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 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.

COUNTABLE BOREL EQUIVALENCE RELATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 1-80 ◽  
Author(s):  
S. JACKSON ◽  
A. S. KECHRIS ◽  
A. LOUVEAU
Keyword(s):  
Set Theory ◽  
Equivalence Relations ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Borel Equivalence Relations ◽  
Universal Equivalence ◽  
Descriptive Set

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.

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).

scholarly journals Set theory and von Neumann algebras

2012 ◽  
pp. 363-396
Author(s):  
Asger Törnquist ◽  
Martino Lupini
Keyword(s):  
Set Theory ◽  
Von Neumann Algebras ◽  
Von Neumann

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.

Borel reductions and cub games in generalised descriptive set theory

2013 ◽  
Vol 78 (2) ◽  
pp. 439-458 ◽  
Author(s):  
Vadim Kulikov
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Regular Cardinal ◽  
Equivalence Relations ◽  
Descriptive Set Theory ◽  
Borel Equivalence Relations ◽  
Power Set ◽  
Borel Reducibility ◽  
Descriptive Set

AbstractIt is shown that the power set of κ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on 2κ under Borel reducibility. Here κ is an uncountable regular cardinal with κ<κ = κ.

scholarly journals JUMP OPERATIONS FOR BOREL GRAPHS

2018 ◽  
Vol 83 (1) ◽  
pp. 13-28
Author(s):  
ADAM R. DAY ◽  
ANDREW S. MARKS
Keyword(s):  
Set Theory ◽  
Equivalence Relations ◽  
Connected Components ◽  
Descriptive Set Theory ◽  
Jump Operator ◽  
Borel Equivalence Relations ◽  
Locally Countable ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory

AbstractWe investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

scholarly journals Kadison–Singer algebras: Hyperfinite case

2010 ◽  
Vol 107 (5) ◽  
pp. 1838-1843 ◽  
Author(s):  
Liming Ge ◽  
Wei Yuan
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Triangular Matrix ◽  
Von Neumann ◽  
Matrix Algebras ◽  
New Class

A new class of operator algebras, Kadison–Singer algebras (KS-algebras), is introduced. These highly noncommutative, non-self-adjoint algebras generalize triangular matrix algebras. They are determined by certain minimally generating lattices of projections in the von Neumann algebras corresponding to the commutant of the diagonals of the KS-algebras. A new invariant for the lattices is introduced to classify these algebras.

Sign in / Sign up

Export Citation Format

Share Document