v 1 8 M ay 1 99 6 TWO APPLICATIONS OF FREE ENTROPY Kenneth

In [8] and [9], Voiculescu introduced free entropy for n–tuples of self–adjoint elements in a II1–factor, and used it to prove that free group factors, L(Fn), lack Cartan subalgebras [9]. In [4], S. Popa introduced a property for II1–factors, called property C. (See [5] for a paper related to [4].) Like Property Γ of Murray and von Neumann, this is an asymptotic commutivity property, but it is formally weaker than property Γ. Factors possessing Cartan subalgebras have property C. After Voiculescu’s striking result, it is a natural question whether the factors L(Fn) have property C. In this note, we show that they do not. Liming Ge [2] used Voiculescu’s free entropy to show that the free group factors L(Fn) for 2 ≤ n < ∞ lack simple (i.e. of multiplicity one) abelian subalgebras. We say that an abelian subalgebra, A, of a II1–factor M with trace τ has finite multiplicity m if there are ξ1, . . . , ξm ∈ L (M, τ) for which