Proof of Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data

Each result cited below is universally quantified over the data in its own statement. For $k\in\mathbb{N}$ the space $\mathcal{P}_{k}$ is a complex vector space by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §vector-space, and finite sums in it are those of Finite Sum Notation in a Vector Space. We use the rules of Properties of Finite Sums of Vectors for such sums (claim 1: recursion; claim 2: additivity; claim 3: homogeneity; claim 4: a linear map commutes with a finite sum; claim 7: a single possibly nonzero summand) and the same rules for finite sums of numbers, claims 1, 2, 3 and 7 of Properties of Finite Sums. For $p\in\mathcal{P}_{k}$ the maps $q\mapsto pq$ and $q\mapsto qp$ of $\mathcal{P}_{k}$ into itself are linear by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra; composing with a linear map $\lambda:\mathcal{P}_{k}\to\mathbb{C}$, so are $q\mapsto\lambda(pq)$ and $q\mapsto\lambda(qp)$. We call this fact (L). For an affine datum $U$ with tuple $a^{U}$, the map $\sigma_{U}=\sigma_{a^{U}}$ is linear by Substitution of Noncommutative Polynomials into the Variables §substitution, and 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 (for the tuple $a^{U}$) it satisfies $\sigma_{U}(1)=1$, $\sigma_{U}(x_{j})=a^{U}_{j}$ and $\sigma_{U}(pq)=\sigma_{U}(p)\sigma_{U}(q)$. We call these facts (H).

Proof of clause 1 (Self-adjointness). Let $i\in[n]$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint, $\mathcal{P}_{m,\mathrm{sa}}$ contains $1$ and $x_{1},\dots,x_{m}$ and is closed under sums and under multiplication by real numbers. Since $c_{i}$ and the $A_{ij}$ are real, $c_{i}1$ and each $A_{ij}x_{j}$ are self-adjoint; by induction on $k\in[m]$ along the recursion of claim 1 of Properties of Finite Sums of Vectors, each partial sum $\sum_{j=1}^{k}A_{ij}x_{j}$ is self-adjoint, and hence so is $a^{T}_{i}$. Thus $a^{T}$ is an $n$-tuple in $\mathcal{P}_{m,\mathrm{sa}}$.

Let $\lambda$ be a tracial state on $\mathcal{P}_{m}$ and put $\lambda_{T}=\lambda\circ\sigma_{T}$, a linear map $\mathcal{P}_{n}\to\mathbb{C}$ as a composite of linear maps. Since $a^{T}$ is self-adjoint, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint gives $\sigma_{T}(p^{*})=\sigma_{T}(p)^{*}$ for $p\in\mathcal{P}_{n}$. Let $p,q\in\mathcal{P}_{n}$. By (H) and the three conditions of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state for $\lambda$: $\lambda_{T}(1)=\lambda(1)=1$; $\lambda_{T}(p^{*}p)=\lambda\bigl(\sigma_{T}(p)^{*}\sigma_{T}(p)\bigr)$ is real and nonnegative; and $\lambda_{T}(pq)=\lambda\bigl(\sigma_{T}(p)\sigma_{T}(q)\bigr)=\lambda\bigl(\sigma_{T}(q)\sigma_{T}(p)\bigr)=\lambda_{T}(qp)$. Hence $\lambda_{T}$ is a tracial state on $\mathcal{P}_{n}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state.

