Ideal amenability of various classes of Banach algebras (Q763694)

Let \(\mathcal{A}\) be a Banach algebra, and let \(n \in \mathbb{N}\). Then \(\mathcal{A}\) is said to be amenable if \(H^1 (\mathcal{A}, X^*) =\{ 0\}\) for all Banach \(\mathcal{A}\)-bimodules \(X\), and \(\mathcal{A}\) is called weakly amenable if \(H^1 (\mathcal{A}, \mathcal{A}^*) =\{ 0\}.\) \(\mathcal{A}\) is called \(n\)-weakly amenable if \(H^1 (\mathcal{A}, \mathcal{A}^{(n)}) =\{ 0\}\), and \(\mathcal{A}\) is said to be ideal amenable if \(H^1 (\mathcal{A}, \mathfrak{T}^{*}) =\{ 0\}\) for every closed two sided ideal \(\mathfrak{T}\) of \(\mathcal{A}\). Now, let \(\mathfrak{T}\) be a closed two sided ideal in \(\mathcal{A}\). Then \(\mathcal{A}\) is called \(n\)-\(\mathfrak{T}\)-weakly amenable if the first cohomology group of \(\mathcal{A}\) with coefficients in the \(n\)-th dual space \(\mathfrak{T}^{(n)}\) is zero, i.e., \(H^1 (\mathcal{A}, \mathfrak{T}^{(n)} ) =\{ 0\}.\) Further, \(\mathcal{A}\) is said to be \(n\)-ideal amenable (ideal amenable) if \(\mathcal{A}\) is \(n\)-\(\mathfrak{T}\)-weakly amenable (\(1\)-\(\mathfrak{T}\)-weakly amenable) for every closed two sided ideal \(\mathfrak{T}\) in \(\mathcal{A}.\) In this paper, the author investigates \((2m+1)\)-\(\mathfrak{T}\)-weakly amenability of Banach algebras for \(m \geq 1,\) and ideal amenability of Segal algebras and triangular Banach algebras.