The mappings of degree 1

The maps of the form f(x)=∑i=1nai⋅x⋅bi,called 1-degree maps, are introduced and investigated. Fornoncommutative algebras and modules over them 1-degree maps givean analogy of linear maps and differentials. Under some conditionson the algebra , contractibility of the group of1-degree isomorphisms is proved for the module l2().It is shown that these conditions are fulfilled for the algebra oflinear maps of a finite-dimensional linear space. The notion of1-degree map gives a possibility to define a nonlinear Fredholmmap of l2() and a Fredholm manifold modelled byl2(). 1-degree maps are also applied to someproblems of Markov chains