scholarly journals Lattice isomorphisms between projection lattices of von Neumann algebras

2020 ◽  
Vol 8 ◽  
Author(s):  
Michiya Mori
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Type Ii ◽  
Measurable Operator ◽  
Von Neumann ◽  
Measurable Operators ◽  
Locally Measurable Operator

Abstract Generalizing von Neumann’s result on type II $_1$ von Neumann algebras, I characterise lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, I give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.

TRACE JENSEN INEQUALITY FOR SELF-ADJOINT OPERATORS IN SEMI-FINITE VON NEUMANN ALGEBRAS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350075
Author(s):  
HIDEKI KOSAKI
Keyword(s):  
Convex Function ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Jensen Inequality ◽  
Measurable Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Measurable Operators ◽  
Adjoint Operators ◽  
Finite Von Neumann Algebras

Let [Formula: see text] be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace τ, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality in our previous work states τ(f(a*xa)) ≤ τ(a*f(x)a) (as long as the both sides are well-defined) for a contraction [Formula: see text] and a semi-bounded τ-measurable operator x. Validity of this inequality for (not necessarily semi-bounded) self-adjoint τ-measurable operators is investigated.

James Cummings and Ernest Schimmerling, editors. Lecture Note Series of the London Mathematical Society, vol. 406. Cambridge University Press, New York, xi + 419 pp. - Paul B. Larson, Peter Lumsdaine, and Yimu Yin. An introduction to Pmax forcing. pp. 5–23. - Simon Thomas and Scott Schneider. Countable Borel equivalence relations. pp. 25–62. - Ilijas Farah and Eric Wofsey. Set theory and operator algebras. pp. 63–119. - Justin Moore and David Milovich. A tutorial on set mapping reflection. pp. 121–144. - Vladimir G. Pestov and Aleksandra Kwiatkowska. An introduction to hyperlinear and sofic groups. pp. 145–185. - Itay Neeman and Spencer Unger. Aronszajn trees and the SCH. pp. 187–206. - Todd Eisworth, Justin Tatch Moore, and David Milovich. Iterated forcing and the Continuum Hypothesis. pp. 207–244. - Moti Gitik and Spencer Unger. Short extender forcing. pp. 245–263. - Alexander S. Kechris and Robin D. Tucker-Drob. The complexity of classification problems in ergodic theory. pp. 265–299. - Menachem Magidor and Chris Lambie-Hanson. On the strengths and weaknesses of weak squares. pp. 301–330. - Boban Veličković and Giorgio Venturi. Proper forcing remastered. pp. 331–362. - Asger ToÖrnquist and Martino Lupini. Set theory and von Neumann algebras. pp. 363–396. - W. Hugh Woodin, Jacob Davis, and Daniel RodrÍguez. The HOD dichotomy. pp. 397–419.

2014 ◽  
Vol 20 (1) ◽  
pp. 94-97
Author(s):  
Natasha Dobrinen
Keyword(s):  
New York ◽  
Set Theory ◽  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
London Mathematical Society ◽  
Equivalence Relations ◽  
Classification Problems ◽  
Von Neumann ◽  
Borel Equivalence Relations

scholarly journals LOCAL DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2011 ◽  
Vol 13 (04) ◽  
pp. 643-657 ◽  
Author(s):  
S. ALBEVERIO ◽  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV ◽  
B. O. NURJANOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Continuity Condition ◽  
Type I ◽  
Atomic Lattice ◽  
Von Neumann ◽  
Measurable Operators ◽  
Finite Trace ◽  
Necessary And Sufficient

The paper is devoted to local derivations on the algebra [Formula: see text] of τ-measurable operators affiliated with a von Neumann algebra [Formula: see text] and a faithful normal semi-finite trace τ. We prove that every local derivation on [Formula: see text] which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra [Formula: see text] for the algebra [Formula: see text] to admit local derivations which are not derivations.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2010 ◽  
Vol 13 (02) ◽  
pp. 305-337 ◽  
Author(s):  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV
Keyword(s):  
Measurable Function ◽  
Von Neumann Algebras ◽  
Special Section ◽  
Type I ◽  
Von Neumann ◽  
Abelian Case ◽  
Measurable Operators

