}V^{(u)})=\lambda_{V}(vu)$. Positive: let $F\subseteq W_{2d}$ be nonempty and finite, and put $\xi_{v}=V^{(v)}\Omega$ for $v\in F$. By (i), (iii), and since $V^{(v)}$ is the adjoint of $(V^{(v)})^{*}$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint),

$$K(v,v'):=\lambda_{V}(v^{*}v')=\langle\Omega,(V^{(v)})^{*}V^{(v')}\Omega\rangle=\langle\xi_{v},\xi_{v'}\rangle\qquad(v,v'\in F).$$

Hence $K(v',v)=\overline{K(v,v')}$ by the conjugate symmetry of the inner product, and for $z:F\to\mathbb{C}$, with $\eta=\sum_{v\in F}z(v)\xi_{v}$, the sesquilinearity of the inner product (conjugate-linear in the first and linear in the second argument, Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces) gives $\sum_{(v,v')\in F\times F}\overline{z(v)}\,z(v')\,K(v,v')=\langle\eta,\eta\rangle=\lVert\eta\rVert^{2}$, a nonnegative real number. So $K$ is a positive semidefinite kernel on $F$. Thus $\lambda_{V}\in\mathcal{L}_{d}$.

Proof of clause 2 (realisation). Fix $\lambda\in\mathcal{L}_{d}$. We use the notation of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation: $\mathcal{P}_{n}$, monomials $x_{v}$, $\Sigma_{n,R}$, $\Sigma_{n}$, and, for a law, its complex GNS space, vacuum vector, classes $\widehat{p}$, multiplication operators $L_{p}$ and tracial algebra (Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §polynomials, Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §gns). The monomials of $\mathcal{P}_{2d}$ are indexed by the words in the letters $1,\dots,2d$, which are the elements of $W_{2d}$: the words and their concatenation are defined by identical clauses in Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal, used for $W_{2d}$, and in the earlier version of the definition of words, used by the polynomial items, so this monomial indexing applies; so a variable index $v\in[2d]$ is also a letter and has a generator $g(v)$ and a sign $\varepsilon(v)$. Every $p\in\mathcal{P}_{n}$ satisfies

$$p=\sum_{v\in\operatorname{supp}p}p(v)\,x_{v}\qquad(p\neq0),\tag{E}$$

by the formula of Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension(b) applied to the identity map of $\mathcal{P}_{n}$, which is linear with value $x_{v}$ at $x_{v}$; and $x_{v}=x_{v_{1}}x_{v_{2}}\cdots x_{v_{k}}$ for $v$ of length $k$, by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials. The set $\{1,-1\}^{k}$ of $k$-tuples of signs has $2^{k}$ elements (induction on $k$), and for maps $f_{m}:\{1,-1\}\to\mathbb{C}$ ($m\in[k]$), multiplying out by distributivity (induction on $k$) gives

$$\sum_{e\in\{1,-1\}^{k}}\ \prod_{m=1}^{k}f_{m}(e_{m})=\prod_{m=1}^{k}\bigl(f_{m}(1)+f_{m}(-1)\bigr);\tag{D}$$

the same multiplying out holds for products of $k$ sums of two terms in $\mathcal{P}_{n}$, 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.

