Non-Isomorphic Non-Hyperfinite Factors

Abstract

A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II 1 ( 4 , Theorem 16’), and of type III ( 8 , Theorem 1); all hyperfinite factors of type Hi are isomorphic ( 4 , Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II 1 ( 10 ), and of type III ( 11 ). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II 1 (Theorem 2), and of type III (Theorem 3). Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.