The present paper is a survey of recent results concerning derivations on various algebras of measurable operators affiliated with von Neumann algebras. A complete description of derivation is obtained in the case of type I von Neumann algebras. A special section is devoted to the Abelian case, namely to the existence of nontrivial derivations on algebras of measurable function. Local derivations on the above algebras are also considered.

scholarly journals NORMALIZERS OF OPERATOR ALGEBRAS AND REFLEXIVITY

2003 ◽  
Vol 86 (2) ◽  
pp. 463-484 ◽  
Author(s):  
A. KATAVOLOS ◽  
I. G. TODOROV
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Spectral Synthesis ◽  
Sufficient Conditions ◽  
Linear Structure ◽  
Linear Spaces ◽  
Von Neumann ◽  
Primary 47L35 ◽  
Borel Functions ◽  
Reflexive Algebras

The set of normalizers between von Neumann (or, more generally, reflexive) algebras $\mathcal{A}$ and $\mathcal{B}$ (that is, the set of all operators $T$ such that $T \mathcal{A} T^{\ast} \subseteq \mathcal{B}$ and $T^{\ast} \mathcal{B} T \subseteq \mathcal{A}$) possesses ‘local linear structure’: it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces $\mathcal{U}$ of operators satisfying $\mathcal{UU}^{\ast} \mathcal{U} \subseteq \mathcal{U}$ (also known as ternary rings of operators). Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form $\mathcal{U} = \{T : TL = \phi (L) T$ for all $L \in \mathcal{L}\}$ where $\mathcal{L}$ is a set of projections and $\phi$ a certain map defined on $\mathcal{L}$. A normalizing space consists of normalizers between appropriate von Neumann algebras $\mathcal{A}$ and $\mathcal{B}$. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian self-adjoint algebras consist of operators ‘supported’ on sets of the form $[f = g]$ where $f$ and $g$ are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.2000 Mathematical Subject Classification: 47L05 (primary), 47L35, 46L10 (secondary).

Ergodic Actions of Compact Groups on Operator Algebras II: Classification of Full Multiplicity Ergodic Actions

1988 ◽  
Vol 40 (6) ◽  
pp. 1482-1527 ◽  
Author(s):  
Antony Wassermann
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Crossed Product ◽  
Ergodic Action ◽  
Compact Groups ◽  
Type I ◽  
List Type ◽  
Von Neumann ◽  
C Algebra ◽  
Set Up

In the first paper of this series [17], we set up some general machinery for studying ergodic actions of compact groups on von Neumann algebras, namely, those actions for which . In particular we obtained a characterisation of the full multiplicity ergodic actions:THEOREM A. If α is an ergodic action of G on , then the following conditions are equivalent:(1) Each spectral subspace has multiplicity dim π for π in .(2) Each π in admits a unitary eigenmatrix in .(3) The W* crossed product is a (Type I) factor.(4) The C* crossed product of the C* algebra of norm continuity is isomorphic to the algebra of compact operators on a Hilbert space.

scholarly journals STRONG SINGULARITY FOR SUBALGEBRAS OF FINITE FACTORS

2003 ◽  
Vol 14 (03) ◽  
pp. 235-258 ◽  
Author(s):  
GUYAN ROBERTSON ◽  
ALLAN M. SINCLAIR ◽  
ROGER R. SMITH
Keyword(s):  
Von Neumann Algebras ◽  
Discrete Groups ◽  
Type Ii ◽  
Von Neumann ◽  
Strong Singularity ◽  
Geometric Objects ◽  
Underlying Geometry

In this paper we develop the theory of strongly singular subalgebras of von Neumann algebras, begun in earlier work. We mainly examine the situation of type II1 factors arising from countable discrete groups. We give simple criteria for strong singularity, and use them to construct strongly singular subalgebras. We particularly focus on groups which act on geometric objects, where the underlying geometry leads to strong singularity.

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