The starred Dixmier's conjecture

Abstract

Dixmier's famous question says the following: Is every algebra endomorphism of the first Weyl algebra, $A_1(F)$, where $F$ is a zero characteristic field, an automorphism? Let $\alpha$ be the exchange involution on $A_1(F)$: $\alpha (x)=y$, $\alpha (y)=x$. An $\alpha$-endomorphism of $A_1(F)$ is an endomorphism which preserves the involution $\alpha$. Then one may ask the following question, which may be called the "$\alpha$-Dixmier's problem $1$" or the "starred Dixmier's problem $1$": Is every $\alpha$-endomorphism of $A_1(F)$ an automorphism?

Revised proof in section 3

References in corpus (4)

• On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic
• An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators
• The Dixmier conjecture and the shape of possible counterexamples II
• A conjecture of Bavula on homomorphisms of the Weyl algebra

Cited by in corpus (2)

• Ideas about the Jacobian Conjecture
• Observations on the two dimensional Jacobian Conjecture