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

Abstract

It is proved that a large class ofII 1 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 as proven by Araki-Smith-Smith [1]. The class includes the approximately finite dimensional factor of typeII 1 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 typeII 1 and typeII ∞. It is further shown that the fundamental group of the automorphism group of the approximately finite dimensional factor of typeIII λ, 0<λ<1, is isomorphic to the integer group ℤ.