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On stability of index of Fredholm complexes on the \(C^*\)-algebra. (English) Zbl 1225.47130

This paper appears to be the same as [Proc.Jangjeon Math.Soc.9, No.2, 151–160 (2006; Zbl 1124.47301)]. The aim of the paper is to generalize parts of the theory of Fredholm complexes to the case of Hilbert C\(^*\)-modules. However, the paper is not carefully written and there are mathematical mistakes in addition to an unusually large number of typographical and grammatical errors. For instance, the proof of Theorem 2.2 requires the maps \(R_i\) to form a complex, a fact not following from the definition of a Fredholm complex given by the author. The proofs in Section 2 rely heavily on the (false) facts that a direct sum of Fredholm operators is Fredholm (not true even at the level of Hilbert spaces) and on the assumption that the compact operators form an ideal (which is not the case in general Hilbert C\(^*\)-modules). In Section 3, Definition 3.1 does not make sense the way it is stated, because the objects being added are morphisms of a compact space \(X\), with no obvious definition of addition in sight.

MSC:

47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
47A53 (Semi-) Fredholm operators; index theories
47C15 Linear operators in \(C^*\)- or von Neumann algebras
55N15 Topological \(K\)-theory

Citations:

Zbl 1124.47301
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References:

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