scholarly journals A COMPACT QUALITATIVE UNCERTAINTY PRINCIPLE FOR SOME NONUNIMODULAR GROUPS

2018 ◽  
Vol 99 (1) ◽  
pp. 114-120
Author(s):  
WASSIM NASSERDDINE
Keyword(s):  
Regular Representation ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compactly Supported ◽  
Finite Dimensional ◽  
Left Regular Representation ◽  
Left Regular ◽  
Compact Abelian Groups ◽  
Qualitative Uncertainty ◽  
The Fourier Transform

Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$. There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\,\hat{f}$ is finite dimensional, then $f=0$. In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.

Pego everywhere

2016 ◽  
Vol 15 (04) ◽  
pp. 1650074 ◽  
Author(s):  
Przemysław Górka ◽  
Tomasz Kostrzewa
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
General Version ◽  
Pontryagin Duality ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform ◽  
Compact Subsets

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of [Formula: see text] in terms of the Fourier transform.

scholarly journals A Finiteness Condition for Locally Compact Abelian Groups

1971 ◽  
Vol 12 (1) ◽  
pp. 115-121 ◽  
Author(s):  
L. C. Grove ◽  
L. J. Lardy
Keyword(s):  
Finite Type ◽  
Abelian Groups ◽  
Finiteness Condition ◽  
Vector Spaces ◽  
Finite Sets ◽  
Locally Compact ◽  
Object A ◽  
Locally Compact Abelian Groups ◽  
Finite Dimensional ◽  
Compact Abelian Groups

A map f: A→B in category is called monic if fg = fh implies that g = h for all maps g, h: C → A; it is called epic if gf = hf implies that g = h for all maps g, h: B → C. An object A ∈ is called an S-object if every monic map f: A → A is also epic; it is called a Q-object if every epic map f: A → A is also monic. If A is both an S-object and a Q-object then A is called an SQ-object. In the category of sets the SQ-sets are the finite sets. In the category of vector spaces over a field F the SQ-spaces are precisely the finite dimensional spaces. In the light of these simple examples, it seems reasonable to view the SQ-objects of a category as being of ‘finite type’. We shall be chiefly concerned with investigating the SQ-objects in certain subcategories of the category of locally compact abelian groups.

scholarly journals On Group Von Neumann Algebras with Vector-Valued Functions

2018 ◽  
Vol 14 (1) ◽  
pp. 7596-7614
Author(s):  
Julien Esse Atto ◽  
Victor Kofi Assiamoua
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Regular Representation ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Left Regular Representation ◽  
Left Regular ◽  
Vector Valued Functions ◽  
Vector Valued

Let G be a locally compact group equipped with a normalized Haar measure , A(G) the Fourier algebraof G and V N(G) the von Neumann algebra generated by the left regular representation of G. In this paper, we introduce the space V N(G;A) associated with the Fourier algebra A(G;A) for vector-valued functions on G, where A is a H-algebra. Some basic properties are discussed in the category of Banach space, and alsoin the category of operator space.

PEGO THEOREM ON LOCALLY COMPACT ABELIAN GROUPS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350143 ◽  
Author(s):  
PRZEMYSłAW GÓRKA
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform

In this paper, we show the version of Pego's theorem on locally compact abelian groups. This theorem, [R. L. Pego, Compactness in L2 and the Fourier transform, Proc. Amer. Math. Soc.95 (1985) 252–254], gives a characterization of precompact sets of L2 in terms of the Fourier transform.

scholarly journals Discrete series of fusion algebras

2002 ◽  
Vol 72 (3) ◽  
pp. 419-426
Author(s):  
R. Srinivasan ◽  
V. S. Sunder ◽  
N. J. Wildberger
Keyword(s):  
Regular Representation ◽  
Discrete Series ◽  
Invariant Subspaces ◽  
Discrete Groups ◽  
Compact Lie Group ◽  
Finite Dimensional ◽  
Fusion Algebra ◽  
Left Regular Representation ◽  
Left Regular ◽  
Countably Infinite

AbstractWe show that the left regular representation of a countably infinite (discrete) group admits no finite-dimensional invariant subspaces. We also discuss a consequence of this fact, and the reason for our interest in this statement.We then formally state, as a ‘conjecture’, a possible generalisation of the above statement to the context of fusion algebras. We prove the validity of this conjecture in the case of the fusion algebra arising from the dual of a compact Lie group.We finally show, by example, that our conjecture is false as stated, and raise the question of whether there is a ‘good’ class of fusion algebras, which contains (a) the two ‘good classes’ discussed above, namely, discrete groups and compact group duals, and (b) only contains fusion algebras for which the conjecture is valid.

Fourier transforms of convolution operators on orlicz spaces

2021 ◽  
Vol 71 (2) ◽  
pp. 369-382
Author(s):  
Seyyed Mohammad Tabatabaie ◽  
AliReza Bagheri Salec
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fourier Transforms ◽  
Orlicz Spaces ◽  
Locally Compact Group ◽  
Convolution Operators ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform

Abstract In this paper, we study convolution operators on an Orlicz space L Φ(G) commuting with left translations, where Φ is an N-function and G is a locally compact group. We also present some basic properties of the Fourier transform of a Φ-convolution operator in the context of locally compact abelian groups.

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