Proof of Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space

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

Preliminaries. Since $\lambda\in\Sigma_{d}$, it is a tracial state on $\mathcal{P}_{d}$, in particular a linear map, and there is a real $r>0$ with $\lambda\in\Sigma_{d,r}$, by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. Write $J=J_{\lambda}$, $\mathcal{H}=\mathcal{H}_{\lambda}$ and $\langle\cdot,\cdot\rangle$ for its inner product. By Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §gns, $J$ is the conjugation of Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation, which therefore gives, for all $\xi,\eta\in\mathcal{H}$ and $q\in\mathcal{P}_{d}$,

$$J\widehat{q}=\widehat{q^{*}},\qquad \langle J\xi,J\eta\rangle=\langle\eta,\xi\rangle .$$

By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ is a tracial W*-probability space whose conjugation is $J$. Hence, by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, a $d$-tuple of vectors of $\mathcal{H}$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ exactly when each of its entries lies in the set $\mathcal{H}^{J}=\{\zeta\in\mathcal{H}:\ J\zeta=\zeta\}$ of fixed vectors of $J$. Moreover, by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint applied to this tracial W*-probability space, $\mathcal{H}^{J}$ is closed under sums and under multiplication by real numbers, and the limit in $\mathcal{H}$ of every convergent sequence in $\mathcal{H}^{J}$ lies in $\mathcal{H}^{J}$.

Claim 1 (Conjugate variables). Suppose that $\lambda$ has conjugate variables $\xi_{\lambda}=(\xi_{\lambda,1},\dots,\xi_{\lambda,d})$, so that, by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate,

$$\langle\xi_{\lambda,j},\widehat{p}\rangle=\partial^{\lambda}_{j}(p)\qquad\text{for every }j\in[d]\text{ and }p\in\mathcal{P}_{d}. \tag{1}$$

Fix $j\in[d]$ and write $\partial=\partial^{\lambda}_{j}$, the linear map of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient.

Step 1: subwords of a reversed word. Let $w\in W_{d}$ be a word of length $k\in\mathbb{N}$. By Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal, $w^{\mathrm{rev}}$ has length $k$ and $(w^{\mathrm{rev}})_{i}=w_{k+1-i}$ for $i\in[k]$; by the uniqueness stated there, $l\mapsto k+1-l$ maps $[k]$ onto itself and is its own inverse, so it is a bijection of $[k]$. Let $l\in[k]$ and put $m=k+1-l\in[k]$. Then $(w^{\mathrm{rev}})_{l}=w_{m}$, and we claim

$$(w^{\mathrm{rev}})_{<l}=(w_{>m})^{\mathrm{rev}},\qquad (w^{\mathrm{rev}})_{>l}=(w_{<m})^{\mathrm{rev}} .$$

For the first identity: if $l=1$ then $m=k$ and both sides are the empty word, as $\varnothing^{\mathrm{rev}}=\varnothing$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal. If $l\ge2$, then $w_{>m}=(w_{m+1},\dots,w_{k})$ has length $k-m=l-1$, as does $(w^{\mathrm{rev}})_{<l}$; for $i\in[l-1]$ the $i$-th letter of $(w^{\mathrm{rev}})_{<l}$ is $(w^{\mathrm{rev}})_{i}=w_{k+1-i}$, and the $i$-th letter of $(w_{>m})^{\mathrm{rev}}$ is the $(l-i)$-th letter of $w_{>m}$, namely $w_{m+l-i}=w_{k+1-i}$. For the second identity: if $l=k$ then $m=1$ and both sides are empty. If $l<k$, then $w_{<m}=(w_{1},\dots,w_{m-1})$ has length $m-1=k-l$, as does $(w^{\mathrm{rev}})_{>l}$; for $i\in[k-l]$ the $i$-th letter of $(w^{\mathrm{rev}})_{>l}$ is $(w^{\mathrm{rev}})_{l+i}=w_{k+1-l-i}=w_{m-i}$, and the $i$-th letter of $(w_{<m})^{\mathrm{rev}}$ is the $(m-i)$-th letter of $w_{<m}$, namely $w_{m-i}$.

Step 2: values on reversed monomials. For every $u\in W_{d}$, the adjoint rules for polynomials give $x_{u^{\mathrm{rev}}}=(x_{u})^{*}$, and the adjoint rule for tracial states gives

$$\lambda(x_{u^{\mathrm{rev}}})=\lambda\bigl((x_{u})^{*}\bigr)=\overline{\lambda(x_{u})}. \tag{2}$$

Let $w$ be a word of length $k$. Since $w^{\mathrm{rev}}$ has length $k$, the defining formula of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient, Step 1, (2), and reindexing the finite sum along the bijection $l\mapsto m=k+1-l$ of $[k]$ give

$$\partial(x_{w^{\mathrm{rev}}})=\sum_{l=1}^{k}\delta_{w_{k+1-l}\,j}\,\overline{\lambda(x_{w_{>k+1-l}})}\;\overline{\lambda(x_{w_{<k+1-l}})} =\sum_{m=1}^{k}\delta_{w_{m}j}\,\overline{\lambda(x_{w_{<m}})\,\lambda(x_{w_{>m}})} =\overline{\partial(x_{w})},$$

where we used that each $\delta_{w_{m}j}\in\{0,1\}$ is real and that conjugation is additive, multiplicative and fixes real numbers, by claim 1 of Properties of Complex Conjugation and Modulus.

Step 3: the identity $\partial(p^{*})=\overline{\partial(p)}$. Define $\varphi:\mathcal{P}_{d}\to\mathbb{C}$ by $\varphi(p)=\overline{\partial(p^{*})}$. By the adjoint rules $(p+q)^{*}=p^{*}+q^{*}$ and $(cp)^{*}=\overline{c}\,p^{*}$, the linearity of $\partial$ and claim 1 of Properties of Complex Conjugation and Modulus, we get $\varphi(p+q)=\varphi(p)+\varphi(q)$ and $\varphi(cp)=\overline{\overline{c}\,\partial(p^{*})}=c\,\varphi(p)$, so $\varphi$ is linear. It agrees with $\partial$ on every monomial: by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words every $w\in W_{d}$ is either $\varnothing$ or a word of some length $k\in\mathbb{N}$; for $w=\varnothing$ we have $x_{\varnothing}=1$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials and $1=1^{*}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, so $\varphi(1)=\overline{\partial(1)}=0=\partial(1)$; and for $w$ of length $k$, $(x_{w})^{*}=x_{w^{\mathrm{rev}}}$ and Step 2 give $\varphi(x_{w})=\overline{\overline{\partial(x_{w})}}=\partial(x_{w})$. 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, $\varphi=\partial$, that is, $\overline{\partial(p^{*})}=\partial(p)$ for every $p\in\mathcal{P}_{d}$.

Step 4: $J$ fixes $\xi_{\lambda,j}$. Let $p\in\mathcal{P}_{d}$. Since $(p^{*})^{*}=p$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, the Preliminaries give $J\widehat{p^{*}}=\widehat{p}$, and therefore

$$\langle J\xi_{\lambda,j},\widehat{p}\rangle=\langle J\xi_{\lambda,j},J\widehat{p^{*}}\rangle=\langle\widehat{p^{*}},\xi_{\lambda,j}\rangle=\overline{\langle\xi_{\lambda,j},\widehat{p^{*}}\rangle}=\overline{\partial(p^{*})}=\partial(p),$$

using the reversal of inner products by $J$, conjugate symmetry (condition 1 of Complex Inner Product Space), (1) and Step 3. This holds for each $j\in[d]$, so the vectors $J\xi_{\lambda,1},\dots,J\xi_{\lambda,d}$ satisfy the defining condition of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate, and by the uniqueness of such vectors stated in that clause, $J\xi_{\lambda,j}=\xi_{\lambda,j}$ for every $j\in[d]$. Thus every $\xi_{\lambda,j}$ lies in $\mathcal{H}^{J}$, and by the Preliminaries $\xi_{\lambda}$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$.

