A noncommutative homogeneous variety in the free $d$-dimensional ball is a subset of the free ball cut out by homogenous free polynomials in $d$ variables. Such a variety comes with a certain pseudo metric that is defined on its similarity envelope: the set of all $d$-tuples of matrices jointly similar to a point in the variety.

In this talk, I will show how this geometric data encodes the topological-algebraic structure of the algebras of bounded nc holomorphic functions over these varieties (equipped with the natural sup norm). More precisely, I will prove that given two nc varieties, their algebras of bounded holomorphic nc functions are continuously isomorphic if and only if there is a bi-Lipschitz invertible linear map sending one variety onto the other.

The talk is based on joint work with Eli Shamovich and Orr Shalit.