scholarly journals The Inner Corona Algebra of a C0(X)-Algebra

2016 ◽  
Vol 60 (2) ◽  
pp. 299-318
Author(s):  
Robert J. Archbold ◽  
Douglas W. B. Somerset
Keyword(s):  
Continuum Hypothesis ◽  
General Setting ◽  
Continuous Functions ◽  
Ideal Space ◽  
Multiplier Algebra ◽  
Infinite Dimensional ◽  
Essential Ideal ◽  
Power Set ◽  
Isolated Points ◽  
Proper Subalgebra

AbstractLet A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let As be the algebra of strong*-continuous functions from X to K(H). Then As/A is the inner corona algebra of A. We show that if X has no isolated points, then As/A is an essential ideal of the corona algebra of A, and Prim(As/A), the primitive ideal space of As/A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of As/A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of As/A. Several of the results are obtained in the more general setting of C0(X)-algebras.

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 .

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

The Maximal Ideal Space of Subalgebras of the Disk Algebra

1975 ◽  
Vol 18 (1) ◽  
pp. 61-65 ◽  
Author(s):  
Bruce Lund
Keyword(s):  
Unit Disk ◽  
Maximal Ideal ◽  
Function Algebra ◽  
Continuous Functions ◽  
Maximal Ideal Space ◽  
Open Unit ◽  
Ideal Space ◽  
Disk Algebra ◽  
Closed Subalgebra ◽  
Uniformly Closed

Let X be a compact Hausdorff space and C(X) the complexvalued continuous functions on X. We say A is a function algebra on X if A is a point separating, uniformly closed subalgebra of C(X) containing the constant functions. Equipped with the sup-norm ‖f‖ = sup{|f(x)|: x ∊ X} for f ∊ A, A is a Banach algebra. Let MA denote the maximal ideal space.Let D be the closed unit disk in C and let U be the open unit disk. We call A(D)={f ∊ C(D):f is analytic on U} the disk algebra. Let T be the unit circle and set C1(T) = {f ∊ C(T): f'(t) ∊ C(T)}.

scholarly journals The Centre of the Spaces of Banach Lattice-Valued Continuous Functions on the Generalized Alexandroff Duplicate

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Faruk Polat
Keyword(s):  
Banach Lattice ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Isolated Points ◽  
Discrete Functions

We characterize the centre of the Banach lattice of Banach lattice -valued continuous functions on the Alexandroff duplicate of a compact Hausdorff space in terms of the centre of , the space of -valued continuous functions on . We also identify the centre of whose elements are the sums of -valued continuous and discrete functions defined on a compact Hausdorff space without isolated points, which was given by Alpay and Ercan (2000).

Existence of unique solution to nonlinear mixed Volterra Fredholm-Hammerstein integral equations in complex-valued fuzzy metric spaces

2020 ◽  
pp. 1-10
Author(s):  
Humaira ◽  
Muhammad Sarwar ◽  
Thabet Abdeljawad
Keyword(s):  
Unique Solution ◽  
Metric Spaces ◽  
General Setting ◽  
Continuous Functions ◽  
Contractive Condition ◽  
Nonlinear Functions ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Hammerstein Integral Equations ◽  
Complex Valued

The purpose of this article is to investigate the existence of unique solution for the following mixed nonlinear Volterra Fredholm-Hammerstein integral equation considered in complex plane; (0.1) ξ ( τ ) = g ( t ) + ρ ∫ 0 τ K 1 ( τ , ℘ ) ϝ 1 ( ℘ , ξ ( ℘ ) ) d ℘ + ϱ ∫ 0 1 K 2 ( τ , ℘ ) ϝ 2 ( ℘ , ξ ( ℘ ) ) d ℘ , such that ξ = ξ 1 + ξ 2 , ξ 1 , ξ 2 ∈ ( C ( [ 0 , 1 ] ) , R ) g = g 1 + g 2 , g l : [ 0 , 1 ] → R , l = 1 , 2 , ϝ l ( ℘ , ξ ( ℘ ) ) = ϝ l 1 * ( ℘ , ξ 1 * ) + i ϝ l 2 * ( ℘ , ξ 2 * ) , ϝ lj * : [ 0 , 1 ] × R → R for l , j = 1 , 2 , and ξ 1 * , ξ 2 * ∈ ( C ( [ 0 , 1 ] ) , R ) K l ( t , ℘ ) = K l 1 * ( t , ℘ ) + iK l 2 * ( t , ℘ ) , for l , j = 1 , 2 and K lj * : [ 0 , 1 ] 2 → R , where ρ and ϱ are constants, g (t), the kernels K l  (τ, ℘) and the nonlinear functions ϝ1 (℘ , ξ (℘)) , ϝ 2 (℘ , ξ (℘)) are continuous functions on the interval 0 ≤ τ ≤ 1. In this direction we apply fixed point results for self mappings with the concept of (ψ, ϕ) contractive condition in the setting of complex-valued fuzzy metric spaces. This study will be useful in the development of the theory of fuzzy fractional differential equations in a more general setting.

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 On weighted spaces without a fundamental sequence of bounded sets

2002 ◽  
Vol 30 (8) ◽  
pp. 449-457 ◽  
Author(s):  
J. O. Olaleru
Keyword(s):  
Weighted Space ◽  
General Setting ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Fundamental Sequence ◽  
Spaces Of Continuous Functions ◽  
Bounded Sets

The problem of countably quasi-barrelledness of weighted spaces of continuous functions, of which there are no results in the general setting of weighted spaces, is tackled in this paper. This leads to the study of quasi-barrelledness of weighted spaces in which, unlike that of Ernst and Schnettler (1986), though with a similar approach, we drop the assumption that the weighted space has a fundamental sequence of bounded sets. The study of countably quasi-barrelledness of weighted spaces naturally leads to definite results on the weighted (DF)-spaces for those weighted spaces with a fundamental sequence of bounded sets.

scholarly journals The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers

2012 ◽  
Vol 154 (1) ◽  
pp. 119-126 ◽  
Author(s):  
SIEGFRIED ECHTERHOFF ◽  
MARCELO LACA
Keyword(s):  
Recent Result ◽  
Orbit Space ◽  
Crossed Product ◽  
Primitive Ideal ◽  
Prime Ideals ◽  
Ideal Space ◽  
C Algebra ◽  
Ring Of Integers ◽  
Affine Semigroup ◽  
Power Set

AbstractThe purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

scholarly journals R*-I-Lc-Continuous Functions in an Ideal Space

2018 ◽  
Vol 7 (4.10) ◽  
pp. 816
Author(s):  
A. Josephine christilda ◽  
R. Manoharan
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Ideal Space ◽  
Continuous Map ◽  
Continuous Maps

  Some new sets like of “R*-LC-sets”, “R*- I-LC-sets” are introduced using R*-topology. Also defined the notions of “R*- I-LC-continuous maps”, “I*R- continuous maps”, “g­-R*- continuous map”s and represent “*-continuous function” as decomposition two newly defined functions.   

Sign in / Sign up

Export Citation Format

Share Document