Step 1 (the polynomials $b_{l}$ and $P_{\omega}$). For a letter $l\in[2d]$ put $b_{l}=x_{g(l)}+i\,\varepsilon(l)\,x_{d+g(l)}\in\mathcal{P}_{2d}$, and for $\omega\in W_{2d}$ let $P_{\omega}=b_{\omega}$ be the product along $\omega$ of the $2d$-tuple $b=(b_{1},\dots,b_{2d})$; thus $P_{\varnothing}=1$, $P_{\omega}=b_{\omega_{1}}\cdots b_{\omega_{k}}$ for $\omega$ of length $k$, and $P_{uv}=P_{u}P_{v}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $x_{j}^{*}=x_{j}$ and $\overline{i\varepsilon(l)}=-i\varepsilon(l)$, so $b_{l}^{*}=x_{g(l)}-i\varepsilon(l)x_{d+g(l)}=b_{l^{-1}}$, since $g(l^{-1})=g(l)$ and $\varepsilon(l^{-1})=-\varepsilon(l)$. Hence $P_{\omega}^{*}=P_{\omega^{*}}$ for every $\omega$: this holds for $\omega=\varnothing$ ($1^{*}=1$) and for letters, and if it holds for $\omega$ then $(P_{\omega l})^{*}=(P_{\omega}b_{l})^{*}=b_{l^{-1}}P_{\omega^{*}}=P_{l^{-1}\omega^{*}}=P_{(\omega l)^{*}}$ by $(pq)^{*}=q^{*}p^{*}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint); induct on the length.

For a variable index $v\in[2d]$ and $s\in\{1,-1\}$ put $c(v,s)=\tfrac12$ if $\varepsilon(v)=1$ and $c(v,s)=\tfrac{s}{2i}$ if $\varepsilon(v)=-1$. Then

$$x_{v}=c(v,1)\,b_{\ell(g(v),1)}+c(v,-1)\,b_{\ell(g(v),-1)}.\tag{1}$$

Indeed, with $j=g(v)$, $b_{\ell(j,1)}=x_{j}+ix_{d+j}$ and $b_{\ell(j,-1)}=x_{j}-ix_{d+j}$; if $\varepsilon(v)=1$ then $v=j$ and the right side is $\tfrac12(2x_{j})=x_{j}$, and if $\varepsilon(v)=-1$ then $v=d+j$ and the right side is $\tfrac{1}{2i}(2i\,x_{d+j})=x_{d+j}$. For a word $v$ of length $k$ and $s\in\{1,-1\}^{k}$ put $c(v,s)=\prod_{m=1}^{k}c(v_{m},s_{m})$ and let $\omega(v,s)\in W_{2d}$ be the word of length $k$ with $m$-th letter $\ell(g(v_{m}),s_{m})$; for $v=\varnothing$ put $c(\varnothing,\cdot)=1$ and $\omega(\varnothing,\cdot)=\varnothing$, with a single (empty) sign tuple. Multiplying out $x_{v}=x_{v_{1}}\cdots x_{v_{k}}$ with (1) gives

$$x_{v}=\sum_{s\in\{1,-1\}^{k}}c(v,s)\,P_{\omega(v,s)}.\tag{2}$$

Consequently, by (E), every $p\in\mathcal{P}_{2d}$ with $p\ne0$ is a finite combination $p=\sum_{a\in G}z(a)P_{\omega_{a}}$, where $G$ is the nonempty finite set of pairs $a=(v,s)$ with $v\in\operatorname{supp}p$ and $s$ a sign tuple of the length of $v$, $z(a)=p(v)c(v,s)$ and $\omega_{a}=\omega(v,s)$ (Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §pairs).

Step 2 (the law $\alpha$). By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension(a) there is exactly one linear map $\alpha:\mathcal{P}_{2d}\to\mathbb{C}$ with $\alpha(1)=\lambda(\varnothing)=1$ and

$$\alpha(x_{v})=\sum_{s\in\{1,-1\}^{k}}c(v,s)\,\lambda\bigl(\omega(v,s)\bigr)\qquad\text{for every word }v\text{ of length }k\in\mathbb{N}.$$