Now let $\lambda\in\Sigma_{m}$; by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there is a real $R>0$ with $\lambda\in\Sigma_{m,R}$. For $i\in[n]$ put $s_{i}=|c_{i}|+R\sum_{j=1}^{m}|A_{ij}|$. Each $|A_{ij}|$ and $|c_{i}|$ is nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field, so $\sum_{j}|A_{ij}|\ge0$ by claim 5 of Properties of Finite Sums, and $s_{i}\ge0$ by claims 5 and 2 of Elementary Arithmetic in an Ordered Field. Put $S=1+\sum_{i=1}^{n}s_{i}$. By claims 5 and 6 of Properties of Finite Sums, $0\le s_{i}\le\sum_{k=1}^{n}s_{k}$, so by claims 1, 2 and 3 of Elementary Arithmetic in an Ordered Field we get $s_{i}\le S$ for every $i\in[n]$ and $1\le S$; hence $S>0$ by claims 6 and 2 of Elementary Order Arithmetic in an Ordered Field. The tuple $a^{T}$ has the form required in The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions §affine (for the law $\lambda\in\Sigma_{m,R}$, with $d=m$, coefficients $c_{i0}=c_{i}$ and $c_{ik}=A_{ik}$, and the bound $S$), so that clause gives $\lambda\circ\sigma_{T}\in\Sigma_{n,S}\subseteq\Sigma_{n}$.

Proof of clause 2 (Composition). Write $S=(B,b)$, so that $a^{S}_{i}=b_{i}1+\sum_{j=1}^{n}B_{ij}x_{j}\in\mathcal{P}_{n}$ for $i\in[r]$. Let $e$ be the $r$-tuple in $\mathcal{P}_{m}$ with $e_{i}=\sigma_{T}(a^{S}_{i})$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition, applied to the $r$-tuple $a^{S}$ in $\mathcal{P}_{n}$ and the $n$-tuple $a^{T}$ in $\mathcal{P}_{m}$, we have $\sigma_{T}(\sigma_{S}(p))=\sigma_{e}(p)$ for every $p\in\mathcal{P}_{r}$. By linearity of $\sigma_{T}$, claim 4 of Properties of Finite Sums of Vectors and (H),

while by Affine Data and Affine Substitutions of Noncommutative Polynomials §composite and Affine Data and Affine Substitutions of Noncommutative Polynomials §tuple

Fix a word $v\in W_{m}$. The map $p\mapsto p(v)$ from $\mathcal{P}_{m}$ to $\mathbb{C}$ is linear, because the linear operations of The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear act coefficientwise. Applying it, with claim 4 of Properties of Finite Sums of Vectors and claims 2 and 3 of Properties of Finite Sums, and writing $\epsilon=1(v)$ and $\epsilon_{l}=x_{l}(v)$, we get

These agree by Interchange of a Finite Double Sum, applied in $\mathbb{C}$ to the $n$-tuple of $m$-tuples with components $B_{ij}A_{jl}\epsilon_{l}$. Polynomials are maps $W_{m}\to\mathbb{C}$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §polynomials, so $e_{i}=a^{S\circ T}_{i}$ for every $i\in[r]$. Hence $\sigma_{T}\circ\sigma_{S}=\sigma_{e}=\sigma_{a^{S\circ T}}=\sigma_{S\circ T}$.

For the identity datum, $a^{\mathrm{id}_{m}}_{i}=0\cdot1+\sum_{j=1}^{m}E_{ij}x_{j}$. The summands with $j\neq i$ are $0x_{j}=0$ and the summand with $j=i$ is $x_{i}$, so claim 7 of Properties of Finite Sums of Vectors gives $a^{\mathrm{id}_{m}}_{i}=x_{i}$. Thus $a^{\mathrm{id}_{m}}$ is the tuple $x=(x_{1},\dots,x_{m})$ of variables, and $\sigma_{\mathrm{id}_{m}}=\sigma_{x}$ is the identity map of $\mathcal{P}_{m}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity. Finally, for a tracial state $\lambda$ on $\mathcal{P}_{m}$, associativity of composition of maps gives $(\lambda\circ\sigma_{T})\circ\sigma_{S}=\lambda\circ(\sigma_{T}\circ\sigma_{S})=\lambda\circ\sigma_{S\circ T}$ and $\lambda\circ\sigma_{\mathrm{id}_{m}}=\lambda$.

