The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n\in\mathbb{N}$, and let $\iota^{1}:\mathcal{P}_{m}\to\mathcal{P}_{m+n}$, $\iota^{2}:\mathcal{P}_{n}\to\mathcal{P}_{m+n}$ and freeness be as in Freeness of Two Groups of Variables under a Noncommutative Law §free, for the split of $m+n$ variables into the first $m$ and the last $n$ (and likewise for any other split named below). Marginals of laws, and laws obtained from a law by substituting a tuple of self-adjoint polynomials, are laws by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law; the tuples substituted below 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 and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint.

1. (Free product)¶ For all $\alpha\in\Sigma_{m}$ and $\beta\in\Sigma_{n}$ there is exactly one free law $\gamma\in\Sigma_{m+n}$ with $\gamma\circ\iota^{1}=\alpha$ and $\gamma\circ\iota^{2}=\beta$. It is written $\alpha\star\beta$ and called the free product of $\alpha$ and $\beta$.

2. (Norm bound)¶ If $R>0$ is real, $\alpha\in\Sigma_{m,R}$ and $\beta\in\Sigma_{n,R}$, then $\alpha\star\beta\in\Sigma_{m+n,R}$.

3. (Substitutions)¶ Let $m',n'\in\mathbb{N}$, let $a$ be an $m'$-tuple in $\mathcal{P}_{m,\mathrm{sa}}$, let $b$ be an $n'$-tuple in $\mathcal{P}_{n,\mathrm{sa}}$, and let $c=(\iota^{1}(a_{1}),\dots,\iota^{1}(a_{m'}),\iota^{2}(b_{1}),\dots,\iota^{2}(b_{n'}))$, an $(m'+n')$-tuple in $\mathcal{P}_{m+n}$. Then for all $\alpha\in\Sigma_{m}$ and $\beta\in\Sigma_{n}$,

$$(\alpha\star\beta)\circ\sigma_{c}=(\alpha\circ\sigma_{a})\star(\beta\circ\sigma_{b}),$$

the right side being the free product for the split of $m'+n'$ variables into the first $m'$ and the last $n'$.

4. (Symmetry)¶ For all $\alpha\in\Sigma_{m}$ and $\beta\in\Sigma_{n}$, $(\alpha\star\beta)\circ\sigma_{s}=\beta\star\alpha$, where $s=(x_{m+1},\dots,x_{m+n},x_{1},\dots,x_{m})$ and the right side is the free product for the split of $n+m$ variables into the first $n$ and the last $m$.

5. (Weak-star continuity)¶ Let $R>0$ be real, and let $(\alpha_{k})_{k\in\mathbb{N}}$ in $\Sigma_{m,R}$ and $(\beta_{k})_{k\in\mathbb{N}}$ in $\Sigma_{n,R}$ converge weak-star to $\alpha\in\Sigma_{m}$ and $\beta\in\Sigma_{n}$. Then $\alpha_{k}\star\beta_{k}\to\alpha\star\beta$ weak-star.