Claim A: $\alpha(P_{\omega})=\lambda(\omega)$ for every $\omega\in W_{2d}$. For $\omega=\varnothing$ this is $\alpha(1)=1=\lambda(\varnothing)$ (Laws of d-Tuples of Unitaries §normalised). Let $\omega$ have length $k$ and $j_{m}=g(\omega_{m})$. Since $x_{\ell(j,1)}=x_{j}$ and $x_{\ell(j,-1)}=x_{d+j}$, we have $b_{\omega_{m}}=\sum_{e\in\{1,-1\}}\beta_{m}(e)\,x_{\ell(j_{m},e)}$ with $\beta_{m}(1)=1$ and $\beta_{m}(-1)=i\varepsilon(\omega_{m})$; multiplying out,

$$P_{\omega}=\sum_{e\in\{1,-1\}^{k}}\Bigl(\prod_{m=1}^{k}\beta_{m}(e_{m})\Bigr)x_{v(e)},$$

where $v(e)$ is the word with $m$-th letter $\ell(j_{m},e_{m})$. Now $g(v(e)_{m})=j_{m}$ and $\varepsilon(v(e)_{m})=e_{m}$, so $\omega(v(e),s)$ is the word $\omega_{s}$ with $m$-th letter $\ell(j_{m},s_{m})$, independent of $e$, and $c(v(e)_{m},s_{m})$ is $\tfrac12$ if $e_{m}=1$ and $\tfrac{s_{m}}{2i}$ if $e_{m}=-1$. By linearity of $\alpha$, exchanging the two finite sums, and (D),

$$\alpha(P_{\omega})=\sum_{s\in\{1,-1\}^{k}}\lambda(\omega_{s})\prod_{m=1}^{k}\Bigl(\tfrac12+i\varepsilon(\omega_{m})\tfrac{s_{m}}{2i}\Bigr)=\sum_{s\in\{1,-1\}^{k}}\lambda(\omega_{s})\prod_{m=1}^{k}\tfrac{1+\varepsilon(\omega_{m})s_{m}}{2}.$$

Each factor $\tfrac{1+\varepsilon(\omega_{m})s_{m}}{2}$ is $1$ if $s_{m}=\varepsilon(\omega_{m})$ and $0$ otherwise. So only the tuple $s=(\varepsilon(\omega_{1}),\dots,\varepsilon(\omega_{k}))$ contributes, with product $1$ (Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing), and for it $\omega_{s}=\omega$ because $\ell(g(l),\varepsilon(l))=l$. Hence $\alpha(P_{\omega})=\lambda(\omega)$.

Claim B: $\alpha\in\Sigma_{2d,2}$. $\alpha$ is linear and $\alpha(1)=1$. Let $p,q\in\mathcal{P}_{2d}$; if $p=0$ or $q=0$ then $pq=qp=0$ and $p^{*}p=0$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra), so the conditions below hold trivially. Otherwise write $p=\sum_{a\in G}z(a)P_{\omega_{a}}$ and $q=\sum_{b\in G'}z'(b)P_{\omega'_{b}}$ as in Step 1. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $P_{u}P_{v}=P_{uv}$, Claim A and Laws of d-Tuples of Unitaries §cyclic,

$$\alpha(pq)=\sum_{(a,b)\in G\times G'}z(a)z'(b)\lambda(\omega_{a}\omega'_{b})=\sum_{(a,b)\in G\times G'}z(a)z'(b)\lambda(\omega'_{b}\omega_{a})=\alpha(qp).$$

