Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $\mathbb{N}$ be the set of natural numbers, let $d\in\mathbb{N}$, let $W_{2d}$ and the generators $g(l)$ and signs $\varepsilon(l)$ of letters $l\in[2d]$ be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words, and let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples.

For every tracial W*-probability space $(H,M,\Omega)$, with trace $\tau_{M}$, and every $d$-tuple $V=(V_{1},\dots,V_{d})$ of unitary operators on $H$ belonging to $M$, the following notation is used. For a letter $l\in[2d]$ put $V^{(l)}=V_{g(l)}$ if $\varepsilon(l)=1$ and $V^{(l)}=V_{g(l)}^{*}$ if $\varepsilon(l)=-1$; put $V^{(\varnothing)}=I$ and, for a word $w$ of length $k\in\mathbb{N}$, $V^{(w)}=V^{(w_{1})}V^{(w_{2})}\cdots V^{(w_{k})}$. These operators belong to $M$, which contains $I$ and is closed under products and adjoints by Cyclic Tracial Operator Algebras and Their Traces §star-algebra. We write $\mathcal{A}_{V}$ for the set of finite linear combinations, with complex coefficients, of the operators $V^{(w)}$, $w\in W_{2d}$; thus $\mathcal{A}_{V}\subseteq M$, since $M$ is closed under sums, complex scalar multiples, products and adjoints by Cyclic Tracial Operator Algebras and Their Traces §star-algebra. Finally, $\lambda_{V}:W_{2d}\to\mathbb{C}$ is the map given by $\lambda_{V}(w)=\tau_{M}\bigl(V^{(w)}\bigr)$.

Then the following hold.

1. (Laws of unitary tuples)¶ For every tracial W*-probability space $(H,M,\Omega)$ and every $d$-tuple $V$ of unitary operators on $H$ belonging to $M$, $\lambda_{V}\in\mathcal{L}_{d}$.

2. (Realisation together with a semicircular family)¶ For every $\lambda\in\mathcal{L}_{d}$ there are a tracial W*-probability space $(H,M,\Omega)$ with trace $\tau_{M}$, a $d$-tuple $U=(U_{1},\dots,U_{d})$ of unitary operators on $H$ belonging to $M$ with $\lambda_{U}=\lambda$, and self-adjoint operators $S_{1},\dots,S_{d}\in M$ with $\lVert S_{j}\rVert_{\mathrm{op}}\le2$ for every $j\in[d]$, such that the following two properties hold, where $\delta_{il}=1$ if $i=l$ and $\delta_{il}=0$ if $i\neq l$.

¶(a) (Centred) $\tau_{M}(S_{i}Z)=0$ for every $i\in[d]$ and every $Z\in\mathcal{A}_{U}$.

¶(b) (Covariance) $\tau_{M}(S_{i}\,Z\,S_{l}\,Z')=\delta_{il}\,\tau_{M}(Z)\,\tau_{M}(Z')$ for all $i,l\in[d]$ and all $Z,Z'\in\mathcal{A}_{U}$.