An Alternative to Free Entropy for Free Group Factors

In this paper we define a very simple invariant η for a k-tuple of unitaries in a finite factor von Neumann algebra, and we show how this invariant can replace free entropy in many of the important applications. We also introduce a notion of metric free entropy and some related concepts. We include proofs, using η, of the theorems of Liming Ge and of D. Voiculescu, respectively, on the primeness of and on the absence of Cartan subalgebras in the noncommutative free group factors. Received November 8, 2002, Accepted March 5, 2003 The second author is supported by the National Science Foundation of USA