By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and $P_{\omega}^{*}=P_{\omega^{*}}$, $p^{*}=\sum_{a\in G}\overline{z(a)}P_{\omega_{a}^{*}}$, so by Claim A $\alpha(p^{*}p)=\sum_{(a,b)\in G\times G}\overline{z(a)}z(b)\lambda(\omega_{a}^{*}\omega_{b})$. Let $F=\{\omega_{a}:a\in G\}$, nonempty and finite, and $Z(\omega)=\sum_{a\in G,\ \omega_{a}=\omega}z(a)$ for $\omega\in F$. Grouping the terms according to $(\omega_{a},\omega_{b})$, that is, reindexing the finite sum over $G\times G$ as the iterated sum over $(\omega,\omega')\in F\times F$ and over the nonempty set of pairs $(a,b)\in G\times G$ with $\omega_{a}=\omega$ and $\omega_{b}=\omega'$ (Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §pairs, together with reindexing along a bijection, claim 2 of Properties of a Sum over a Finite Index Set), and using distributivity and Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §conjugate, gives $\alpha(p^{*}p)=\sum_{(\omega,\omega')\in F\times F}\overline{Z(\omega)}Z(\omega')\lambda(\omega^{*}\omega')$, which is real and nonnegative by Laws of d-Tuples of Unitaries §positive and Positive Semidefinite Kernel on a Finite Set §kernel. So $\alpha$ is a tracial state on $\mathcal{P}_{2d}$. For $v$ of length $k$, $|c(v,s)|=(\tfrac12)^{k}$ by claim 4 of Properties of Complex Conjugation and Modulus (as $|\tfrac{s}{2i}|=\tfrac12$), and $|\lambda(\omega(v,s))|\le1$ by Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §bound; by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §modulus and the count $2^{k}$ of sign tuples, $|\alpha(x_{v})|\le2^{k}(\tfrac12)^{k}=1\le2^{k}$. So $\alpha$ has norm bound $2$, i.e. $\alpha\in\Sigma_{2d,2}\subseteq\Sigma_{2d}$.

Step 3 (the joint law $\gamma$). Let $\mathrm{sc}_{d}$ be the semicircular law of $d$ variables. It satisfies the Schwinger-Dyson equation, so it equals the vacuum law $\lambda_{S}$ of The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §unique, which belongs to $\Sigma_{d,2}$ by The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §vacuum-law. Let $\gamma=\alpha\star\mathrm{sc}_{d}\in\Sigma_{3d}$ be the free product for the split of $3d=2d+d$ variables into the first $2d$ and the last $d$, and let $\iota^{1}:\mathcal{P}_{2d}\to\mathcal{P}_{3d}$ be the substitution of $(x_{1},\dots,x_{2d})$ (Freeness of Two Groups of Variables under a Noncommutative Law §free). Then $\gamma\circ\iota^{1}=\alpha$ by The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §product, $\gamma\in\Sigma_{3d,2}$ by The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity §bound with $R=2$, and, by Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations §characterisation (with $m=2d$, $n=d$) and The Schwinger-Dyson Equation for a Noncommutative Law §relative,

$$\gamma(x_{2d+j}\,q)=\partial^{\gamma}_{2d+j}(q)\qquad(j\in[d],\ q\in\mathcal{P}_{3d}),\tag{SD}$$

with $\partial^{\gamma}_{2d+j}$ the free difference quotient. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $\iota^{1}$ is linear and multiplicative with $\iota^{1}(1)=1$ and $\iota^{1}(x_{v})=x_{v}$ for every $v\in W_{2d}$, the word $v$ being read as a word in the letters $1,\dots,3d$ all of whose letters are at most $2d$; and $\iota^{1}(p^{*})=\iota^{1}(p)^{*}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint, the variables being self-adjoint. Put $y_{l}=\iota^{1}(b_{l})=x_{g(l)}+i\varepsilon(l)x_{d+g(l)}\in\mathcal{P}_{3d}$ and $Q_{\omega}=\iota^{1}(P_{\omega})$; thus $Q_{\varnothing}=1$, $Q_{\omega}=y_{\omega_{1}}\cdots y_{\omega_{k}}$, $Q_{u}Q_{v}=Q_{uv}$, $Q_{\omega}^{*}=Q_{\omega^{*}}$, and, by Claim A,

$$\gamma(Q_{\omega})=\alpha(P_{\omega})=\lambda(\omega)\qquad(\omega\in W_{2d}).\tag{3}$$

