Affine density, von Neumann dimension and a problem of Perelomov

Affine Connections, Duality and Divergences for a Von Neumann Algebra

On the predual of a von Neumann algebra, we define a dif-ferentiable manifold structure and affine connections by embeddings into non-commutative L p –spaces. Using the geometry of uniformly convex Ba-nach spaces and duality of the L p and L q spaces for 1/p + 1/q = 1, we show that we can introduce the α-divergence, for α ∈ (−1, 1), in a similar manner as Amari in the classical case. If restric...

Semisimplicity and Global Dimension of a Finite von Neumann Algebra

We prove that a finite von Neumann algebra A is semisimple if the algebra of affiliated operators U of A is semisimple. When A is not semisimple, we give the upper and lower bounds for the global dimensions of A and U . This last result requires the use of the Continuum Hypothesis.

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(...

Mean Dimension, Mean Rank, and Von Neumann-lück Rank

We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Lück rank of M all coincide. As applications, we establish an addition formula for mean dimension o...

On the Von Neumann Economic Growth Problem