In this talk, I will focus on the similarity envelope of the noncommutative ball, namely the set of all $d$-tuples of matrices that are simultaneously similar to a strict row contraction. In a sense, similarity invariant nc sets are the natural domain of definition of nc functions. In our case, the similarity envelope also coincides with the weak-* continuous finite-dimensional representations of the free semi-group algebra (also known as the algebra of bounded nc functions on the ball). Our key problem is to extend several classical results such as the Schwarz lemma and Cartan's uniqueness theorem to this setting. The problem is of course, that the similarity envelope is inherently unbounded. To deal with this issue we will use the joint spectral radius of a row introduced by Popescu and some algebraic machinery arising from finite-dimensional representations of the free algebra. I will also discuss an application to the study of automorphisms of the free semi-group algebra and the noncommutative ball.

This talk is based on joint work with Guy Salomon and Orr Shalit.