We prove that every rigid C⁎-bicategory with finite-dimensional centers (finitely decomposable horizontal units) can be realized as Connes' bimodules over finite direct sums of II1 factors. In ...particular, we realize every multitensor C⁎-category as bimodules over a finite direct sum of II1 factors.
Let M be a type II1 von Neumann algebra, S(M) be the Murray–von Neumann algebra associated with M and let A be a ⁎-subalgebra in S(M) with M⊆A. We prove that any ring derivation D from A into S(M) is ...necessarily inner. Further, we prove that if A is an EW⁎-algebra such that its bounded part Ab is a W⁎-algebra without finite type I direct summands, then any ring derivation D from A into LS(Ab) — the algebra of all locally measurable operators affiliated with Ab, is an inner derivation. We also give an example showing that the condition M⊆A is essential. At the end of this paper, we provide several criteria for an abelian extended W⁎-algebra such that all ring derivations on it are linear.
We study bounded bilinear maps on a C⁎-algebra A having product property at c∈A. This leads us to the question of when a C⁎-algebra is determined by products at c. In the first part of our paper, we ...investigate this question for compact C⁎-algebras, and in the second part, we deal with von Neumann algebras having non-trivial atomic part. Our results are applicable to descriptions of homomorphism-like and derivation-like maps at a fixed point on such algebras.
The aim of this paper is to study the logarithmic submajorizations inequalities for operators in a finite von Neumann algebra. Firstly, some logarithmic submajorizations inequalities due to Garg and ...Aulja are extended to the case of operators in a finite von Neumann algebra. As an application, we get some new Fuglede-Kadison determinant inequalities of operators in that circumstance. Secondly, we improve and generalize to the setting of finite von Neumann algebras, a generalized Hölder type generalized singular numbers inequality.
The paper contains a probabilistic proof of the L^p-boundedness of the Hilbert transform in the context of a free group von Neumann algebra VN(\mathbb {F}_q). The argument rests on noncommutative ...version of good-\lambda inequalities and yields a tight L\log L order of the L^p norms as p\to 1^+ and p\to \infty.
We construct the first II1 factors having exactly two group measure space decompositions up to unitary conjugacy. Also, for every positive integer n, we construct a II1 factor M that has exactly n ...group measure space decompositions up to conjugacy by an automorphism.
On construit les premiers facteurs de type II1 ayant exactement deux décompositions groupe – espace mesuré à conjugaison unitaire près. Pour chaque entier positif n nous construisons également un facteur de type II1 qui a exactement n décompositions groupe – espace mesuré à conjugaison par un automorphisme près.
In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full unitary group, the unitary Schatten perturbations of the identity ...and unitary groups of finite von Neumann algebras. The Finsler structures are defined by translation of different norms on the tangent space at the identity. We first prove a convexity result for the metric derived from the operator norm on the full unitary group. We also prove strong convexity results for the squared metrics in Hilbert-Schmidt unitary groups and unitary groups of finite von Neumann algebras. In both cases the tangent spaces are endowed with an inner product defined with a trace. These results are applied to fixed point properties and to quantitative metric bounds in certain rigidity problems. Radius bounds for all convexity and fixed point results are shown to be optimal.
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