The mappings of degree 1

The maps of the form f ( x ) = ∑ i = 1 n a i ⋅ x ⋅ b i , 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 l 2 ( 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 l 2 ( 𝒜 ) and a Fredholm manifold modelled by l 2 ( 𝒜 ) . 1-degree maps are also applied to some problems of Markov chains.