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Almost Fredholm operators in von Neumann algebras

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Abstract

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF + be the left-Fredholm class ofA. We call almost left-Fredholm the class\(\tilde F_ + \) = {A εA: if P εA is a projection and AP εJ then P εJ}. Then\(F_ + \subset \tilde F_ + \) and the inclusion is proper unlessA is semifinite and has a non-large center.\(\tilde F_ +\) satisfies all of the algebraic properties ofF + but it is generally not open. IfA is semifinite then A ε\(\tilde F_ +\) iff there are central projectionsG γ with ∑Gγ = I such that AGγ ε F+(AGγ). Let π:A → A/J. Then the left almost essential spectrum ofA ε A,\(\tilde \sigma _\ell (A) = \{ \gamma \varepsilon \mathbb{C}: A - \gamma {\rm I} \varepsilon | \tilde F_ +\), coincides with the set of eigenvalues of π(A)

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, 45221, Cincinnati, OH, USA

    Victor Kaftal

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  1. Victor Kaftal
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Kaftal, V. Almost Fredholm operators in von Neumann algebras. Integr equ oper theory 5, 50–70 (1982). https://doi.org/10.1007/BF01694029

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