scholarly journals The Fourier Algebra for Locally Compact Groupoids

2004 ◽  
Vol 56 (6) ◽  
pp. 1259-1289 ◽  
Author(s):  
Alan L. T. Paterson
Keyword(s):  
Duality Theorem ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Fourier Algebra ◽  
Hilbert Modules ◽  
Locally Compact ◽  
Classical Duality ◽  
Special Case

AbstractWe introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

scholarly journals Completely bounded homomorphisms of the Fourier algebra revisited

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Piecewise Affine ◽  
Affine Maps

Abstract Assume that A ⁢ ( G ) A(G) and B ⁢ ( H ) B(H) are the Fourier and Fourier–Stieltjes algebras of locally compact groups 𝐺 and 𝐻, respectively. Ilie and Spronk have shown that continuous piecewise affine maps α : Y ⊆ H → G \alpha\colon Y\subseteq H\to G induce completely bounded homomorphisms Φ : A ⁢ ( G ) → B ⁢ ( H ) \Phi\colon A(G)\to B(H) and that, when 𝐺 is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure-theoretic ideas, following more closely the original ideas of Cohen.

Strong and Extremely Strong Ditkin sets for the Banach Algebras Apr(G) = Ap ⋂ Lr(G)

2011 ◽  
Vol 63 (1) ◽  
pp. 123-135 ◽  
Author(s):  
Edmond E. Granirer
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Discrete Groups ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Strict Inclusion ◽  
Sidon Sets ◽  
Sequential Completeness

Abstract Let Ap(G) be the Figa-Talamanca, Herz Banach Algebra on G; thus A2(G) is the Fourier algebra. Strong Ditkin (SD) and Extremely Strong Ditkin (ESD) sets for the Banach algebras Apr (G) are investigated for abelian and nonabelian locally compact groups G. It is shown that SD and ESD sets for Ap(G) remain SD and ESD sets for Apr(G), with strict inclusion for ESD sets. The case for the strict inclusion of SD sets is left open.A result on the weak sequential completeness of A2(F) for ESD sets F is proved and used to show that Varopoulos, Helson, and Sidon sets are not ESD sets for A2r(G), yet they are such for A2(G) for discrete groups G, for any 1 ≤ r ≤ 2.A result is given on the equivalence of the sequential and the net definitions of SD or ESD sets for σ-compact groups.The above results are new even if G is abelian.

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