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

Abstract

We show that for n ≥ 2, n-weak amenability of the second dual A * * of a Banach algebra A implies that of A. We also provide a positive answer for the case n = 1, which sharpens some older results. Our method of proof also provides a unified approach to give short proofs for some known results in the case where n = 1.2000 Mathematics subject classification: primary 46H20; secondary 46H25.

Characterization of Some Biderivations on Triangular Banach Algebras

“…Indeed, each abelian group G is biamenable while the commutative group algebra L 1 (G) is not biamenable [2]. Of course, the situation of weak biamenability is better than biamenability, and many similar results of [4] and [7] are valid for weak biamenability of Banach algebras. Also, for each locally compact abelian group G, L 1 (G) is weakly biamenable.…”

m−WEAK AMENABILITY OF (2n)−TH DUALS OF BANACH ALGEBRAS

“…It was shown in [4,11] that the n−weak amenability of A ′′ implies the n−weak amenability of A. In [13] it was shown that if the Banach algebra A is complete Arens regular and every derivation D : A → A ′ is weakly compact, the weak amenability of A (2n) for some n ≥ 1 implies of A.…”

“…) are bounded nets in A ′′ . For the left A (4) −module action on A (5) = ((A ′ ) ′′ ) ′′ we have a (4) • a (5) , b (4) = lim…”

“…and for the left A (4) −module action on A (5) = ((A ′′ ) ′′ ) ′ we have a (4) .a (5) , b (4) = a (5) , b (4) a (4) = lim…”

“…But two right A (4) −module actions a (5) • a (4) and a (5) • a (4) are equal. To prove the equality of the left A (4) −module actions on A (5) , we need the equality of two left A ′′ −module actions on A (3) = (A ′′ ) ′ and A (3) = (A ′ ) ′′ by the following lemma, whose proof is straightforward.…”