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

  • Anwar A. Irmatov (a1) and Alexandr S. Mishchenko (a2)

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).

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3. M. Frank , A set of maps from K to EndA(l2(A)) isomorphic to EndA(K)(l2(A(K))). Applications, Ann. Global Anal. Geom. 3 (1985), 155171

4. A. Irmatov , On a New Topology in the Space of Fredholm Operators, Ann. Global. Anal. Geom. 7 (2) (1989), 93106

5. K. Jänich , Vektorraumbündel und der Raum der Fredholm-Operatoren, Math.Ann. 161 (1965), 129142

9. M. Reed and B. Simon , Methods of Modern Mathematical Phisics, V.1. Academic Press, New York, London, (1972)

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Journal of K-Theory
  • ISSN: 1865-2433
  • EISSN: 1865-5394
  • URL: /core/journals/journal-of-k-theory
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