Given a d-dimensional operator space E with basis Q₁, ..., Q_d, consider the corresponding noncommutative (nc) operator ball D_Q determined by Q. In this talk, we discuss the problem of extending certain biholomorphic maps between subvarieties V₁ and V₂ of nc operator balls D_Q1 and D_Q2 .

For trivial reasons, such an extension cannot exist in general, and we discuss several examples to showcase the obstructions. When the operator spaces E₁ and E₂ are both injective, and the subvarieties V₁ and V₂ are both homogeneous, we show that a biholomorphism between V₁ and V₂ can be extended to a biholomorphism between D_Q1 and D_Q2. Moreover, we show that if such an extension exists then there exists a linear isomorphism between D_Q1 and D_Q2 that sends V₁ and V₂.