Symmetric functions of two noncommuting variables

We prove a noncommutative analogue of the fact that every symmetric analytic function of $(z,w)$ in the bidisc $\mathbb{D}^2$ can be expressed as an analytic function of the variables $z+w$ and $zw$. We construct an analytic nc-map $S$ from the biball to an infinite-dimensional nc-domain $\Omega$ with the property that, for every bounded symmetric function $\varphi$ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function $\Phi$ on $\Omega$ such that $\varphi=\Phi\circ S$. We also establish a realization formula for $\Phi$, and hence for $\varphi$, in terms of operators on Hilbert space.