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On Compact and Fredholm Operators over C*-algebras and a New Topology in the Space of Compact Operators

Published online by Cambridge University Press:  17 April 2008

Anwar A. Irmatov  and
Alexandr S. Mishchenko
Anwar A. Irmatov
Affiliation:
irmatov@mech.math.msu.suDept. of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia
Alexandr S. Mishchenko
Affiliation:
asmish@higeom.math.msu.suDept. of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia
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Abstract

It is well-known that bounded operators in Hilbert C*-modules over C*-algebras may not be adjointable and the same is true for compact operators. So, there are two analogs for classical compact operators in Hilbert C*-modules: adjointable compact operators and all compact operators, i.e. those not necessarily having an adjoint.

Classical Fredholm operators are those that are invertible modulo compact operators. When the notion of a Fredholm operator in a Hilbert C*-module was developed in [6], the first analog was used: Fredholm operators were defined as operators that are invertible modulo adjointable compact operators.

In this paper we use the second analog and develop a more general version of Fredholm operators over C*-algebras. Such operators are defined as bounded operators that are invertible modulo the ideal of all compact operators. The main property of this new class is that a Fredholm operator still has a decomposition into a direct sum of an isomorphism and a finitely generated operator.

The special case of Fredholm operators (in the sense of [6]) over the commutative C*-algebra C(K) of continuous functions on a compact topological space K was also considered in [2]. In order to describe general Fredholm operators (invertible modulo all compact operators over C(K)) we construct a new IM-topology on the space of compact operators on a Hilbert space such that continuous families of compact operators generate the ideal of all compact operators over C(K).

Keywords

C*-algebrasFredholm operatorsCompact operatorsIndex of Fredholm operator
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Special Issue 2: In Memory of Yurii Petrovich Solovyev October 8, 1944 – September 11, 2003 , October 2008 , pp. 329 - 351
Copyright
Copyright © ISOPP 2008

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