Finite dimensional zero Jordan product determined algebras are generated by idempotents

Brešar showed that a finite dimensional unital associative algebra is zero product determined if and only if it is generated by idempotents. For the analogue of zero Jordan product determined algebras, only one direction was known: over a field of characteristic not 2, every algebra generated by idempotents is zero Jordan product determined. Whether the converse holds has remained an open problem. In this paper, we answer this question affirmatively in the finite dimensional case. Some related open problems are stated at the end.