Bernstein algebras that are algebraic and the Kurosh problem

TL;DR

This paper investigates algebraic Bernstein algebras, characterizes their properties, explores the Kurosh problem regarding their finite-dimensionality, and connects these findings to the Jacobian conjecture.

Contribution

It provides new characterizations of algebraic Bernstein algebras, links the Kurosh problem to associative algebra questions, and proves the generalized Jacobian conjecture for quadratic mappings in this context.

Findings

Algebraic Bernstein algebras have bounded degree.

Counter-examples exist for the Kurosh problem in general.

The generalized Jacobian conjecture holds for Bernstein algebras.

Abstract

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of train Bernstein algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train). Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the barideal is nil, the Kurosh problem asks whether a finitely generated Bernstein-train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case. By results of Yagzhev, the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras.