Abstract view
Extreme Version of Projectivity for Normed Modules Over Sequence Algebras


Published:20120517
Printed: Jun 2013
A. Ya. Helemskii,
Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992 Russia
Abstract
We define and study the socalled 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 $A$ is just
$\mathbb{C}$, we concentrate on the case of the next degree of complication,
where $A$ 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 nondegenerate
normed $A$modules, consisting of the socalled homogeneous modules.
We consider two cases, `noncomplete' and `complete', and the
respective answers turn out to be essentially different.
In particular, all Banach nondegenerate homogeneous modules,
consisting of sequences, are extremely projective within the category
of Banach nondegenerate homogeneous modules. However, neither of
them, provided it is infinitedimensional, is extremely projective
within the category of all normed nondegenerate homogeneous modules.
On the other hand, submodules of these modules, consisting of finite
sequences, are extremely projective within the latter category.