von Neumann regular modules

In this paper, we introduce von Neumann regular modules and give many characterizations of von Neumann regular modules. Further, we investigate the relations between von Neumann regular modules and other classical modules. Finally, we characterize Noetherian von Neumann regular modules.     

Applications of the complex interpolation method to a von Neumann algebra: Non-commutative Lp-spaces

Non-commutative L^p-spaces, 1 < p < ∞, associated with a von Neumann algebra are considered. The paper consists of two parts. In part I, by making use of the complex interpolation method, non-commutative L^p-spaces are defined as interpolation spaces between the von Neumann algebra in question and its predual. Also, all expected properties (such as duality and uniform convexity) are proved in the frame of interpolaton theory and relative modular theory. In part II, these L^p-spaces are compared with Haagerup's L^p-spaces. Based on this comparison, a non-commutative analogue of the classical Stein-Weiss interpolation theorem is obtained.     

On the geometry of von Neumann algebra preduals

Let \(M\) be a von Neumann algebra and let \(M_\star \) be its (unique) predual. We study when for every \(\varphi \in M_\star \) there exists \(\psi \in M_\star \) solving the equation \(\Vert \varphi \pm \psi \Vert =\Vert \varphi \Vert =\Vert \psi \Vert \) . This is the case when \(M\) does not contain type I nor type III \(_1\) factors as direct summands and it is false at least for the unique hyperfinite type III \(_1\) factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of \(M_\star \) of length \(4\) . An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.     

Non injectivity of the q-deformed von Neumann algebra

In this paper we prove that the von Neumann algebra generated by q-gaussians is not injective as soon as the dimension of the underlying Hilbert space is greater than 1. Our approach is based on a suitable vector valued Khintchine type inequality for Wick products. The same proof also works for the more general setting of a Yang-Baxter deformation. Our techniques can also be extended to the so called q-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter case, we obtain the non injectivity under some asssumption on the spectral set of the positive operator associated with the deformation.Mathematics Subject Classification (2000): 46L65, 46L54Revised version: 13 January 2004     

An inner amenable group whose von Neumann algebra does not have property Gamma

We construct inner amenable groups G with infinite conjugacy classes and such that the associated II₁ factor has no non-trivial asymptotically central elements, i.e. does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.     

A Note on the Mackey topology of a von Neumann algebra

Some of the properties of the upper bound of the spectrum of a quasilinear eigenvalue problem, subject to a positivity requirement, are derived. It is shown that, as a function of the surface heat-transfer coefficient, this parameter is a continuous, monotonic increasing function and is bounded above.     

A note on faithful traces on a von Neumann algebra

In this short note we give some techniques for constructing, starting from asufficient family ℱ of semifinite or finite traces on a von Neumann algebraM, a new trace which is faithful.     

Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

Let M be a type I von Neumann algebra with the center Z and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Z-linear derivation on LS(M) is inner. In particular, all Z-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements of LS(M). The text was submitted by the authors in English.     

Metric geometry of partial isometries in a finite von Neumann algebra

We study the geometry of the set

     

Trace and integrable operators affiliated with a semifinite von Neumann algebra

New properties of the space of integrable (with respect to the faithful normal semifinite trace) operators affiliated with a semifinite von Neumann algebra are found. A trace inequality for a pair of projections in the von Neumann algebra is obtained, which characterizes trace in the class of all positive normal functionals on this algebra. A new property of a measurable idempotent are determined. A useful factorization of such an operator is obtained; it is used to prove the nonnegativity of the trace of an integrable idempotent. It is shown that if the difference of two measurable idempotents is a positive operator, then this difference is a projection. It is proved that a semihyponormal measurable idempotent is a projection. It is also shown that a hyponormal measurable tripotent is the difference of two orthogonal projections.