Step 4 (the operators). Let $(H,M,\Omega)=(\mathcal{H}_{\gamma},\mathcal{M}_{\gamma},\Omega_{\gamma})$, a tracial W*-probability space by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, with $\mathcal{M}_{\gamma}=\mathcal{A}_{\gamma}''$ where $\mathcal{A}_{\gamma}=\{L_{p}:p\in\mathcal{P}_{3d}\}$. Every $L_{p}$ belongs to $M$: it commutes with every element of $\mathcal{A}_{\gamma}'$ by The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant, so $L_{p}\in\mathcal{A}_{\gamma}''$. The trace is $\tau_{M}(T)=\langle\Omega,T\Omega\rangle$ (Tracial W*-Probability Spaces §trace, Cyclic Tracial Operator Algebras and Their Traces §trace). Since $\gamma\in\Sigma_{3d,2}$, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation applies with $r=2$: for $p,q\in\mathcal{P}_{3d}$ and $c\in\mathbb{C}$, $L_{p+q}=L_{p}+L_{q}$, $L_{cp}=cL_{p}$, $L_{pq}=L_{p}L_{q}$, $L_{1}=I$ (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), $L_{p}^{*}=L_{p^{*}}$ (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), $\lVert L_{x_{j}}\rVert_{\mathrm{op}}\le2$ (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), and $L_{p}\Omega=\widehat{p}$, $\langle\widehat{p},\widehat{q}\rangle=\gamma(p^{*}q)$, $\tau_{M}(L_{p})=\gamma(p)$ (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).

For $j\in[d]$ put $U_{j}=L_{y_{j}}$, where $y_{j}=x_{j}+ix_{d+j}$, and $S_{j}=L_{x_{2d+j}}$; both belong to $M$. Each $S_{j}$ is self-adjoint (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, $x_{2d+j}$ being self-adjoint) with $\lVert S_{j}\rVert_{\mathrm{op}}\le2$. For a letter $l$ we have $U^{(l)}=L_{y_{l}}$ in the notation of the statement: if $\varepsilon(l)=1$ then $l=g(l)$ and this is the definition; if $\varepsilon(l)=-1$ then $l=d+j$ with $j=g(l)$ and $U^{(l)}=U_{j}^{*}=L_{y_{j}^{*}}$ with $y_{j}^{*}=x_{j}-ix_{d+j}=y_{d+j}$. By $L_{pq}=L_{p}L_{q}$ and $L_{1}=I$ it follows that

$$U^{(\omega)}=L_{Q_{\omega}}\qquad(\omega\in W_{2d}).\tag{4}$$

Unitarity. Let $l$ be a letter and $r=Q_{l\,l^{-1}}-1$. Since $(l\,l^{-1})^{*}=(l^{-1})^{-1}l^{-1}=l\,l^{-1}$, we get $r^{*}=r$ and $r^{*}r=Q_{l\,l^{-1}\,l\,l^{-1}}-2Q_{l\,l^{-1}}+1$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra). By (3) and Laws of d-Tuples of Unitaries §cancellation (with $u=\varnothing$ and $v=l\,l^{-1}$, resp. $u=v=\varnothing$) and Laws of d-Tuples of Unitaries §normalised, $\gamma(Q_{l\,l^{-1}\,l\,l^{-1}})=\lambda(l\,l^{-1})=\lambda(\varnothing)=1$, so $\lVert L_{r}\Omega\rVert^{2}=\langle\widehat{r},\widehat{r}\rangle=\gamma(r^{*}r)=1-2+1=0$. As $L_{r}\in M$ and $(H,M,\Omega)$ is a cyclic tracial operator algebra, $L_{r}=0$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §separating. By (4), $U^{(l)}U^{(l^{-1})}=L_{Q_{l\,l^{-1}}}=L_{r}+L_{1}=I$. For $l=j$ this reads $U_{j}U_{j}^{*}=I$, and for $l=d+j$ it reads $U_{j}^{*}U_{j}=I$. By Fact U, every $U_{j}$ is unitary. By (4) and (3), $\lambda_{U}(\omega)=\tau_{M}(L_{Q_{\omega}})=\gamma(Q_{\omega})=\lambda(\omega)$ for every $\omega$, i.e. $\lambda_{U}=\lambda$.

