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1-amenability of 𝒜(X) for Banach spaces with 1-unconditional bases

A. Blanco (2012)

Studia Mathematica

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property 𝔸 has in fact the property. Some further ideas on the problem of whether or not amenability (in...

A generalized notion of n -weak amenability

Abasalt Bodaghi, Behrouz Shojaee (2014)

Mathematica Bohemica

In the current work, a new notion of n -weak amenability of Banach algebras using homomorphisms, namely ( ϕ , ψ ) - n -weak amenability is introduced. Among many other things, some relations between ( ϕ , ψ ) - n -weak amenability of a Banach algebra 𝒜 and M m ( 𝒜 ) , the Banach algebra of m × m matrices with entries from 𝒜 , are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra L 1 ( G ) is ( ϕ , ψ )- n -weakly amenable for any...

A note on a construction of J. F. Feinstein

M. J. Heath (2005)

Studia Mathematica

In [6] J. F. Feinstein constructed a compact plane set X such that R(X), the uniform closure of the algebra of rational functions with poles off X, has no non-zero, bounded point derivations but is not weakly amenable. In the same paper he gave an example of a separable uniform algebra A such that every point in the character space of A is a peak point but A is not weakly amenable. We show that it is possible to modify the construction in order to produce examples which are also regular.

A spectral mapping theorem for Banach modules

H. Seferoğlu (2003)

Studia Mathematica

Let G be a locally compact abelian group, M(G) the convolution measure algebra, and X a Banach M(G)-module under the module multiplication μ ∘ x, μ ∈ M(G), x ∈ X. We show that if X is an essential L¹(G)-module, then σ ( T μ ) = μ ̂ ( s p ( X ) ) ¯ for each measure μ in reg(M(G)), where T μ denotes the operator in B(X) defined by T μ x = μ ∘ x , σ(·) the usual spectrum in B(X), sp(X) the hull in L¹(G) of the ideal I X = f ∈ L ¹ ( G ) | T f = 0 , μ̂ the Fourier-Stieltjes transform of μ, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains all...

Amenability and weak amenability of l¹-algebras of polynomial hypergroups

Rupert Lasser (2007)

Studia Mathematica

We investigate amenability and weak amenability of the l¹-algebra of polynomial hypergroups. We derive conditions for (weak) amenability adapted to polynomial hypergroups and show that these conditions are often not satisfied. However, we prove amenability for the hypergroup induced by the Chebyshev polynomials of the first kind.

Amenability for dual Banach algebras

V. Runde (2001)

Studia Mathematica

We define a Banach algebra 𝔄 to be dual if 𝔄 = (𝔄⁎)* for a closed submodule 𝔄⁎ of 𝔄*. The class of dual Banach algebras includes all W*-algebras, but also all algebras M(G) for locally compact groups G, all algebras ℒ(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable...

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

Nico Spronk (2010)

Banach Center Publications

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...

An additivity formula for the strict global dimension of C(Ω)

Seytek Tabaldyev (2014)

Open Mathematics

Let A be a unital strict Banach algebra, and let K + be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K +), the algebra of continuous functions on K +. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .

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