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Approximate amenability of semigroup algebras and Segal algebras

100%
Dales H., Loy R.
Instytut Matematyczny Polskiej Akademii Nauk
EN
In recent years, there have been several studies of various 'approximate' versions of the key notion of amenability, which is defined for all Banach algebras; these studies began with work of Ghahramani and Loy in 2004. The present memoir continues such work: we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability. There are a number of open questions on these relationships; these will be considered. In Chapter 1, we shall give all the relevant definitions and a number of basic results, partly surveying existing work; we shall concentrate on the case of Banach function algebras. In Chapter 2, we shall discuss these properties for the semigroup algebra ℓ¹(S) of a semigroup S. In the case where S has only finitely many idempotents, ℓ¹(S) is approximately amenable if and only if it is amenable. In Chapter 3, we shall consider the class of weighted semigroup algebras of the form $ℓ¹(ℕ_{∧},ω)$, where ω: ℤ → [1,∞) is an arbitrary function. We shall determine necessary and sufficient conditions on ω for these Banach sequence algebras to have each of the various approximate amenability properties that interest us. In this way we shall illuminate the implications between these properties. In Chapter 4, we shall discuss Segal algebras on 𝕋 and on ℝ. It is a conjecture that every proper Segal algebra on 𝕋 fails to be approximately amenable; we shall establish this conjecture for a wide class of Segal algebras.
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Approximate and weak amenability of certain Banach algebras

94%
Bharucha P., Loy R.
Studia Mathematica
|
2010
|
tom 197
|
nr 2
195-204
EN
The notions of approximate amenability and weak amenability in Banach algebras are formally stronger than that of approximate weak amenability. We demonstrate an example confirming that approximate weak amenability is indeed actually weaker than either approximate or weak amenability themselves. As a consequence, we examine the (failure of) approximate amenability for $ℓ^{p}$-sums of finite-dimensional normed algebras.
3
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Approximate amenability for Banach sequence algebras

76%
Dales H., Loy R., Zhang Y.
Studia Mathematica
|
2006
|
tom 177
|
nr 1
81-96
EN
We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A = ℓ^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras $ℓ^{p}(ω)$.
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