Proof of Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound

This proof uses Weak-Star Convergence of Noncommutative Laws §weak-star; conditions (a), (b), (c) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law; The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §polynomials, The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials; Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension; Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words and Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §countable; Countable Set; Sum over a Finite Index Set and Finite Sum Notation in a Field; Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits §limit; Real and Imaginary Parts of a Complex Number, claims 3 and 4 of Canonical Form and Arithmetic of Complex Numbers, Modulus of a Complex Number, claims 6 and 8 of Properties of Complex Conjugation and Modulus, Absolute Value in an Ordered Field; Limit of a Sequence of Real Numbers, claims 1, 2, 3 of Arithmetic of Limits of Real Sequences, claim 1 of Order Properties of Limits of Real Sequences, claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences; The Absolute Value Metric on the Real Line, Convergent Sequence in a Metric Space, A Subsequence of a Convergent Sequence Has the Same Limit; Subsequence of a Sequence in a Set, claims 2 and 3 of A Subsequence of a Subsequence is a Subsequence, The Diagonal Subsequence Lemma for Bounded Real Arrays; claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; claim 5 of Properties of Natural Number Powers in a Field.

Step 0 (Preliminaries). (0a) Every $z\in\mathbb{C}$ satisfies $z=\operatorname{Re}z+(\operatorname{Im}z)i$ (Real and Imaginary Parts of a Complex Number), and $z$ is determined by $\operatorname{Re}z$ and $\operatorname{Im}z$ (claim 3 of Canonical Form and Arithmetic of Complex Numbers). By claims 3 and 4 of Canonical Form and Arithmetic of Complex Numbers, for $z,\zeta\in\mathbb{C}$: $\operatorname{Re}(z+\zeta)=\operatorname{Re}z+\operatorname{Re}\zeta$, $\operatorname{Im}(z+\zeta)=\operatorname{Im}z+\operatorname{Im}\zeta$, $\operatorname{Re}(\zeta z)=\operatorname{Re}\zeta\operatorname{Re}z-\operatorname{Im}\zeta\operatorname{Im}z$ and $\operatorname{Im}(\zeta z)=\operatorname{Re}\zeta\operatorname{Im}z+\operatorname{Im}\zeta\operatorname{Re}z$; and a real $x$ has $\operatorname{Re}x=x$, $\operatorname{Im}x=0$. By Modulus of a Complex Number, $|z|\ge0$ and $|z|^{2}=(\operatorname{Re}z)^{2}+(\operatorname{Im}z)^{2}$. By claim 6 of Properties of Complex Conjugation and Modulus and Absolute Value in an Ordered Field, the absolute values of $\operatorname{Re}z$ and $\operatorname{Im}z$ are at most $|z|$. (0b) A constant real sequence converges to its value, directly by Limit of a Sequence of Real Numbers. A real sequence converges to $L$ in the sense of Limit of a Sequence of Real Numbers exactly when it converges to $L$ in the real line $(\mathbb{R},d_{\mathbb{R}})$ of The Absolute Value Metric on the Real Line in the sense of Convergent Sequence in a Metric Space, since $d_{\mathbb{R}}(s,t)=|s-t|$ and both definitions then state the same condition. Hence, by A Subsequence of a Convergent Sequence Has the Same Limit, every subsequence (Subsequence of a Sequence in a Set) of a real sequence converging to $L$ converges to $L$. (0c) Real limits are unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences.

