Addendum to “Amenability and weak amenability of second conjugate Banach algebras”

By F. Ghahramani, R. J. Loy, George A. Willis. Proceedings of the American Mathematical Society, 2020

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For a Banach algebra A, amenability of A∗∗ necessitates amenability of A, and similarly for weak amenability provided A is a left ideal in A∗∗. For G a locally compact group, indeed more generally, L1(G)∗∗ is amenable if and only if G is finite. If L1(G)∗∗ is weakly amenable, then M(G) is weakly amenable. 0. Introduction For a Banach algebra A,A∗∗ is a Banach algebra under two Arens products, of which we will always take the first, or left, product. For further details see the survey article [8]. This product can be characterized as the extension to A∗∗×A∗∗ of the bilinear map A×A→ A : (x, y) 7→ xy with the following continuity properties: for fixed y ∈ A∗∗, x 7→ xy is weak*-continuous on A∗∗; for fixed y ∈ A, x 7→ yx is weak*-continuous on A∗∗. Here, as elsewhere, we identify A with its canonical image in A∗∗. In terms of the asymmetry here, define the topological centre of A∗∗ by Zt(A ∗∗) = {y ∈ A∗∗ : x 7→ yx is weak∗-continuous}. Clearly, Zt(A ∗∗) contains the (algebraic) centre Z(A∗∗) of A∗∗; it also contains A. In the case that Zt(A ∗∗) = A∗∗, A∗∗ is said to be Arens regular . Any C∗-algebra is Arens regular [6, Theorem 7.1], but for a locally compact group G, L(G) is Arens regular if and only if G is finite [25]. A Banach algebra A is amenable if every derivation D : A → X∗ is inner, for every Banach A-bimodule X . If one only considers the bimodule X = A, one has the notion of weak amenability. There are many alternative formulations of the notion of amenability, of which we need the following two, first introduced in [18]. For further details see [3, 16, 7]. The Banach algebra A is amenable if and only if either, and hence both, of the following hold: (i) A has an approximate diagonal , that is, a bounded net (mi) ⊂ A⊗A such that for each x ∈ A, mix− xmi → 0, π(mi)x→ x; (ii) A has a virtual diagonal , that is, an element M ∈ (A⊗A)∗∗ such that for each x ∈ A, xM = Mx, (π∗∗M)x = x. Received by the editors June 27, 1994 and, in revised form, October 19, 1994. 1991 Mathematics Subject Classification. Primary 46H20; Secondary 43A20.

Subfields: Algebra and Number Theory, Mathematical Physics

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