Krein, Maria N.
January 2006
Abstract & Applied Analysis;2006, p1
Academic Journal
The maps of the form f (x) = ∑i=1n ai · 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.


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