Step 1 (Convergence on monomials propagates). Let $(\mu_{m})_{m\in\mathbb{N}}$ be a sequence of linear maps $\mathcal{P}_{d}\to\mathbb{C}$ and $\mu:\mathcal{P}_{d}\to\mathbb{C}$ linear, such that for every $w\in W_{d}$ the real sequences $(\operatorname{Re}\mu_{m}(x_{w}))_{m}$ and $(\operatorname{Im}\mu_{m}(x_{w}))_{m}$ converge to $\operatorname{Re}\mu(x_{w})$ and $\operatorname{Im}\mu(x_{w})$. We show the same holds with $x_{w}$ replaced by any $p\in\mathcal{P}_{d}$. If $p=0$, then $\mu_{m}(p)=0=\mu(p)$ by linearity, and both real sequences are constant $0$, so (0b) applies. Let $p\ne0$; then $\operatorname{supp}p$ is nonempty and finite (The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §polynomials), say with $N$ elements, and let $\varphi:[N]\to\operatorname{supp}p$ be a bijection. By the uniqueness in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension (a), applied to $c(w)=\mu_{m}(x_{w})$ and to $c(w)=\mu(x_{w})$, the linear maps $\mu_{m}$ and $\mu$ are given by the formula there, so by Sum over a Finite Index Set

By induction on $j\in[N]$, using the recursion of Finite Sum Notation in a Field and the additivity in (0a), $\operatorname{Re}\sum_{k=1}^{j}y_{k}=\sum_{k=1}^{j}\operatorname{Re}y_{k}$ and likewise for $\operatorname{Im}$, for any $y:[N]\to\mathbb{C}$. For each $k\in[N]$, by (0a) and claims 3 and 1 of Arithmetic of Limits of Real Sequences (scalar multiples, sums and differences),

and in the same way $\operatorname{Im}(\zeta_{k}z_{m,k})\to\operatorname{Im}(\zeta_{k}z_{k})$ as $m\to\infty$. By Finite Sums of Real Numbers: Nonnegativity, Domination by the Sum, and Limits §limit (with $j=N$), $\operatorname{Re}\mu_{m}(p)=\sum_{k=1}^{N}\operatorname{Re}(\zeta_{k}z_{m,k})\to\sum_{k=1}^{N}\operatorname{Re}(\zeta_{k}z_{k})=\operatorname{Re}\mu(p)$, and likewise $\operatorname{Im}\mu_{m}(p)\to\operatorname{Im}\mu(p)$.

Step 2 (Monomials suffice). If $\lambda_{m}\to\lambda$ weak-star, the condition of Weak-Star Convergence of Noncommutative Laws §weak-star for $p=x_{w}$ is the monomial condition. Conversely, tracial states are linear (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state), so Step 1 with $\mu_{m}=\lambda_{m}$, $\mu=\lambda$ gives the condition for every $p\in\mathcal{P}_{d}$, i.e. $\lambda_{m}\to\lambda$ weak-star. This proves Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §monomials.

Step 3 (Uniqueness of limits). Let $\lambda_{m}\to\lambda$ and $\lambda_{m}\to\lambda'$ weak-star. For $w\in W_{d}$, by (0c) $\operatorname{Re}\lambda(x_{w})=\operatorname{Re}\lambda'(x_{w})$ and $\operatorname{Im}\lambda(x_{w})=\operatorname{Im}\lambda'(x_{w})$, so $\lambda(x_{w})=\lambda'(x_{w})$ by (0a). Thus $\lambda$ and $\lambda'$ are linear maps with the same values on all monomials, and $\lambda=\lambda'$ by the uniqueness in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension (a). Next let $(n_{k})_{k\in\mathbb{N}}$ be strictly increasing; $(\lambda_{n_{k}})_{k}$ is a sequence in $\Sigma_{d}$, and for every $p$ the real sequences $(\operatorname{Re}\lambda_{n_{k}}(p))_{k}$ and $(\operatorname{Im}\lambda_{n_{k}}(p))_{k}$ are subsequences of $(\operatorname{Re}\lambda_{m}(p))_{m}$ and $(\operatorname{Im}\lambda_{m}(p))_{m}$, so they converge to $\operatorname{Re}\lambda(p)$ and $\operatorname{Im}\lambda(p)$ by (0b). Hence $\lambda_{n_{k}}\to\lambda$ weak-star. This proves Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §unique.

