The Starred Dixmier Conjecture for A1

Let A1(K) = K ⟨ X, Y | YX − XY = 1 ⟩ be the first Weyl algebra over a characteristic zero field K, and let α be the exchange involution on A1(K) given by α(X) = Y and α(Y) = X. The Dixmier conjecture of Dixmier (1968) asks the following question: Is every algebra endomorphism of the Weyl algebra A1(K) an automorphism? The aim of this paper is to prove that each α-endomorphism of A1(K) is an automorphism. Here an α-endomorphism of A1(K) is an endomorphism which preserves the involution α. We also prove an analogue result for the Jacobian conjecture in dimension 2, called α −JC2.