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 D2 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 Ω with the property that, for every bounded symmetric function φ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function Φ on Ω such that φ=Φ∘S. We also establish a realization formula for Φ, and hence for φ, in terms of operators on Hilbert space.