Step 4 (The norm bound passes to limits). Let $R>0$, let $(\mu_{m})_{m}$ be a sequence in $\Sigma_{d,R}$ and $\mu:\mathcal{P}_{d}\to\mathbb{C}$ linear such that $\operatorname{Re}\mu_{m}(x_{w})\to\operatorname{Re}\mu(x_{w})$ and $\operatorname{Im}\mu_{m}(x_{w})\to\operatorname{Im}\mu(x_{w})$ for every $w\in W_{d}$. Let $k\in\mathbb{N}$ and $w$ a word of length $k$. By (0a) and claims 2 and 1 of Arithmetic of Limits of Real Sequences,

Since $0\le|\mu_{m}(x_{w})|\le R^{k}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound) and $R^{k}\ge0$ (claim 5 of Properties of Natural Number Powers in a Field), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $|\mu_{m}(x_{w})|^{2}\le(R^{k})^{2}$ for all $m$; comparing with the constant sequence $(R^{k})^{2}$ (claim 1 of Order Properties of Limits of Real Sequences and (0b)) gives $|\mu(x_{w})|^{2}\le(R^{k})^{2}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field again gives $|\mu(x_{w})|\le R^{k}$.

Step 5 (Closedness). Let $\lambda_{m}\in\Sigma_{d,R}$ for all $m$ and $\lambda_{m}\to\lambda$ weak-star. Then $\lambda\in\Sigma_{d}$ is a tracial state (Weak-Star Convergence of Noncommutative Laws §weak-star, Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law), and Step 4 with $\mu_{m}=\lambda_{m}$, $\mu=\lambda$ (the hypothesis holds by taking $p=x_{w}$ in weak-star convergence) shows $|\lambda(x_{w})|\le R^{k}$ for every word $w$ of length $k$. Hence $\lambda\in\Sigma_{d,R}$, proving Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §closed.

Step 6 (Sequential compactness). Let $(\lambda_{m})_{m\in\mathbb{N}}$ be a sequence in $\Sigma_{d,R}$. The set $W_{d}$ contains $\varnothing$ (Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words), so it is nonempty, and it is countable by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §countable; by Countable Set there is a sequence $(w_{k})_{k\in\mathbb{N}}$ in $W_{d}$ such that every $w\in W_{d}$ equals $w_{k}$ for some $k$. For $k\in\mathbb{N}$ put $B_{k}=1$ if $w_{k}=\varnothing$, and $B_{k}=R^{l}$ if $w_{k}$ has length $l$ (unique by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words). Then $|\lambda_{m}(x_{w_{k}})|\le B_{k}$ for all $m,k$: if $w_{k}=\varnothing$ then $x_{w_{k}}=1$ (The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials) and $|\lambda_{m}(1)|=|1|=1$ by (a) and claim 8 of Properties of Complex Conjugation and Modulus; otherwise this is Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound. By (0a), the real numbers $a_{m,k}=\operatorname{Re}\lambda_{m}(x_{w_{k}})$ satisfy $|a_{m,k}|\le B_{k}$, so The Diagonal Subsequence Lemma for Bounded Real Arrays gives a strictly increasing $(n_{j})_{j\in\mathbb{N}}$ such that $(a_{n_{j},k})_{j}$ converges for every $k$. Again by (0a), $b_{j,k}=\operatorname{Im}\lambda_{n_{j}}(x_{w_{k}})$ satisfies $|b_{j,k}|\le B_{k}$, so The Diagonal Subsequence Lemma for Bounded Real Arrays gives a strictly increasing $(l_{i})_{i\in\mathbb{N}}$ such that $(b_{l_{i},k})_{i}$ converges for every $k$. Put $m_{i}=n_{l_{i}}$; by claims 2 and 3 of A Subsequence of a Subsequence is a Subsequence, $(m_{i})_{i}$ is strictly increasing, $(\lambda_{m_{i}})_{i}$ is a subsequence of $(\lambda_{m})_{m}$, and $(a_{m_{i},k})_{i}$ is the subsequence of $(a_{n_{j},k})_{j}$ determined by $(l_{i})$, hence convergent by (0b). Thus for every $k$ the sequences $(\operatorname{Re}\lambda_{m_{i}}(x_{w_{k}}))_{i}=(a_{m_{i},k})_{i}$ and $(\operatorname{Im}\lambda_{m_{i}}(x_{w_{k}}))_{i}=(b_{l_{i},k})_{i}$ converge; call their limits $\alpha_{k}$ and $\beta_{k}$.

