Abstract and Applied Analysis
Volume 2006 (2006), Article ID 90837, 14 pages
doi:10.1155/AAA/2006/90837
The mappings of degree 1
Faculty of Physics and
Mathematics, Lipetsk State Pedagogical University, ul.
Lenina 42, Lipetsk 398020, Russia
Received 26 June 2005; Accepted 1 July 2005
Copyright © 2006 Maria N. Krein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The maps of the form f(x)=∑i=1nai⋅x⋅bi,
called 1-degree maps, are introduced and investigated. For
noncommutative algebras and modules over them 1-degree maps give
an analogy of linear maps and differentials. Under some conditions
on the algebra 𝒜, contractibility of the group of
1-degree isomorphisms is proved for the module l2(𝒜).
It is shown that these conditions are fulfilled for the algebra of
linear maps of a finite-dimensional linear space. The notion of
1-degree map gives a possibility to define a nonlinear Fredholm
map of l2(𝒜) and a Fredholm manifold modelled by
l2(𝒜). 1-degree maps are also applied to some
problems of Markov chains.