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

Let $d\in\mathbb{N}$, let $\mathcal{P}_{d}=\mathbb{C}\langle x_{1},\dots,x_{d}\rangle$ be the noncommutative polynomials in $d$ variables with monomials $x_{w}$ indexed by the set $W_{d}$ of words, let $\Sigma_{d,R}$ be the set of tracial states with norm bound $R$ and $\Sigma_{d}$ the set of noncommutative laws, and let weak-star convergence be that of Weak-Star Convergence of Noncommutative Laws §weak-star. Real and imaginary parts are those of Real and Imaginary Parts of a Complex Number, and subsequences are those of Subsequence of a Sequence in a Set.

1. (Monomials suffice)¶ Let $(\lambda_{m})_{m\in\mathbb{N}}$ be a sequence in $\Sigma_{d}$ and $\lambda\in\Sigma_{d}$. Then $\lambda_{m}\to\lambda$ weak-star if and only if, for every $w\in W_{d}$, the real sequences $(\operatorname{Re}\lambda_{m}(x_{w}))_{m}$ and $(\operatorname{Im}\lambda_{m}(x_{w}))_{m}$ converge to $\operatorname{Re}\lambda(x_{w})$ and $\operatorname{Im}\lambda(x_{w})$.

2. (Uniqueness of limits)¶ If $(\lambda_{m})$ is a sequence in $\Sigma_{d}$, $\lambda,\lambda'\in\Sigma_{d}$, $\lambda_{m}\to\lambda$ and $\lambda_{m}\to\lambda'$ weak-star, then $\lambda=\lambda'$. If $\lambda_{m}\to\lambda$ weak-star, then every subsequence of $(\lambda_{m})$ converges weak-star to $\lambda$.

3. (Closedness)¶ Let $R>0$ be real. If $\lambda_{m}\in\Sigma_{d,R}$ for every $m\in\mathbb{N}$ and $\lambda_{m}\to\lambda$ weak-star, then $\lambda\in\Sigma_{d,R}$.

4. (Sequential compactness)¶ Let $R>0$ be real. Every sequence in $\Sigma_{d,R}$ has a subsequence that converges weak-star to an element of $\Sigma_{d,R}$.