The topological structure of the unitary and automorphism groups of a factor

It is proved that a large class of II1 factors have unitary group which is contractible in the strong operator topology, but whose fundamental group in the norm topology is isomorphic to the additive real numbers. The class includes the approximately finite dimensional factor of type II1 and the group factor associated with the free group on infinitely many generators. This contractibility is used to prove the contractibility of the automorphism group of the approximately finite dimensional factor of type II1 and type II∞. It is further shown that the fundamental group of the automorphism group of the approximately finite dimensional factor of type IIIλ, 0<λ<1, is isomorphic to the integer group Z. (orig.)