Which notation is better(or more standard) for the idele of a number field $K$?
$\mathbb{I}_K$ or $\mathbb{A}^\times_K$?
$\mathbb{I}_K$ is used in Milne's class field theory and has the advantage that it is a "thick" fractional ideal group $I_K$.
$\mathbb{A}^\times_K$ has the advantage that it is the group of units of adele ring.