  In this article, we introduce a completion
$\widehat{U}^+_v(\mathcal{L}\mathfrak{g})$ of the positive half of the quantum
affinization $U^+_v(\mathcal{L}\mathfrak{g})$ of a symmetrizable Kac-Moody
algebra $\mathfrak{g}$. On $\widehat{U}^+_v(\mathcal{L}(\mathfrak{g}))$, we
define a new "bar-involution" and construct the analogue Kashiwara's operators.
We conjecture that the resulting pair
$(\widehat{\mathcal{L}},\widehat{\mathcal{B}})$ is a crystal basis which
provides the existence of the "canonical basis" on the (completion of the) of
the positive half of the quamtum affinization.
