Abstract and Applied Analysis
Volume 2006 (2006), Article ID 90837, 14 pages
doi:10.1155/AAA/2006/90837

The mappings of degree 1

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.