Abstract:

Given a d-dimensional operator space E with basis Q

_{1}, ..., 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 V_{1}and V_{2}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 E

_{1}and E_{2 }are both injective, and the subvarieties V_{1}and V_{2 }are both homogeneous, we show that a biholomorphism between V_{1}and V_{2}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 V_{1}and V_{2}.