Convolutions on compact groups and Fourier algebras of coset spaces

Brian E. Forrest, Ebrahim Samei, and Nico Spronk. "Convolutions on compact groups and Fourier algebras of coset spaces." Studia Mathematica 196.3 (2010): 223-249. <http://eudml.org/doc/285825>.

@article{BrianE2010,
abstract = { We study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(G × G) given for u,v in A(G) by u × v ↦ u*v̌ (v̌(s) = v(s^-1)) and u × v ↦ u*v? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain, for compact groups which do not admit an abelian subgroup of finite index, some new subalgebras of A(G). Using those algebras we can find many instances in which A(G/K) fails the most rudimentary amenability property: operator weak amenability. However, using different techniques, we show that if the connected component of the identity of G is abelian, then A(G/K) always satisfies the stronger property that it is hyper-Tauberian, which is a concept developed by the second named author. We also establish a criterion which characterises operator amenability of A(G/K) for a class of groups which includes the maximally almost periodic groups. Underlying our calculations are some refined techniques for studying spectral synthesis properties of sets for Fourier algebras. We even find new sets of synthesis and nonsynthesis for Fourier algebras of some classes of groups. },
author = {Brian E. Forrest, Ebrahim Samei, Nico Spronk},
journal = {Studia Mathematica},
keywords = {convolution; coset space; Fourier algebra; amenability; hyper-Tauberian},
language = {eng},
number = {3},
pages = {223-249},
title = {Convolutions on compact groups and Fourier algebras of coset spaces},
url = {http://eudml.org/doc/285825},
volume = {196},
year = {2010},
}

TY - JOUR
AU - Brian E. Forrest
AU - Ebrahim Samei
AU - Nico Spronk
TI - Convolutions on compact groups and Fourier algebras of coset spaces
JO - Studia Mathematica
PY - 2010
VL - 196
IS - 3
SP - 223
EP - 249
AB - We study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(G × G) given for u,v in A(G) by u × v ↦ u*v̌ (v̌(s) = v(s^-1)) and u × v ↦ u*v? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain, for compact groups which do not admit an abelian subgroup of finite index, some new subalgebras of A(G). Using those algebras we can find many instances in which A(G/K) fails the most rudimentary amenability property: operator weak amenability. However, using different techniques, we show that if the connected component of the identity of G is abelian, then A(G/K) always satisfies the stronger property that it is hyper-Tauberian, which is a concept developed by the second named author. We also establish a criterion which characterises operator amenability of A(G/K) for a class of groups which includes the maximally almost periodic groups. Underlying our calculations are some refined techniques for studying spectral synthesis properties of sets for Fourier algebras. We even find new sets of synthesis and nonsynthesis for Fourier algebras of some classes of groups.
LA - eng
KW - convolution; coset space; Fourier algebra; amenability; hyper-Tauberian
UR - http://eudml.org/doc/285825
ER -