What question did this study set out to answer?

The research aims to explore the conditions under which a nonlinear map in unital algebras behaves predictably.

February 12, 2026Open Access

Nonlinear Lie n-centralizers at zero-products on unital algebras

Key Points

The research aims to explore the conditions under which a nonlinear map in unital algebras behaves predictably.
Investigated maps that satisfy a specific polynomial condition in unital algebras.
Applied findings to triangular algebras and certain von Neumann algebras.
Analyzed the structure of the map to derive its form.
Established that if a nonlinear map satisfies a given polynomial condition, it has a specific structure.
Demonstrated that the difference between the map on a sum and the sum of maps lies in the center of the algebra.
Identified that the nonlinear map can be expressed as a linear component plus an almost additive component.

Abstract

Let A be a unital algebra with nontrivial idempotents. In this article, it is shown that under certain conditions if a map (not necessarily linear): A A satisfies (pₙ (a₁, a₂, , aₙ) ) = pₙ ( (a₁), a₂, , aₙ) for all a₁, a₂, , aₙ A with a₁ a₂ aₙ = 0, then (a + b) - (a) - (b) Z (A) for all a, b A. Moreover, is of the form (a) = a + (a) for all a A, where Z (A) and: A Z (A) is an almost additive map such that (pₙ (a₁, a₂, , aₙ) ) = 0 for all a₁, a₂, , aₙ A with a₁ a₂ aₙ = 0. Moreover, this result can also be applied to triangular algebras, von Neumann algebras without central summands of type I₁ and so on.

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