Let B be an abstract Segal algebra with respect to A. For a nonzero character φ on A, we study φ-amenability, and φ-contractibility of A and B. We then apply these results to abstract Segal algebras related to locally compact groups.2000 Mathematics subject classification: primary 46H05; secondary 43A07.
Let H be an ultraspherical hypergroup associated to a locally compact group G
and let A(H) be the Fourier algebra of H. For a left Banach A(H)-submodule X
of VN(H), define QX to be the norm closure of the linear span of the set {u
f : u ?A(H), f ? X} in BA(H)(A(H),X
For two Banach algebras A and B, an interesting product A × θ B, called the θ-Lau product, was recently introduced and studied for some nonzero characters θ on B. Here, we characterize some notions of amenability as approximate amenability, essential amenability, n-weak amenability and cyclic amenability between A and B and their θ-Lau product.
Introduction. LetA and B be two Banach algebras and θ ∈ σ(B), the spectrum of B of all nonzero characters on B. Then the θ-Lau product of A and B, denoted by A × θ B, is defined as the space A × B equipped with the multiplication (a, b)(a , b ) = (aa + θ(b)a + θ(b )a, bb ), and the norm (a, b) = a + b , for all a, a ∈ A and b, b ∈ B. The θ-Lau product A × θ B is a Banach algebra.This product was first introduced by Lau [L1] for Lau algebras; recall that a Lau algebra is a Banach algebra which is the predual of a von Neumann algebra for which the identity of the dual is a multiplicative linear functional. The study of this large class of Banach algebras originated with a paper published in 1983 by Lau [L1] in which he referred to them as "F-algebras"; see also Lau [L2]. Later on, in his useful monograph Pier [Pi] introduced the name "Lau algebra". Examples of Lau algebras include the group algebra and the measure algebra of a locally compact group or hypergroup (see Lau [L1]), and also the Fourier algebra and
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