On the Computation of Spectra in Free Probability

We use free probability techniques to compute borders of spectra of non-hermitian operators in finite von Neumann algebras which arise as “free sums” of “simple” operators. To this end, the resolvent is analyzed with the aid of the Haagerup inequality. Concrete examples coming from reduced C*-algebras of free product groups and leading to systems of polynomial equations illustrate the approach.