Characterization of algebras supporting pointwise algebras of holomorphic functions

Determine which finite-dimensional real algebras carry an algebra of holomorphic functions under pointwise multiplication.

Background

The paper characterizes finite-dimensional real algebras in which the familiar Product Rule holds for holomorphic functions, showing that, in the unital case, this is equivalent to associativity and commutativity. However, the authors explicitly state that this result does not settle the broader question of which algebras support an algebra of holomorphic functions under pointwise multiplication. The unresolved problem is therefore to obtain a complete characterization beyond the specific derivative rule addressed in the paper.