Extreme Version of Projectivity for Normed Modules Over Sequence Algebras

We define and study the so-called extreme version of the notion of a projective normed module. The relevant definition takes into account the exact value of the norm of the module in question, in contrast with the standard known definition that is formulated in terms of norm topology. After the discussion of the case where our normed algebra is just , we concentrate on the case of the next degree of complication, where is a sequence algebra satisfying some natural conditions. The main results give a full characterization of extremely projective objects within the subcategory of the category of non-degenerate normed -modules, consisting of the so-called homogeneous modules. We consider two cases, ‘non-complete’ and ‘complete’, and the respective answers turn out to be essentially different. In particular, all Banach non-degenerate homogeneous modules consisting of sequences are extremely projective within the category of Banach non-degenerate homogeneous modules. However, neither of them, provided it is infinite-dimensional, is extremely projective within the category of all normed non-degenerate homogeneous modules. On the other hand, submodules of these modules consisting of finite sequences are extremely projective within the latter category.