Second cohomology of Beurling algebras (Q2719789)

The concept of weak amenability was introduced by \textit{W. G. Bade}, \textit{P. C. Curtis, jun.} and \textit{H. G. Dales} [Proc. Lond. Math. Soc., III. Ser. 55, 359-377 (1987; Zbl 0634.46042)] for commutative Banach algebras and by \textit{B. E. Johnson} [Lect. Notes Math. 1359, 191-198 (1988; Zbl 0668.43003)] for noncommutative Banach algebras. \textit{N. Grønbæk} investigates the weak amenability of the weighted discrete convolution algebra \(\ell^1(G,a)\), where \(G\) is an Abelian group. So, in [Stud. Math. 94, No. 2, 149-162 (1989; Zbl 0704.46030)] he proves the following important result:NEWLINENEWLINENEWLINELet \(G\) be an Abelian group and let \(\omega: G\to R^+\) be a weight. Then the Beurling algebra \(\ell^1(G,\omega)\) is weakly amenable if and only if NEWLINE\[NEWLINE\sup\Biggl\{{|f(g)|\over \omega(g)\omega(g^{-1})}: g\in G\Biggr\}= \inftyNEWLINE\]NEWLINE from all \(f\in\Hom_2(G, C)\setminus\{0\}\).NEWLINENEWLINENEWLINEIn the present paper the author generalizes the ``only if'' part of previous Grønbæk's result for a not necessarily Abelian group \(G\).NEWLINENEWLINENEWLINEFinally, in the last part of the paper, the second cohomology group of the Beurling algebra \(\ell^1(Z,\omega_\alpha)\), where \(\omega_\alpha(\eta)= (1+|n|)^\alpha\), \(\alpha> 0\), with coefficients in \(C\) is studied and its non-vanishes is proved.