## The Atkinson theorem in Hilbert $$C^{\ast}$$-modules over $$C^{\ast}$$-algebras of compact operators.(English)Zbl 1155.46026

Summary: The concept of unbounded Fredholm operators on Hilbert $$C^{\ast}$$-modules over an arbitrary $$C^{\ast}$$-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Fredholm operators on Hilbert $$C^{\ast}$$-modules over $$C^{\ast}$$-algebras of compact operators. In the framework of Hilbert $$C^{\ast}$$-modules over $$C^{\ast}$$-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same.

### MSC:

 46L08 $$C^*$$-modules 47A53 (Semi-) Fredholm operators; index theories
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### References:

 [1] J. M. Garcia-Bondía, J. C. Várilly, and H. Figueroa, Elements of Non-Commutative Geometry, Birkhäuser, Boston, Mass, USA, 2000. [2] N. E. Wegge-Olsen, k-Theory and C*-Algebras: A Friendly Approach, The Clarendon Press, Oxford University Press, New York, NY, USA, 1993. · Zbl 0780.46038 [3] E. C. Lance, Hilbert C*-Modules, vol. 210 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1995. · Zbl 0822.46080 [4] D. Bakić and B. Gulja\vs, “Hilbert C*-modules over C*-algebras of compact operators,” Acta Scientiarum Mathematicarum (Szegediensis), vol. 68, no. 1-2, pp. 249-269, 2002. · Zbl 1026.46039 [5] S. Baaj and P. Julg, “Théorie bivariante de Kasparov et opérateurs non bornés dans les C*-modules hilbertiens,” Comptes Rendus des Séances de l/Académie des Sciences. Série I. Mathématique, vol. 296, no. 21, pp. 875-878, 1983. · Zbl 0551.46041 [6] B. Booss-Bavnbek, M. Lesch, and J. Phillips, “Unbounded fredholm operators and spectral flow,” Canadian Journal of Mathematics, vol. 57, no. 2, pp. 225-250, 2005. · Zbl 1085.58018 [7] M. Joachim, “Unbounded Fredholm operators and k-theory,” in High-Dimensional Manifold Topology, pp. 177-199, World Science Publishing, River Edge, NJ, USA, 2003. · Zbl 1049.46051 [8] M. Magajna, “Hilbert C*-modules in which all closed submodules are complemented,” Proceedings of the American Mathematical Society, vol. 125, no. 3, pp. 849-852, 1997. · Zbl 0865.46038 [9] G. J. Murphy, C*-Algebras and Operator Theory, Academic Press, Boston, Mass, USA, 1990. · Zbl 0714.46041
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