THE TOPOLOGICAL-STRUCTURE OF THE UNITARY AND AUTOMORPHISM-GROUPS OF AFACTOR

It is proved that a large class of II1 factors have unitary group whic h is contractible in the strong operator topology, but whose fundament al group in the norm topology is isomorphic to the additive real numbe rs as proven by Araki-Smith-Smith [1]. The class includes the approxim ately finite dimensional factor of type II1 and the group factor assoc iated with the free group on infinitely many generators. This contract ibility is used to prove the contractibility of the automorphism group of the approximately finite dimensional factor of type II1 and type I I(infinity). It is further shown that the fundamental group of the aut omorphism group of the approximately finite dimensional factor of type III(lambda), 0 < lambda < 1, is isomorphic to the integer group Z.