Zero-product balanced algebras
Eusebio Gardella, Hannes Thiel · arXiv (Cornell University) · 2022
We say that an algebra is zero-product balanced if $ab\otimes c$ and $a\otimes bc$ agree modulo tensors of elements with zero-product. This is closely related to but more general than the notion of a zero-product determined algebra of Brešar, Grašič and Ortega. Every surjective, zero-product preserving map from a zero-product balanced algebra is automatically a weighted epimorphism, and this implies that zero-product balanced algebras are determined by their linear and zero-product structure. Further, the commutator subspace of a zero-product balanced algebra can be described in terms of square-zero elements. We show that a semiprime, commutative algebra is zero-product balanced if and only if it is generated by idempotents. It follows that every commutative, zero-product balanced algebra is spanned by nilpotent and idempotent elements. We deduce a dichotomy for unital, zero-product balanced algebras: They either admit a character or are generated by nilpotents.