Elements of $\mathcal{A}_{U}$. Let $Z\in\mathcal{A}_{U}$, say $Z=\sum_{r=1}^{N}c_{r}U^{(\omega_{r})}$. By (4) and the algebra rules, $Z=L_{q}$ with $q=\sum_{r}c_{r}Q_{\omega_{r}}=\iota^{1}(\tilde q)$, $\tilde q=\sum_{r}c_{r}P_{\omega_{r}}\in\mathcal{P}_{2d}$. If $\tilde q\neq0$, then by (E) and the properties of $\iota^{1}$,

$$q=\sum_{v\in\operatorname{supp}\tilde q}\tilde q(v)\,x_{v},\qquad\tau_{M}(Z)=\gamma(q)=\sum_{v\in\operatorname{supp}\tilde q}\tilde q(v)\,\gamma(x_{v}),\tag{5}$$

where every $v$ is a word all of whose letters are at most $2d$; if $\tilde q=0$ then $Z=L_{0}=0$.

(a) Centred. Let $i\in[d]$ and $Z=L_{q}$ as above. Then $\tau_{M}(S_{i}Z)=\tau_{M}(L_{x_{2d+i}q})=\gamma(x_{2d+i}q)=\partial^{\gamma}_{2d+i}(q)$ by (SD). If $\tilde q=0$ this is $0$. Otherwise, by (5) and linearity, it is $\sum_{v}\tilde q(v)\,\partial^{\gamma}_{2d+i}(x_{v})$; here $\partial^{\gamma}_{2d+i}(x_{\varnothing})=\partial^{\gamma}_{2d+i}(1)=0$, and for $v$ of length $k$ every term of the sum defining $\partial^{\gamma}_{2d+i}(x_{v})$ in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient carries the factor $\delta_{v_{m},2d+i}=0$, since $v_{m}\le2d<2d+i$. So $\tau_{M}(S_{i}Z)=0$.

(b) Covariance. Let $i,l\in[d]$ and $Z=L_{q}$, $Z'=L_{q'}$ as above. Then $\tau_{M}(S_{i}ZS_{l}Z')=\gamma(x_{2d+i}\,q\,x_{2d+l}\,q')=\partial^{\gamma}_{2d+i}(q\,x_{2d+l}\,q')$ by (SD). If $\tilde q=0$ or $\tilde q'=0$, both this and $\delta_{il}\tau_{M}(Z)\tau_{M}(Z')$ vanish. Otherwise, by (5), the algebra rules and $x_{v}x_{2d+l}x_{v'}=x_{v\,(2d+l)\,v'}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials),

$$\partial^{\gamma}_{2d+i}(q\,x_{2d+l}\,q')=\sum_{v\in\operatorname{supp}\tilde q}\ \sum_{v'\in\operatorname{supp}\tilde q'}\tilde q(v)\tilde q'(v')\,\partial^{\gamma}_{2d+i}\bigl(x_{v\,(2d+l)\,v'}\bigr).$$

The word $w=v\,(2d+l)\,v'$ has the letter $2d+l$ at the position $n_{0}$ following the letters of $v$, while all its other letters are at most $2d$ and so differ from $2d+i$. Hence in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient only the term at $n_{0}$ can be nonzero; it carries $\delta_{2d+l,2d+i}=\delta_{il}$, and there $w_{<n_{0}}=v$ and $w_{>n_{0}}=v'$. So $\partial^{\gamma}_{2d+i}(x_{v(2d+l)v'})=\delta_{il}\gamma(x_{v})\gamma(x_{v'})$, and summing with (5),

$$\tau_{M}(S_{i}ZS_{l}Z')=\delta_{il}\Bigl(\sum_{v}\tilde q(v)\gamma(x_{v})\Bigr)\Bigl(\sum_{v'}\tilde q'(v')\gamma(x_{v'})\Bigr)=\delta_{il}\,\tau_{M}(Z)\,\tau_{M}(Z').$$

This proves clause 2 with $(H,M,\Omega)$, $U=(U_{1},\dots,U_{d})$ and $S_{1},\dots,S_{d}$. $\blacksquare$