Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $k\in\mathbb{N}$, let $f:\Sigma^{2}_{k}\to\mathbb{R}$, and let $(H,M,\Omega)$ be a tracial W*-probability space; $L^{2}$ tuples, pairs and laws are those of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws.

¶The lift of $f$ to $(H,M,\Omega)$ is the function $f_{M}$ that assigns to every $L^{2}$ $k$-tuple $X$ of $(H,M,\Omega)$ the real number $f_{M}(X)=f(\mathrm{law}(X))$. When $k=m+n$ with $m,n\in\mathbb{N}$, and $X$ and $P$ are $L^{2}$ tuples of $(H,M,\Omega)$ of lengths $m$ and $n$, we write $f_{M}(X,P)=f_{M}((X,P))=f(\mathrm{law}(X,P))$.