  Let $\{\Gamma_t, \, t\ge 0\}$ be the Gamma subordinator. Using a moment
identification due to Bertoin-Yor (2002), we observe that for every $t > 0$ and
$\alpha\in (0,1)$ the random variable $\Gamma_t^{-\alpha}$ is distributed as
the exponential functional of some spectrally negative L\'evy process. This
entails that all size-biased samplings of Fr\'echet distributions are
self-decomposable and that the extreme value distribution $F_\xi$ is infinitely
divisible if and only if $\xi\not\in (0,1),$ solving problems raised by Steutel
(1973) and Bondesson (1992). We also review different analytical and
probabilistic interpretations of the infinite divisibility of
$\Gamma_t^{-\alpha}$ for $t,\alpha > 0.$
