scholarly journals REDUCED PRODUCTS OF METRIC STRUCTURES: A METRIC FEFERMAN–VAUGHT THEOREM

2016 ◽  
Vol 81 (3) ◽  
pp. 856-875 ◽  
Author(s):  
SAEED GHASEMI
Keyword(s):  
Continuum Hypothesis ◽  
C Algebras ◽  
Reduced Products ◽  
Metric Structures ◽  
The Continuum

AbstractWe extend the classical Feferman–Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum Hypothesis. We also prove the existence of two separable C*-algebras of the form ⊕iMk(i) (ℂ) such that the assertion that their coronas are isomorphic is independent from ZFC, which gives the first example of genuinely noncommutative coronas of separable C*-algebras with this property.

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.

scholarly journals Set Theory and C*-Algebras

2007 ◽  
Vol 13 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Nik Weaver
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
C Algebra ◽  
Calkin Algebra ◽  
C Algebras ◽  
Basic Object ◽  
The Continuum

AbstractWe survey the use of extra-set-theoretic hypotheses, mainly the continuum hypothesis, in the C*-algebra literature. The Calkin algebra emerges as a basic object of interest.

On the compactness of some Boolean algebras

1984 ◽  
Vol 49 (1) ◽  
pp. 63-67
Author(s):  
Jacek Cichoń
Keyword(s):  
Boolean Algebra ◽  
Continuum Hypothesis ◽  
Cardinal Number ◽  
Boolean Algebras ◽  
Cardinal Numbers ◽  
Martin’S Axiom ◽  
Reduced Products ◽  
Standard Set ◽  
General Continuum ◽  
The Continuum

We say that the Boolean algebra B is λ-compact, where λ is a cardinal number, if for every family Z ⊆ B∖{0} of power at most λ, if inf Z = 0 then for some finite subfamily Z0 ⊆ Z we have inf Z0 = 0.On the set of all finite subsets of a cardinal number κ, which is denoted [κ]<ω, the sets of the form for any p Є [κ]<ω generate the filter Tκ.This filter is a standard example of a κ-regular filter (see [2]). Because of the importance of κ-regular filters in studying the saturatedness of ultraproducts and reduced products by model-theoretic methods, the question of compactness of the algebra Bκ = P([κ]<ω/Tκ was natural. This question in the most optimistical way was formulated by M. Benda [1, Problem 5c]: is the algebra Bκω-compact for every uncountable κ?In this paper we show that for most of the cardinal numbers which are greater or equal to 2ω the algebra Bκ is not ω-compact. Hence, in view of obtained results, the following question appears: does there exist an uncountable κ such that the algebra Bκ is κ-compact?We use standard set-theoretical notations. CH denotes the Continuum Hypothesis, GCH denotes the General Continuum Hypothesis and MA denotes Martin's Axiom.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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