Interpolated free group factors

Abstract

The interpolated free group factors L(¥ r ) for 1 < r < oo (also defined by F. Radulescu) 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*ί L(¥ 2 ) where R is the hyperfinite Hi -factor.Introduction. The free group factors L(F n ) for n -2, 3, ... , oo (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 isomorphism question, first raised by R. V. Kadison of whether L(F n ) = L(F m ) for n Φ m. In [7], F. Radulescu introduced Hi -factors L(F r ) for 1 < r < oo, equalling the free group factor L(¥ n ) when r = n e N\{0, 1} and satisfying

Stable equivalence of the weak closures of free groups convolution algebras

“…(F (r _ 1)t -2 + 1 ) and j^(F r ) * j^(F p ) £ &(F r+p ), t, r, p e R, r, p > 1, ί > 0. This was independently discovered by K. Dykema (in [8]) and the author (in [10]). …”

On spectra and Brown's spectral measures of elements in free products of matrix algebras

“…In section 5, we find * -free generators h, u, v of L(F 3 ) (different from the free generators given in [4]) so that we may explicitly write out x 1 , x 2 , x 3 , x 4 in terms of h, u, v. In section 6, we compute miscellaneous examples of Brown measures of operators A+B and AB, where A ∈ (M 2 (C), 1 2 Tr) * 1 and B ∈ 1 * (M 2 (C), 1 2 Tr). As a corollary, we show that A + B is an R-diagonal operator if and only if A + B = 0.…”

“…We will identify M with E 11 ME 11 ⊗M 2 (C) (1) by the canonical isomorphism . In [4], K. Dykema proved that E 11 ME 11 ∼ = L(F 3 ). For B ∈ M 2 (C) (2) , we may write…”

“…Let X ∈ 1 * (M 2 (C), 1 2 Tr). With respect to the matrix units of (M 2 (C), 1 2 Tr) * 1, X = x 1 x 2 x 3 x 4 . By [4], (M 2 (C), 1 2 Tr) * (M 2 (C),…”

A relation between certain interpolated Cuntz algebras and interpolated free group factors

“…In this article we shall show that some of these von Neumann algebras F β are isomorphic to interpolated free group factors L(F r ) introduced by K. Dykema [4] and F. Rȃdulescu [14]. For example, in case of the golden number β =…”