Arens Regularity of Module Actions and the Second Adjoint of a Derivation

Abstract

In this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions, and then investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.2000 Mathematics subject classification: 46H20, 46H25.

LIFTING DERIVATIONS ANDn-WEAK AMENABILITY OF THE SECOND DUAL OF A BANACH ALGEBRA

“…It should be mentioned that these two actions on A (2n+1) do not coincide, in general. For more information on the equality of these actions in the case where n = 1, see [4,10].…”

“…Theorem 7.1] and also[10, Theorem 4.2] for a more general case. As Lemma 4 demonstrates, (J 0 ) *• D * * is a derivation if and only if π * * * r * (D * * (A * * ), A) ⊆ A * .…”

On extension of bi-derivations to the bidual of Banach algebras

“…In this paper we study the (Arens) extensions of a bi-derivation. The main motivation comes from the papers [6,9,13], where the extension of derivations from a Banach algebra A to certain Banach modules have been investigated. Let δ : A −→ X be a derivation, the question "is the second adjoint δ * * of δ a derivation?"…”

“…They showed that δ is a derivation if and only if δ * * (A * * ).A * * ⊆ A * (see [9, Theorem 7.1]). Then Mohammadzadeh and Vishki answered the question for a general dual Banach A −module X * , (see [13,Theorem 4.2]). Barootkoob and Vishki [3] have also generalized the results of [6,9,13] for derivations into the various duals of a Banach A −module X .…”

Topological centers of module actions induced by unitary representations

“…In [16], Ulger showed that the Arens regularity of a bounded bilinear map can be characterized by its weakly compactness or its reflexiveness and simplified proofs of some old results. For more recent results, the reader is referred to [6] and [14]. On the other hand, special attention has been focused on the bilinear maps arisen from Banach algebras.…”