The values assumed by the weak homological bidimension in certain classes of Banach algebras

Some properties of the weak homological bidimension of Banach algebras are studied and important examples of its calculation are given. In particular, this characteristic is calculated for all semisimple biprojective Banach algebras with the approximation property, all so-called tensor algebras generated by bilinear forms, and all infinite-dimensional Hilbert algebras. In addition, the additivity formula for weak bidimension is proved and it is shown that, in the class of semisimple Banach algebras, this homological characteristic can take any natural values, as well as the values $0$ and $\infty$.