Weak-Star Convergence of Noncommutative Laws

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, and let $\Sigma_{d}$ be the set of noncommutative laws of $d$ variables. For a complex number $z$ let $\operatorname{Re}z$ and $\operatorname{Im}z$ be its real and imaginary parts; sequences are indexed by $\mathbb{N}$ as in Sequence in a Set, and limits of real sequences are those of Limit of a Sequence of Real Numbers.

1. (Weak-star convergence)¶ A sequence $(\lambda_{m})_{m\in\mathbb{N}}$ in $\Sigma_{d}$ converges weak-star to $\lambda\in\Sigma_{d}$ if, for every $p\in\mathcal{P}_{d}$, the real sequences $(\operatorname{Re}\lambda_{m}(p))_{m\in\mathbb{N}}$ and $(\operatorname{Im}\lambda_{m}(p))_{m\in\mathbb{N}}$ converge to $\operatorname{Re}\lambda(p)$ and $\operatorname{Im}\lambda(p)$ respectively. One then writes $\lambda_{m}\to\lambda$ weak-star.