The starred Dixmier conjecture for $A_1$

Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over a characteristic zero field $K$ and let $α$ be the exchange involution on $A_1(K)$ given by $α(x)= y$ and $α(y)= x$. The Dixmier conjecture of Dixmier (1968) asks: Is every algebra endomorphism of the Weyl algebra $A_1(K)$ an automorphism? The aim of this paper is to prove that each $α$-endomorphism of $A_1(K)$ is an automorphism. Here an $α$-endomorphism of $A_1(K)$ is an endomorphism which preserves the involution $α$. We also prove an analogue result for the Jacobian conjecture in dimension 2, called $α-JC_2$.

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