Proof of clause 3 (Moments). Let $i,j\in[m]$. The variables are 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 $\mathrm{m}_{i}(\lambda)$ is real by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint, and $\mathrm{m}_{ij}(\lambda)$ is real by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §pairing (both for the tracial state $\lambda$). Condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state gives $\mathrm{m}_{ij}(\lambda)=\mathrm{m}_{ji}(\lambda)$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $1^{*}=1$, $x_{i}^{*}=x_{i}$, $1x_{i}=x_{i}$ and $1^{*}1=1$. So Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §cauchy-schwarz with $p=x_{i}$ and $q=1$ gives $|\lambda(x_{i})|^{2}\le\lambda(x_{i}x_{i})\lambda(1)=\mathrm{m}_{ii}(\lambda)$. Since $\mathrm{m}_{i}(\lambda)$ is real, its modulus is its absolute value by claim 8 of Properties of the Absolute Value in an Ordered Field. That absolute value equals $\mathrm{m}_{i}(\lambda)$ or $-\mathrm{m}_{i}(\lambda)$ by claim 1 of the same lemma, so its square is $\mathrm{m}_{i}(\lambda)^{2}$, and $\mathrm{m}_{i}(\lambda)^{2}\le\mathrm{m}_{ii}(\lambda)$.

For real $\xi_{1},\dots,\xi_{m}$ put $a=\sum_{i=1}^{m}\xi_{i}x_{i}$, which is self-adjoint by the argument of clause 1. By Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §pairing, $\beta_{\lambda}(u,v)=\lambda(uv)$ is symmetric and bilinear over $\mathbb{R}$ on $\mathcal{P}_{m,\mathrm{sa}}$. Applying claim 4 of Properties of Finite Sums of Vectors to the linear maps $u\mapsto\beta_{\lambda}(u,a)$ and $v\mapsto\beta_{\lambda}(x_{i},v)$, and then claim 3 of Properties of Finite Sums, gives

The left side is nonnegative by condition (b) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state.

Now let $i,k\in[n]$, and write $a_{i}=a^{T}_{i}=c_{i}1+u_{i}$ with $u_{i}=\sum_{j=1}^{m}A_{ij}x_{j}$. By (H), $\sigma_{T}(x_{i})=a_{i}$ and $\sigma_{T}(x_{i}x_{k})=a_{i}a_{k}$. Linearity of $\lambda$, claim 4 of Properties of Finite Sums of Vectors and $\lambda(1)=1$ give $\lambda(u_{i})=\sum_{j}A_{ij}\mathrm{m}_{j}(\lambda)$, and therefore $\mathrm{m}_{i}(\lambda\circ\sigma_{T})=\lambda(a_{i})=c_{i}+\sum_{j=1}^{m}A_{ij}\mathrm{m}_{j}(\lambda)$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $a_{i}a_{k}=c_{i}c_{k}1+c_{i}u_{k}+c_{k}u_{i}+u_{i}u_{k}$. By (L) and claim 4 of Properties of Finite Sums of Vectors, applied first to $q\mapsto\lambda(qu_{k})$ and then to $q\mapsto\lambda(x_{j}q)$, and then claim 3 of Properties of Finite Sums,

Applying $\lambda$ to the expansion of $a_{i}a_{k}$ and inserting $\lambda(u_{k})$, $\lambda(u_{i})$ and $\lambda(u_{i}u_{k})$ gives the stated formula for $\mathrm{m}_{ik}(\lambda\circ\sigma_{T})$.

Proof of clause 4 (Coordinate data). Let $i\in[d]$. By Affine Data and Affine Substitutions of Noncommutative Polynomials §tuple, $a^{\mathrm{pr}^{1}}_{i}=0\cdot1+\sum_{j=1}^{2d}P^{1}_{ij}x_{j}$, where the summands with $j\neq i$ are $0$ and the summand with $j=i$ is $x_{i}$. So $a^{\mathrm{pr}^{1}}_{i}=x_{i}$ by claim 7 of Properties of Finite Sums of Vectors; likewise $a^{\mathrm{pr}^{2}}_{i}=x_{d+i}$. Hence $\sigma_{\mathrm{pr}^{1}}=\sigma_{(x_{1},\dots,x_{d})}=\iota^{1}$ and $\sigma_{\mathrm{pr}^{2}}=\sigma_{(x_{d+1},\dots,x_{2d})}=\iota^{2}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals.

