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1
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Amenability properties of Fourier algebras and Fourier-Stieltjes algebras: a survey

100%
Spronk N.
Banach Center Publications
|
2010
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tom 91
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nr 1
365-383
EN
Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, $L^{-1}(G)$ and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for $L^{-1}(G)$ and M(G). For us, "amenability properties" refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisimple commutative Banach algebras. We wish to emphasize that the theory of operator spaces and completely bounded maps plays an indispensable role when studying A(G) and B(G). We also show some applications of amenability theory to problems of complemented ideals and homomorphisms.
2
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Convolutions on compact groups and Fourier algebras of coset spaces

51%
Forrest B., Samei E., Spronk N.
Studia Mathematica
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2010
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tom 196
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nr 3
223-249
EN
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.
3
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Beurling-Figà-Talamanca-Herz algebras

51%
Öztop S., Runde V., Spronk N.
Studia Mathematica
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2012
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tom 210
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nr 2
117-135
EN
For a locally compact group G and p ∈ (1,∞), we define and study the Beurling-Figà-Talamanca-Herz algebras $A_{p}(G,ω)$. For p = 2 and abelian G, these are precisely the Beurling algebras on the dual group Ĝ. For p = 2 and compact G, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that a locally compact group G is amenable if and only if one-and, equivalently, every-Beurling-Figà-Talamanca-Herz algebra $A_{p}(G,ω)$ has a bounded approximate identity.
4
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Operator Segal algebras in Fourier algebras

51%
Forrest B., Spronk N., Wood P.
Studia Mathematica
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2007
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tom 179
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nr 3
277-295
EN
Let G be a locally compact group, A(G) its Fourier algebra and L¹(G) the space of Haar integrable functions on G. We study the Segal algebra S¹A(G) = A(G) ∩ L¹(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S¹A(G). We use it to show that the restriction operator $u ↦ u|_{H}: S¹A(G) → A(H)$, for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup, then the averaging operator $τ_{N}: S¹A(G) → L¹(G/N)$, $τ_{N}u(sN) = ∫_{N} u(sn)dn$, is a surjective complete quotient map. This puts an operator space perspective on the philosophy that S¹A(G) is "locally A(G) while globally L¹". Also, using the operator space structure we can show that S¹A(G) is operator amenable exactly when when G is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.
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