The second duals of Beurling algebras

Abstract

Chapter 1. Introduction Chapter 2. Definitions and Preliminary Results Chapter 3. Repeated Limit Conditions Chapter 4. Examples Chapter 5. Introverted Subspaces Chapter 6. Banach Algebras of Operators Chapter 7. Beurling Algebras Chapter 8. The Second Dual of 1 (G, ω) Chapter 9. Algebras on Discrete, Abelian Groups Chapter 10. Beurling Algebras on F 2 Chapter 11. Topological Centres of Duals of Introverted Subspaces Chapter 12. The Second Dual of L 1 (G, ω) Chapter 13. Derivations into Second Duals Chapter 14.

Arens regularity of module actions

“…A Banach algebra A is Arens regular when Z(A * * ) = A * * , or equivalently, Z t (A * * ) = A * * ; and according to [5], A is strongly Arens irregular when Z(A * * ) = Z t (A * * ) = A.…”

“…This class includes the recent example given by Dales and Lau in [5,Example 4.5]. Another type of example was also provided by Neufang in [16].…”

Weak amenability of the second dual of a Banach algebra

“…We may recall that any C * -algebra is Arens regular, and that the group algebra L 1 (G) of a locally compact group G is strongly Arens irregular, i.e., Z(L 1 (G) * * ) = L 1 (G) (see [27], or [29] and [14] for different proofs). For more details, the reader is directed for example to [15], [5] or [7]. As already done and throughout the rest of the paper, we shall identify every Banach space with its canonical image in its second dual.…”

On the topological centre problem for weighted convolution algebras and semigroup compactifications

“…Although covering a more general case, our proofs are not of higher complexity, if not even simpler, than the ones given in [8] and [7]. We note that (i) and (ii) above have very recently been obtained independently 1 and via different methods by Dales and Lau in [1] where the present work is mentioned (see [1,Thm. 11.9 and Thm.…”

“…12.3] with remark thereafter). To be more precise, (ii) is shown in [1] under the additional assumption that the weight function w satisfies w ≥ 1, and by an indirect argument, the authors indicate that their proof can be modified to yield (i). Our proofs for both (i) and (ii) are direct and do not assume that w ≥ 1.…”