For $i,j\in[d]$, the value $Q_{ij}$ is $1$ exactly when $j=i$, because $i=d+j$ is impossible since $i\le d<d+j$. Likewise $Q_{d+i,j}$ is $1$ exactly when $j=i$, because $d+i=j$ is impossible since $j\le d<d+i$. So claim 7 of Properties of Finite Sums of Vectors gives $a^{\mathrm{diag}}_{i}=a^{\mathrm{diag}}_{d+i}=x_{i}$, and $\sigma_{\mathrm{diag}}$ is the substitution of $(x_{1},\dots,x_{d},x_{1},\dots,x_{d})$. The same description, together with claim 7 of Properties of Finite Sums, gives for $i,l\in[d]$

and hence, by claims 2 and 3 of Properties of Finite Sums, $((P^{1}-P^{2})Q)_{il}=(P^{1}Q)_{il}-(P^{2}Q)_{il}=0$. In each composite of Affine Data and Affine Substitutions of Noncommutative Polynomials §composite with $\mathrm{diag}$ the second component is $i\mapsto0+\sum_{k}P_{ik}\cdot0=0$. Hence $\mathrm{pr}^{1}\circ\mathrm{diag}=\mathrm{pr}^{2}\circ\mathrm{diag}=(E,0)=\mathrm{id}_{d}$, and both maps of $D\circ\mathrm{diag}$ have value $0$.

Let $\gamma$ be a tracial state on $\mathcal{P}_{2d}$. By claims 2, 3 and 7 of Properties of Finite Sums of Vectors, $a^{D}_{i}=\sum_{j}P^{1}_{ij}x_{j}-\sum_{j}P^{2}_{ij}x_{j}=x_{i}-x_{d+i}$. Hence (H) gives $\sigma_{D}(x_{j}^{2})=(x_{j}-x_{d+j})^{2}$ for $j\in[d]$. By clause 1, $\gamma\circ\sigma_{D}$ is a tracial state on $\mathcal{P}_{d}$. By claim 4 of Properties of Finite Sums of Vectors for $\gamma$ and Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost,

By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $(x_{j}-x_{d+j})^{2}=x_{j}^{2}-x_{j}x_{d+j}-x_{d+j}x_{j}+x_{d+j}^{2}$. By condition (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, $\gamma(x_{d+j}x_{j})=\gamma(x_{j}x_{d+j})$, so $\gamma((x_{j}-x_{d+j})^{2})=\gamma(x_{j}^{2})+\gamma(x_{d+j}^{2})-2\,\mathrm{m}_{j,d+j}(\gamma)$. Applying (H) to $\iota^{1}=\sigma_{\mathrm{pr}^{1}}$ and $\iota^{2}=\sigma_{\mathrm{pr}^{2}}$ gives $\iota^{1}(x_{j}^{2})=x_{j}^{2}$ and $\iota^{2}(x_{j}^{2})=x_{d+j}^{2}$. Hence $\sum_{j}\gamma(x_{j}^{2})=M(\gamma\circ\iota^{1})$ and $\sum_{j}\gamma(x_{d+j}^{2})=M(\gamma\circ\iota^{2})$, where $\gamma\circ\iota^{1}$ and $\gamma\circ\iota^{2}$ are tracial states by clause 1. Summing over $j$ with claims 2 and 3 of Properties of Finite Sums gives $\gamma(\Delta_{d})=M(\gamma\circ\iota^{1})+M(\gamma\circ\iota^{2})-2\sum_{j=1}^{d}\mathrm{m}_{j,d+j}(\gamma)$.