scholarly journals The Pedersen Rigidity Problem

2019 ◽  
Vol 53 (supl) ◽  
pp. 237-244
Author(s):  
S. Kaliszewski ◽  
Tron Omland ◽  
John Quigg
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Morita Equivalence ◽  
Crossed Product ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
C Algebra ◽  
Dual Action ◽  
Continuous Trace

If is an action of a locally compact abelian group G on a C*-algebra A, Takesaki-Takai duality recovers (A, α) up to Morita equivalence from the dual action of Ĝ on the crossed product A × α G. Given a bit more information, Landstad duality recovers (A, α) up to isomorphism. In between these, by modifying a theorem of Pedersen, (A, α) is recovered up to outer conjugacy from the dual action and the position of A in M(A ×α G). Our search (still unsuccessful, somehow irritating) for examples showing the necessity of this latter condition has led us to formulate the "Pedersen Rigidity problem". We present numerous situations where the condition is redundant, including G discrete or A stable or commutative. The most interesting of these "no-go theorems" is for locally unitary actions on continuous-trace algebras.

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 bridge theorem for the entropy of semigroup actions

2020 ◽  
Vol 8 (1) ◽  
pp. 46-57
Author(s):  
Anna Giordano Bruno
Keyword(s):  
Abelian Group ◽  
Topological Entropy ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Finitely Generated ◽  
Semigroup Action ◽  
Locally Compact ◽  
Semigroup Actions ◽  
Dual Action ◽  
Totally Disconnected

AbstractThe topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

scholarly journals A singular convolution kernel without pseudo-periods

1981 ◽  
Vol 83 ◽  
pp. 1-4
Author(s):  
Jesper Laub
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Convolution Kernel ◽  
Locally Compact ◽  
Excessive Measures

Let G be a locally compact abelian group and N a non-zero convolution kernel on G satisfying the domination principle. We define the cone of N-excessive measures E(N) to be the set of positive measures ξ for which N satisfies the relative domination principle with respect to ξ. For ξ ∈ E(N) and Ω ⊆ G open the reduced measure of ξ over Ω is defined as.

scholarly journals Pure point/continuous decomposition of translation-bounded measures and diffraction

2018 ◽  
Vol 40 (2) ◽  
pp. 309-352
Author(s):  
JEAN-BAPTISTE AUJOGUE
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Diffraction Theory ◽  
Natural Generalization ◽  
Pure Point ◽  
Locally Compact Abelian Group ◽  
Uniqueness Property ◽  
Locally Compact ◽  
Continuous Decomposition

In this work we consider translation-bounded measures over a locally compact Abelian group$\mathbb{G}$, with a particular interest in their so-called diffraction. Given such a measure$\unicode[STIX]{x1D714}$, its diffraction$\widehat{\unicode[STIX]{x1D6FE}}$is another measure on the Pontryagin dual$\widehat{\mathbb{G}}$, whose decomposition into the sum$\widehat{\unicode[STIX]{x1D6FE}}=\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}+\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$of its atomic and continuous parts is central in diffraction theory. The problem we address here is whether the above decomposition of$\widehat{\unicode[STIX]{x1D6FE}}$lifts to$\unicode[STIX]{x1D714}$itself, that is to say, whether there exists a decomposition$\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}+\unicode[STIX]{x1D714}_{\text{c}}$, where$\unicode[STIX]{x1D714}_{\text{p}}$and$\unicode[STIX]{x1D714}_{\text{c}}$are translation-bounded measures having diffraction$\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}$and$\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$, respectively. Our main result here is the almost sure existence, in a sense to be made precise, of such a decomposition. It will also be proved that a certain uniqueness property holds for the above decomposition. Next, we will be interested in the situation where translation-bounded measures are weighted Meyer sets. In this context, it will be shown that the decomposition, whether it exists, also consists of weighted Meyer sets. We complete this work by discussing a natural generalization of the considered problem.

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