Large families of singular measures having absolutely continuous convolution squares

1968 ◽  
Vol 64 (4) ◽  
pp. 1015-1022 ◽  
Author(s):  
Karl Stromberg

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).

1997 ◽  
Vol 40 (2) ◽  
pp. 261-266 ◽  
Author(s):  
H. Seferoǧlu

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.


2019 ◽  
Vol 192 (4) ◽  
pp. 935-938
Author(s):  
László Székelyhidi

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.


1968 ◽  
Vol 20 ◽  
pp. 291-297
Author(s):  
Robert Kaufman

Suppose that G is a locally compact abelian group, u an element of infinite order, and w a complex number of modulus 1. It is a familiar fact that there is a complex homomorphism Ψ of the measure algebra M of G, which maps ϵu (the unit mass concentrated at u) to w. Beyond this, one may specify an element μ of M, and require a homomorphism Ψ which does not annihilate μ. The resolution of this problem leads to an abstract lemma on measurable transformations, derived in some generality in the first section.


1982 ◽  
Vol 5 (3) ◽  
pp. 503-512
Author(s):  
U. B. Tewari ◽  
M. Dutta ◽  
Shobha Madan

LetA1,A2be commutative semisimple Banach algebras andA1⊗∂A2be their projective tensor product. We prove that, ifA1⊗∂A2is a group algebra (measure algebra) of a locally compact abelian group, then so areA1andA2. As a consequence, we prove that, ifGis a locally compact abelian group andAis a comutative semi-simple Banach algebra, then the Banach algebraL1(G,A)ofA-valued Bochner integrable functions onGis a group algebra if and only ifAis a group algebra. Furthermore, ifAhas the Radon-Nikodym property, then the Banach algebraM(G,A)ofA-valued regular Borel measures of bounded variation onGis a measure algebra only ifAis a measure algebra.


Author(s):  
Edwin Hewitt ◽  
Herbert S. Zuckerman

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.


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