Define $c:W_{d}\to\mathbb{C}$ by $c(w)=\alpha_{k}+\beta_{k}i$ for any $k$ with $w_{k}=w$. This is well defined: if $w_{k}=w_{k'}$, the sequences defining $\alpha_{k},\alpha_{k'}$ coincide, as do those defining $\beta_{k},\beta_{k'}$, so $\alpha_{k}=\alpha_{k'}$ and $\beta_{k}=\beta_{k'}$ by (0c). By (0a), $\operatorname{Re}c(w)=\alpha_{k}$ and $\operatorname{Im}c(w)=\beta_{k}$, so for every $w\in W_{d}$

Let $\lambda:\mathcal{P}_{d}\to\mathbb{C}$ be the unique linear map with $\lambda(x_{w})=c(w)$ for all $w$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension (a)). By Step 1 with $\mu_{i}=\lambda_{m_{i}}$ and $\mu=\lambda$,

We check that $\lambda$ is a tracial state. (a): $\lambda_{m_{i}}(1)=1$, so $\operatorname{Re}\lambda_{m_{i}}(1)=1$ and $\operatorname{Im}\lambda_{m_{i}}(1)=0$ for all $i$ (0a); by $(\ast)$, (0b) and (0c), $\operatorname{Re}\lambda(1)=1$ and $\operatorname{Im}\lambda(1)=0$, so $\lambda(1)=1$ by (0a). (b): for $p\in\mathcal{P}_{d}$, $\lambda_{m_{i}}(p^{*}p)$ is real and $\ge0$, so $\operatorname{Im}\lambda_{m_{i}}(p^{*}p)=0$ and $\operatorname{Re}\lambda_{m_{i}}(p^{*}p)=\lambda_{m_{i}}(p^{*}p)\ge0$ for all $i$. By $(\ast)$, (0b), (0c), $\operatorname{Im}\lambda(p^{*}p)=0$, so $\lambda(p^{*}p)=\operatorname{Re}\lambda(p^{*}p)$ is real by (0a); and $\operatorname{Re}\lambda(p^{*}p)\ge0$ by claim 1 of Order Properties of Limits of Real Sequences, comparing with the constant sequence $0$. (c): for $p,q\in\mathcal{P}_{d}$, $\lambda_{m_{i}}(pq)=\lambda_{m_{i}}(qp)$ for all $i$, so the sequences $(\operatorname{Re}\lambda_{m_{i}}(pq))_{i}$ and $(\operatorname{Re}\lambda_{m_{i}}(qp))_{i}$ are equal; by $(\ast)$ and (0c) their limits $\operatorname{Re}\lambda(pq)$ and $\operatorname{Re}\lambda(qp)$ are equal, likewise for $\operatorname{Im}$, and $\lambda(pq)=\lambda(qp)$ by (0a). Finally Step 4, applied to the sequence $(\lambda_{m_{i}})_{i}$ in $\Sigma_{d,R}$ and $\mu=\lambda$ (its hypothesis is $(\ast)$ for $p=x_{w}$), shows that $\lambda$ has norm bound $R$. Hence $\lambda\in\Sigma_{d,R}\subseteq\Sigma_{d}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law), and by $(\ast)$ and Weak-Star Convergence of Noncommutative Laws §weak-star the subsequence $(\lambda_{m_{i}})_{i}$ of $(\lambda_{m})_{m}$, a sequence in $\Sigma_{d}$, converges weak-star to $\lambda$. This proves Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §compact.