An Etymological Dictionary of the German Language/Mütze
Mütze, f., from the equiv. late MidHG. mütze, mutze, f., ‘cap, bonnet,’ which is a shortened form of the equiv. armuȥ, almuȥ. It is derived from MidLat. almutia, armutia, almutium, prop. ‘amictus quo Canonici caput humerosque tegebant,’ worn also at a later period by laymen; the development of meaning is similar to that of ModHG. Kappe. MidLat. almutia, the origin of which is entirely obscure (al is thought to be the Arab. article), appears in Rom. Comp. Fr. aumusse or aumuce, ‘amess’ (fur cap worn by officials in Roman Catholic churches).