  Let $f: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be a $\mathbb{C}$-algebra
endomorphism having an invertible Jacobian. We show that for such $f$, if, in
addition, the group of invertible elements of $\mathbb{C}[f(x),f(y),x][1/v]
\subset \mathbb{C}(x,y)$ is contained in $\mathbb{C}(f(x),f(y))-0$, then $f$ is
an automorphism. Here $v \in \mathbb{C}[f(x),f(y)]-0$ is such that $y = u/v$,
with $u \in \mathbb{C}[f(x),f(y),x]-0$. Keller's theorem (in dimension two)
follows immediately, since Keller's condition
$\mathbb{C}(f(x),f(y))=\mathbb{C}(x,y)$ implies that the group of invertible
elements of $\mathbb{C}[f(x),f(y),x][1/v]$ is contained in $\mathbb{C}(x,y)-0 =
\mathbb{C}(f(x),f(y))-0$.
