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Abstract:
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In the first section we present the theory on uniform spaces and measures of noncompactness in metric and uniform spaces. Next, we recall the basic concepts and properties of C∗ and W∗-algebras and Hilbert modules over these algebras with some known topologies on Hilbert W∗-module. In the second section we construct a local convex topology on the standard Hilbert module l2(A), such that any compact” operator (i.e., any operator in the norm closure of the linear span of the operators of the form maps bounded sets into totally bounded sets. In the biginning A presents unital W∗-algebra, leter on A presents unital C∗-algebra. The converse is true in the special case where A = B(H) is the full algebra of all bounded linear operators on a Hilbert space H. In the third section we define a measure of noncompactness λ on the standard Hilbert C∗-module l2(A) over a unital C∗-algebra, such that λ(E) = 0 if and only if E is A-precompact (i.e. it is ε-close to a finitely generated projective submodule for any ε > 0) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istratescu measure of noncompactnes on l2(A) regarded as a locally convex space with respect to a suitable topology. We obtain their properties as well as some relationships between them and above introduced measure of noncompactness. In the forth section we generalize the notion of a Fredholm operator to an arbitrary C∗-algebra. Namely, we define finite type elements in an axiomatic way, and also we define a Fredholm type element a as such an element of a given C∗-algebra for which there are finite type elements p and q such that (1−q)a(1−p) is invertible. We derive an index theorem for such operators. In subsection Corollaries we show that many well-known operators are special cases of our theory. Those include: classical Fredholm operators on a Hilbert space, Fredholm operators in the sense of Breuer, Atiyah and Singer on a properly infinite von Neumann algebra, and Fredholm operators on Hilbert C∗-modules over a unital C∗-algebra in the sense of Mishchenko and Fomenko. |