Lower bounds for homological dimensions of Banach algebras and radical Banach algebras of power series

It is well known that the homological dimensions of Banach algebras possess some peculiar properties that have no analogues in abstract algebra. This is associated with some specific features of Banach structures, and first of all with the existence of non-complemented closed subspaces of Banach spaces. The talk deals with Banach algebras $\ell^1(\omega)$, where $\omega$ is a radical weight on ${\mathbb{Z}^+=\{n\in\mathbb{Z}: n\geq0\}}$. Our aim is to obtain the estimate ${\mathop{\mathrm{dg}}\ell^1(\omega)\geq3}$ for all weights satisfying the condition $\lim\limits_{n\to\infty}\,\omega_{n+1}/\omega_n=0$. (Here $\mathop{\mathrm{dg}} \ell^1(\omega)$ is the global (homological) dimension of the Banach algebra $\ell^1(\omega)$.) Then we shall have ${\mathcal{H}^3(\ell^1(\omega),X)\neq 0}$ for some Banach $\ell^1(\omega)$-bimodule $X$. The notion of a strongly non-complemented subspace of a Banach space plays a substantial role in the proof of the main result.