Claim 2 (Wall force). Let $R>0$ be real and suppose that $\lambda$ has a square-integrable wall force of radius $R$; fix $j\in[d]$. First, $(x_{j}^{s})^{*}=x_{j}^{s}$ for every $s\in\mathbb{N}$, by the argument recorded in the preamble of The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law: let $v$ be the word of length $s$ all of whose letters are $1$ and $a=(x_{j})$, a $1$-tuple in $\mathcal{P}_{d}$ with $x_{j}^{*}=x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; then $x_{j}^{s}=a_{v}=\sigma_{a}(x_{v})$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $v^{\mathrm{rev}}=v$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §reversal, so $(x_{v})^{*}=x_{v}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint gives $(x_{j}^{s})^{*}=\sigma_{a}(x_{v})^{*}=\sigma_{a}((x_{v})^{*})=\sigma_{a}(x_{v})=x_{j}^{s}$.

Next, $(F^{R}_{n,j})^{*}=F^{R}_{n,j}$ for every $n\in\mathbb{N}$, where $F^{R}_{n,j}=\sum_{k=1}^{n}2k\,R^{-2k}\,x_{j}^{2k-1}$ is the polynomial of The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law. Indeed each coefficient $c_{k}=2k\,R^{-2k}$ is real, so $\overline{c_{k}}=c_{k}$ by claim 1 of Properties of Complex Conjugation and Modulus, and the adjoint rules give $(c_{k}x_{j}^{2k-1})^{*}=\overline{c_{k}}\,(x_{j}^{2k-1})^{*}=c_{k}x_{j}^{2k-1}$; since the adjoint is additive by the same clause, induction on $n$ gives the assertion: for $n=1$, $F^{R}_{1,j}=c_{1}x_{j}^{1}$ is self-adjoint by what was just shown, and if $(F^{R}_{n,j})^{*}=F^{R}_{n,j}$, then $F^{R}_{n+1,j}=F^{R}_{n,j}+c_{n+1}x_{j}^{2n+1}$ is a sum of two self-adjoint polynomials, hence self-adjoint. By the Preliminaries, $J\widehat{F^{R}_{n,j}}=\widehat{(F^{R}_{n,j})^{*}}=\widehat{F^{R}_{n,j}}$, so $\widehat{F^{R}_{n,j}}\in\mathcal{H}^{J}$ for every $n\in\mathbb{N}$.

By The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law §force, the sequence $(\widehat{F^{R}_{n,j}})_{n\in\mathbb{N}}$ converges in $\mathcal{H}$ to $F^{R}_{j}(\lambda)$. It is a sequence in $\mathcal{H}^{J}$, so by the closedness of $\mathcal{H}^{J}$ recorded in the Preliminaries (from Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint), $F^{R}_{j}(\lambda)\in\mathcal{H}^{J}$. As $j$ was arbitrary, $F^{R}(\lambda)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$.

Claim 3 (Score). Let $(\mathcal{D}_{0},\mathcal{E}_{0})$ be a free entropy penalty, $R>0$ real and $\lambda\in\mathcal{D}_{\Xi}$. By The Wall-Confined Free Energy and Its Score §score, $\lambda$ has conjugate variables $\xi_{\lambda}$ and a square-integrable wall force of radius $R$, and $\Xi(\lambda)_{j}=F^{R}_{j}(\lambda)-\xi_{\lambda,j}=F^{R}_{j}(\lambda)+(-1)\xi_{\lambda,j}$ for $j\in[d]$. By Claims 1 and 2, $F^{R}_{j}(\lambda)$ and $\xi_{\lambda,j}$ lie in $\mathcal{H}^{J}$; since $\mathcal{H}^{J}$ is closed under multiplication by the real number $-1$ and under sums (Preliminaries, from Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint), $\Xi(\lambda)_{j}\in\mathcal{H}^{J}$ for every $j\in[d]$. Hence $\Xi(\lambda)$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples. $\blacksquare$