Free Products of Hyperfinite Von Neumann Algebras and Free Dimension

Introduction. Voiculescu’s theory of freeness in noncommutative probability spaces (see [10,11,12,13,14,15], especially the latter for an overview) has made possible the recent surge of results about and related to the free group factors L(Fn) [13,3,7,8,9,4]. One hopes to eventually be able to solve the old isomorphism question, first raised by R.V. Kadison in the 1960’s, of whether L(Fn) ∼= L(Fm) for n 6= m. In Voiculescu’s theory, (see also [1]), one takes free products of finite von Neumann algebras, denoted A∗B, and one has L(G) ∗ L(H) ∼= L(G ∗H), (where L(G) for G a discrete group is the group von Neumann algebra, which is generated by the left regular representation of G on l(G)). It is of intrinsic interest to decide when A∗B is a factor, and to determine its isomorphism class. It may be that such results or the techniques used will give insight into the isomorphism problem. Moreover, such free products are related to amalgamated free products, which have arisen in connection with results about irreducible subfactors [6,9,1]. In [4] and [9], the interpolated free group factors L(Fr) (1 < r ≤ ∞) were found, that have equality with the free group factor on n generators if r = n ∈ N\{0, 1} and that satisfy