Interpolated Free Group Factors

The interpolated free group factors L(Fr) for 1 < r ≤ ∞, (also defined by F. Rădulescu) are given another (but equivalent) definition as well as proofs of their properties with respect to compression by projections and free products. In order to prove the addition formula for free products, algebraic techniques are developed which allow us to show R∗R ∼= L(F2) where R is the hyperfinite II1–factor. Introduction. The free group factors L(Fn) for n = 2, 3, . . . ,∞ (introduced in [4]) have recently been extensively studied [11,2,5,6,7] using Voiculescu’s theory of freeness in noncommutative probability spaces (see [8,9,10,11,12,13], especially the latter for an overview). 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 [7], F. Rădulescu introduced II1–factors L(Fr) for 1 < r ≤ ∞, equaling the free group factor L(Fn) when r = n ∈ N\{0, 1} and satisfying L(Fr) ∗ L(Fr′) = L(Fr+r′), (1 < r, r ′ ≤ ∞) (1) and L(Fr)γ = L(F(1 + r − 1 γ2 )), (1 < r ≤ ∞, 0 < γ < ∞). (2) Where for a II1–factor M, Mγ means the algebra [4] defined as follows: for 0 < γ ≤ 1, Mγ = pMp, where p ∈ M is a self–adjoint projection of trace γ; for γ = n = 2, 3, . . . one has Mγ = M⊗Mn(C); for 0 < γ1, γ2 < ∞ one has