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Abstract and Applied Analysis
Volume 2006 (2006), Article ID 90837, 14 pages

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.


The maps of the form f(x)=i=1naixbi, 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.