Functional interpretation of elements of M_d(R) as bona fide NC functions

Ascertain whether every element of the universal skew field M_d(R) of “NC meromorphic expressions” can be identified with an actual non-commutative function on a uniformly open NC subset of the row-ball R·B, rather than merely as an equivalence class of rational expressions.

Background

Elements of M_d(R) are constructed as equivalence classes of non-commutative rational expressions in entries from O_d(R). While this algebraic construction yields a universal skew field, it remains unclear whether such elements define genuine NC functions with well-defined domains and evaluations on uniformly open NC sets.

The authors emphasize this as a central unresolved issue and frame it through two concrete questions: honesty of embeddings O_d(R)→O_d(r) and absence of meromorphic identities.