A direct proof of a theorem of Jech and Shelah on PCF algebras

2018 ◽  
Vol 52 (2) ◽  
pp. 131-137
Author(s):  
Juan Carlos Martínez
Keyword(s):  
Direct Proof ◽  
Locally Compact ◽  
Limit Ordinal ◽  
Scattered Spaces ◽  
The Ideal

By using an argument based on the structure of the locally compact scattered spaces, we prove in a direct way the following result shown by Jech and Shelah: there is a family {Bα : α < ω1} of subsets of ω1 such that the following conditions are satisfied: (a) max Bα - α, (b) if α ∈ Bβ then Bα ⊆ Bβ, (c) if δ ≤ α and δ is a limit ordinal then Bα ∩ δ is not in the ideal generated by the sets Bβ, β < α, and by the bounded subsets of δ, (d) there is a partition {An : n ∈ ω} of ω1 such that for every α and every n, Bα ∩An is finite.

TOPOLOGICAL QUIVERS

2005 ◽  
Vol 16 (07) ◽  
pp. 693-755 ◽  
Author(s):  
PAUL S. MUHLY ◽  
MARK TOMFORDE
Keyword(s):  
Uniqueness Theorem ◽  
Directed Graphs ◽  
Algebra Structure ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Gauge Invariant ◽  
C Algebras ◽  
Natural Analogues ◽  
General Perspective ◽  
The Ideal

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver [Formula: see text] is a C*-correspondence, and from this correspondence one may construct a Cuntz–Pimsner algebra [Formula: see text]. In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of [Formula: see text] can be determined from [Formula: see text]. We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the gauge-invariant uniqueness theorem, the Cuntz–Krieger uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.

scholarly journals Completely simple endomorphism rings of modules

2018 ◽  
Vol 19 (2) ◽  
pp. 223
Author(s):  
Victor Bovdi ◽  
Mohamed Salim ◽  
Mihail Ursul
Keyword(s):  
Commutative Rings ◽  
Compact Ring ◽  
Endomorphism Rings ◽  
Locally Compact ◽  
Ring Topology ◽  
Simple Ring ◽  
Finite Topology ◽  
Completely Simple ◽  
The Ideal

<p>It is proved that if A<sub>p</sub> is a countable elementary abelian p-group, then: (i) The ring End (A<sub>p</sub>) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (A<sub>p</sub>)/I, where I is the ideal of End (A<sub>p</sub>) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (A<sub>p</sub>) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained.</p>

On the Ideal-Triangularizability of Semigroups of Quasinilpotent Positive Operators on C(K)

1998 ◽  
Vol 41 (3) ◽  
pp. 298-305
Author(s):  
M. T. Jahandideh
Keyword(s):  
Banach Lattice ◽  
Compact Hausdorff Space ◽  
Borel Measure ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Integral Operators ◽  
Positive Operators ◽  
Locally Compact ◽  
Regular Borel Measure ◽  
The Ideal

AbstractIt is known that a semigroup of quasinilpotent integral operators, with positive lower semicontinuous kernels, on L2(X, μ), where X is a locally compact Hausdorff-Lindelöf space and μ is a σ-finite regular Borel measure on X, is triangularizable. In this article we use the Banach lattice version of triangularizability to establish the ideal-triangularizability of a semigroup of positive quasinilpotent integral operators on C(K) where K is a compact Hausdorff space.

scholarly journals The Connes spectrum for actions of Abelian groups on continuous-trace algebras

1986 ◽  
Vol 6 (4) ◽  
pp. 541-560 ◽  
Author(s):  
Steven Hurder ◽  
Dorte Olesen ◽  
Iain Raeburn ◽  
Jonathan Rosenberg
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Crossed Product ◽  
Topological Dynamics ◽  
Locally Compact Abelian Group ◽  
Ideal Structure ◽  
Locally Compact ◽  
The Common ◽  
Continuous Trace ◽  
The Ideal

AbstractWe study the various notions of spectrum for an action α of a locally compact abelian groupGon a typeIC*-algebraA, and discuss how these are related to the structure of the crossed productA⋊αG. In the case whereAhas continuous trace and the action ofGon  is minimal, we completely describe the ideal structure of the crossed product. A key role is played by the restriction of α to a certain ‘symmetrizer subgroup’Sof the common stabilizer inGof the points of Â. We show by example that, contrary to a conjecture of Bratteli, it is possble forA⋊Gto be primitive but not simple, provided thatSis not discrete. In such cases, the Connes spectrum Γ(α) differs from the strong Connes spectrumof Kishimoto. The counterexamples come from subtle phenomena in topological dynamics.

scholarly journals A generalization of the m-topology on C(X) finer than the m-topology

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2509-2515
Author(s):  
F. Azarpanah ◽  
F. Manshoor ◽  
R. Mohamadian
Keyword(s):  
Topological Properties ◽  
Compact Sets ◽  
Empty Interior ◽  
Topological Ring ◽  
Locally Compact ◽  
Regular Elements ◽  
Zero Function ◽  
The Ideal

It is well known that the component of the zero function in C(X) with the m-topology is the ideal C?(X). Given any ideal I ? C?(X), we are going to define a topology on C(X) namely the mI-topology, finer than the m-topology in which the component of 0 is exactly the ideal I and C(X) with this topology becomes a topological ring. We show that compact sets in C(X) with the mI-topology have empty interior if and only if X n T Z[I] is infinite. We also show that nonzero ideals are never compact, the ideal I may be locally compact in C(X) with the mI-topology and every Lindel?f ideal in this space is contained in C?(X). Finally, we give some relations between topological properties of the spaces X and Cm(X). For instance, we show that the set of units is dense in Cm(X) if and only if X is strongly zero-dimensional and we characterize the space X for which the set r(X) of regular elements of C(X) is dense in Cm(X).

Integral Formula for Spectral Flow for -Summable Operators

2019 ◽  
Vol 71 (2) ◽  
pp. 337-379
Author(s):  
Magdalena Cecilia Georgescu
Keyword(s):  
Fredholm Operator ◽  
Unbounded Operator ◽  
Von Neumann Algebra ◽  
Integral Formula ◽  
Direct Proof ◽  
Spectral Flow ◽  
Fredholm Operators ◽  
Von Neumann ◽  
Bounded Perturbations ◽  
The Ideal

AbstractFix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

The ordertype of β-R.E. sets

1990 ◽  
Vol 55 (2) ◽  
pp. 573-576
Author(s):  
Klaus Sutner
Keyword(s):  
Finite Sets ◽  
Limit Ordinal ◽  
The Ideal

AbstractLet β be an arbitrary limit ordinal. A β-r.e. set is l-finite iff all its β-r.e. subsets are β-recursive. The l-finite sets correspond to the ideal of finite sets in the lattice of r.e. sets. We give a characterization of l-finite sets in terms of their ordertype: a β-r.e. set is l-finite iff it has ordertype less than β*, the Σ1, projectum of β).

Possible PCF algebras

1996 ◽  
Vol 61 (1) ◽  
pp. 313-317 ◽  
Author(s):  
Thomas Jech ◽  
Saharon Shelah
Keyword(s):  
Limit Ordinal ◽  
The Ideal

AbstractThere exists a family of sets of countable ordinals such that:(1) max Bα = α,(2) if α ∈ Bβ then Bα ⊆ Bβ,(3) if λ ≤ α and λ is a limit ordinal then Bα ∩ λ is not in the ideal generated by the Bβ, β < α, and by the bounded subsets of λ,(4) there is a partition of ω1 such that for every α and every n, Bα ∩ An is finite.

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