  We study algebras of bounded noncommutative (nc) functions on unit balls of
operator spaces (nc operator balls) and on their subvarieties. Considering the
example of the nc unit polydisk we show that these algebras, while having a
natural operator algebra structure, might not be the multiplier algebra of any
reasonable nc reproducing kernel Hilbert space (RKHS). After examining
additional subtleties of the nc RKHS approach, we turn to study the structure
and representation theory of these algebras using function theoretic and
operator algebraic tools. We show that the underlying nc variety is a complete
invariant for the algebra of uniformly continuous nc functions on a homogeneous
subvariety, in the sense that two such algebras are completely isometrically
isomorphic if and only if the subvarieties are nc biholomorphic. We obtain
extension and rigidity results for nc maps between subvarieties of nc operator
balls corresponding to injective spaces that imply that a biholomorphism
between homogeneous varieties extends to a biholomorphism between the ambient
balls, which can be modified to a linear isomorphism. Thus, the algebra of
uniformly continuous nc functions on nc operator balls, and even its
restriction to certain subvarieties, completely determine the operator space up
to completely isometric isomorphism.
