scholarly journals Amenability properties of the central Fourier algebra of a compact group

2016 ◽  
Vol 60 (2) ◽  
pp. 505-527
Author(s):  
Mahmood Alaghmandan ◽  
Nico Spronk
Keyword(s):  
Compact Group ◽  
Fourier Algebra

scholarly journals Completely bounded bimodule maps and spectral synthesis

2017 ◽  
Vol 28 (10) ◽  
pp. 1750067 ◽  
Author(s):  
M. Alaghmandan ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Closed Subset ◽  
Spectral Synthesis ◽  
Locally Compact Group ◽  
Product Formula ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

Uniformly Continuous Functionals and M-Weakly Amenable Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1005-1019 ◽  
Author(s):  
Brian Forrest ◽  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Amenable Groups ◽  
Invariant Means ◽  
Continuous Functionals ◽  
Uniformly Continuous

AbstractLet G be a locally compact group. Let AM(G) (A0(G))denote the closure of A(G), the Fourier algebra of G in the space of bounded (completely bounded) multipliers of A(G). We call a locally compact group M-weakly amenable if AM(G) has a bounded approximate identity. We will show that when G is M-weakly amenable, the algebras AM(G) and A0(G) have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of topologically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra.

scholarly journals Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm

2007 ◽  
Vol 59 (5) ◽  
pp. 966-980 ◽  
Author(s):  
Brian E. Forrest ◽  
Volker Runde ◽  
Nico Spronk
Keyword(s):  
Compact Group ◽  
Free Group ◽  
Discrete Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Residually Finite ◽  
Finite Dimensional

AbstractLet G be a locally compact group, and let Acb(G) denote the closure of A(G), the Fourier algebra of G, in the space of completely boundedmultipliers of A(G). If G is a weakly amenable, discrete group such that C*(G) is residually finite-dimensional, we show that Acb(G) is operator amenable. In particular, Acb() is operator amenable even though , the free group in two generators, is not an amenable group. Moreover, we show that if G is a discrete group such that Acb(G) is operator amenable, a closed ideal of A(G) is weakly completely complemented in A(G) if and only if it has an approximate identity bounded in the cb-multiplier norm.

Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra A(G)

2009 ◽  
Vol 61 (2) ◽  
pp. 382-394
Author(s):  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Arens Product ◽  
The Right ◽  
Topological Centers ◽  
The Relationship

Abstract. Let 𝒜 be a Banach algebra with a bounded right approximate identity and let ℬ be a closed ideal of 𝒜. We study the relationship between the right identities of the double duals ℬ** and 𝒜** under the Arens product. We show that every right identity of ℬ** can be extended to a right identity of 𝒜** in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra A(G) of a locally compact group G, an element ϕ ∈ A(G)** is in A(G) if and only if A(G)ϕ ⊆ A(G) and Eϕ = ϕ for all right identities E of A(G)**. We also prove some results about the topological centers of ℬ** and 𝒜**.

On $${\phi}$$ -contractibility of the Lebesgue–Fourier algebra of a locally compact group

2010 ◽  
Vol 95 (4) ◽  
pp. 373-379 ◽  
Author(s):  
Mahmood Alaghmandan ◽  
Rasoul Nasr-Isfahani ◽  
Mehdi Nemati
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Fourier Algebra ◽  
Locally Compact

Subalgebras of the dual of the Fourier algebra of a compact group

1972 ◽  
Vol 71 (2) ◽  
pp. 329-333 ◽  
Author(s):  
Charles F. Dunkl ◽  
Donald E. Ramirez
Keyword(s):  
Compact Group ◽  
Dual Space ◽  
Continuous Functions ◽  
Left Translation ◽  
Fourier Algebra ◽  
Bounded Operators

We let G denote an infinite compact group and G its dual. We use the notation of our book ((l), Chapters 7 and 8). Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and ℒ∞(G) denotes its dual space under the pairing 〈ƒ,φ〉 (ƒ ∈ A(G), φ ∈ ℒ∞(G)). Further, note ℒ∞(G) is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. The module action of A(G) of ℒ∞(G) is defined by the following: for ƒ ∈ A(G), φ ℒ∞(G), ƒ. φ ∈ ℒ∞(G) by 〈g, ƒ . φ〉 = 〈 ƒg, φ〉, g ∈ A(G) Also ‖ƒ . φ‖∞ ≥ ‖ƒ‖A ‖φ‖∞.

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