  We shall give some criteria for the existence of the Peter-Weyl type
decomposition for modified quantum algebra, which is motivated by the fact that
modified quantum algebra has commutative two crystal structures. We shall apply
the criteria to the crystal base of the level 0 part of the modified quantum
affine algebra $\widetilde U_q(\widehat{sl_2})_0$ and give more precise
description of Peter-Weyl type decomposition.
