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Callias-Type Operators in von Neumann Algebras

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Abstract

We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorem for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah’s \(L^2\)-index theorem, which states that the index of a Callias-type operator on a non-compact manifold M is equal to the \(\Gamma \)-index of its lift to a Galois cover of M. We also prove the cobordism invariance of the index of Callias-type operators. In particular, we give a new proof of the cobordism invariance of the von Neumann index of operators on compact manifolds.

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Notes

  1. In other words in this section we assume that \(A=\mathbb {C}\). We use the notation S (rather than \(\Sigma \)) for the Dirac bundle, to stress the difference from the other sections where \(\Sigma \) was a Hilbert A-bundle.

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Acknowledgements

M. Braverman was supported in part by the NSF Grant DMS-1005888.

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  1. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Maxim Braverman & Simone Cecchini

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  1. Maxim Braverman
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  2. Simone Cecchini
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Correspondence to Maxim Braverman.

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Braverman, M., Cecchini, S. Callias-Type Operators in von Neumann Algebras. J Geom Anal 28, 546–586 (2018). https://doi.org/10.1007/s12220-017-9832-1

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