An alternative to free entropy for free group factors

In this paper we define a very simple invariant eta((V) over right arrow) for a k-tuple (V) over right arrow 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 eta, 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.