  We define the extremal projector of the $q$-boson Kashiwara algebra
$B_q(\ge)$ and study their basic properties. Applying their proerties to the
representation theory of the category ${\cal O}(B_q(\ge))$, whose objects are
''upper bounded '' $B_q(\ge)$-modules, we obtain its semi-simplicity and the
classification of simple modules.
