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arXiv · 1602.06873

Callias-type operators in von Neumann algebras

Maxim Braverman·Simone Cecchini
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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 theorems 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 $Γ$-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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Maxim Braverman, Simone Cecchini. 2016-02-22. Callias-type operators in von Neumann algebras. https://arxiv.org/abs/1602.06873

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