# The starred Dixmier's conjecture

Research paper by **Vered Moskowicz**

Indexed on: **18 Feb '14**Published on: **18 Feb '14**Published in: **Mathematics - Rings and Algebras**

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#### 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?