Spectra of independent power measures

1972 ◽  
Vol 72 (1) ◽  
pp. 27-35 ◽  
Author(s):  
W. J. Bailey ◽  
G. Brown ◽  
W. Moran
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Elementary Proof ◽  
Natural Generalization ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Power Measure ◽  
Lca Groups

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.

scholarly journals When are dense subgroups of LCA groups closed under intersections?

1994 ◽  
Vol 49 (1) ◽  
pp. 59-67
Author(s):  
M.A. Khan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Torsion Subgroup ◽  
Dense Subgroup ◽  
Locally Compact ◽  
Dense Subgroups ◽  
Lca Groups

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:(1) The intersection of any two dense subgroups of G is dense in G.(2) The intersection of all dense subgroups of G is dense in G.(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

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.

On the convergence of iterates of convolution operators in Banach spaces

2020 ◽  
Vol 126 (2) ◽  
pp. 339-366
Author(s):  
Heybetkulu Mustafayev
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Continuous Representation ◽  
Convolution Operators ◽  
Locally Compact ◽  
Bounded Linear ◽  
Almost Everywhere ◽  
Power Bounded

Let $G$ be a locally compact abelian group and let $M(G)$ be the measure algebra of $G$. A measure $\mu \in M(G)$ is said to be power bounded if $\sup _{n\geq 0}\lVert \mu ^{n} \rVert _{1}<\infty $. Let $\mathbf {T} = \{ T_{g}:g\in G\}$ be a bounded and continuous representation of $G$ on a Banach space $X$. For any $\mu \in M(G)$, there is a bounded linear operator on $X$ associated with µ, denoted by $\mathbf {T}_{\mu }$, which integrates $T_{g}$ with respect to µ. In this paper, we study norm and almost everywhere behavior of the sequences $\{ \mathbf {T}_{\mu }^{n}x\}$ $(x\in X)$ in the case when µ is power bounded. Some related problems are also discussed.

scholarly journals Isometric representation of M(G) on B(H)

1982 ◽  
Vol 23 (2) ◽  
pp. 119-122 ◽  
Author(s):  
F. Ghahramani
Keyword(s):  
Hilbert Space ◽  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Isometric Isomorphism ◽  
Isometric Representation ◽  
University Of Edinburgh ◽  
The University ◽  
Algebra Of Operators

In a recent paper, E. Størmer, among other things, proves the existence of an isometric isomorphism from the measure algebra M(G) of a locally compact abelian group G into BB(L2(G)), ([6], Proposition 4.6). Here we give another proof for this result which works for non-commutative G as well as commutative G. We also prove that the algebra L1(G, λ), with λ the left (or right) Haar measure, is not isometrically isomorphic with an algebra of operators on a Hilbert space. The proofs of these two results are taken from the author's Ph.D. thesis [4], submitted to the University of Edinburgh before Størmer's paper. The author wishes to thank Dr. A. M. Sinclair for his help and encouragement.

scholarly journals Spectral mapping theorem for representations of measure algebras

1997 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
H. Seferoǧlu
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Homomorphism ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Stieltjes Transform ◽  
Locally Compact ◽  
Unital Banach Algebra

Let G be a locally compact abelian group, M0(G) be a closed regular subalgebra of the convolution measure algebra M(G) which contains the group algebra L1(G) and ω: M0(G) → B be a continuous homomorphism of M0(G) into the unital Banach algebra B (possibly noncommutative) such that ω(L1(G)) is without order with respect to B in the sense that if for all b ∈ B, b.ω(L1(G)) = {0} implies b = 0. We prove that if sp(ω) is a synthesis set for L1(G) then the equality holds for each μ ∈ M0(G), where sp(ω) denotes the Arveson spectrum of ω, σB(.) the usual spectrum in B, the Fourier-Stieltjes transform of μ.

scholarly journals Note on the semi-simplicity of measure algebras

2019 ◽  
Vol 192 (4) ◽  
pp. 935-938
Author(s):  
László Székelyhidi
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Discrete Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Group Case

AbstractIn this paper we prove that the measure algebra of a locally compact abelian group is semi-simple. This result extends the corresponding result of S. A. Amitsur in the discrete group case using a completely different approach.

scholarly journals Jackson's Theorem for locally compact abelian groups

1974 ◽  
Vol 10 (1) ◽  
pp. 59-66 ◽  
Author(s):  
Walter R. Bloom
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Integrable Function ◽  
Locally Compact Abelian Group ◽  
Classical Theorem ◽  
Extended Version ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Approximate Unit ◽  
Compact Abelian Groups

If f is a p–th integrable function on the circle group and ω(p; f; δ) is its mean modulus of continuity with exponent p then an extended version of the classical theorem of Jackson states the for each positive integer n, there exists a trigonometric polynomial tn of degree at most n for which‖f-tn‖p ≤(p; f; 1/n).In this paper it will be shewn that for G a Hausdorff locally compact abelian group, the algebra L1(G) admits a certain bounded positive approximate unit which, in turn, will be used to prove an analogue of the above result for Lp(G).

Large families of singular measures having absolutely continuous convolution squares

1968 ◽  
Vol 64 (4) ◽  
pp. 1015-1022 ◽  
Author(s):  
Karl Stromberg
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Regular Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Singular Measures ◽  
Large Families ◽  
Nikodym Derivative

In 1966, Hewitt and Zuckerman(3,4) proved that if G is a non-discrete locally compact Abelian group with Haar measure λ, then there exists a non-negative, continuous regular measure μon G that is singular to λ(μ ┴ λ) such that μ(G)= 1, μ * μ is absolutely continuous with respect to λ(μ * μ ≪ λ), and the Lebesgue-Radon-Nikodym derivative of μ * μ with respect to λ is in (G, λ) for all real p > 1. They showed also that such a μ can be chosen so that the support of μ * μ contains any preassigned σ-compact subset of G. It is the purpose of the present paper to extend this result to obtain large independent sets of such measures. Among other things the present results show that, for such groups, the radical of the measure algebra modulo the -algebra has large dimension. This answers a question (6.4) left open in (3).

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