The Range of Invariant Means on Locally Compact Abelian Groups

AbstractIt has been shown by E. Granirer that for certain infinite amenable discrete groups G there exists a nested family of left almost convergent subsets of G on which every left invariant mean on m(G) attains as its range the entire [0,1] interval. This paper examines the range of left invariant means on L∞(G) for infinite locally compact abelian groups G and demonstrates the existence in every such group of a nested family of left almost convergent Borel subsets on which every left invariant mean on L∞ (G) attains as its range the interval [0,1],

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The Helson–Reiter theorem for a class of nilpotent discrete groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004196001600 ◽

1997 ◽

Vol 122 (1) ◽

pp. 95-103 ◽

Cited By ~ 2

Author(s):

EBERHARD KANIUTH

Keyword(s):

Abelian Groups ◽

Group Algebras ◽

Discrete Groups ◽

Locally Compact ◽

Locally Compact Abelian Groups ◽

Compact Abelian Groups

It is shown that the analogue of the Helson–Reiter theorem for ideals in group algebras of locally compact abelian groups holds for a class of groups including all divisible nilpotent discrete groups.

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Scaling sets and generalized scaling sets on Cantor dyadic group

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691320500198 ◽

2020 ◽

Vol 18 (04) ◽

pp. 2050019

Author(s):

Prasadini Mahapatra ◽

Divya Singh

Keyword(s):

Abelian Groups ◽

Real Case ◽

Locally Compact ◽

Wavelet Sets ◽

Locally Compact Abelian Groups ◽

Dyadic Group ◽

Compact Abelian Groups ◽

Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

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Semisimple Banach Algebras Generated by Strongly Continuous Representations of Locally Compact Abelian Groups

Journal of Functional Analysis ◽

10.1006/jfan.1994.1138 ◽

1994 ◽

Vol 126 (1) ◽

pp. 1-6 ◽

Cited By ~ 3

Author(s):

G. Muraz ◽

V.Q. Phong

Keyword(s):

Abelian Groups ◽

Banach Algebras ◽

Locally Compact ◽

Locally Compact Abelian Groups ◽

Compact Abelian Groups ◽

Continuous Representations

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Singular measures with absolutely continuous convolution squares

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100039992 ◽

1966 ◽

Vol 62 (3) ◽

pp. 399-420 ◽

Cited By ~ 38

Author(s):

Edwin Hewitt ◽

Herbert S. Zuckerman

Keyword(s):

Abelian Groups ◽

Natural Order ◽

Continuous Measure ◽

Locally Compact ◽

Absolutely Continuous ◽

Character Group ◽

Locally Compact Abelian Groups ◽

Singular Measures ◽

Compact Abelian Groups ◽

Nikodym Derivative

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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