On Hilbert C*-modules with Hilbert dual and C*-Fredholm operators

Vladimir Manuilov; Evgenij Troitsky

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arXiv · 2302.03760

On Hilbert C-modules with Hilbert dual and C-Fredholm operators

Vladimir Manuilov·Evgenij Troitsky

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Abstract

We study such Hilbert C*-modules over a C*-algebra $A$, that the Banach $A$-dual module carries a natural structure of Hilbert $A$-module. In this direction we prove that if $A$ is monotone complete, $M$ and $N$ are Hilbert $A$-modules, $M$ is self-dual, and both $T:M\to N$ and its Banach $A$-dual $T':N'\to M'$ have trivial kernels and cokernels then $M\cong N'$. With the help of this result, for a monotone complete $C^*$-algebra $A$, we prove that the index of any $A$-Fredholm operator can be calculated as the difference of its kernel and cokernel, as in the Hilbert space case.

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BibTeX RIS

Vladimir Manuilov, Evgenij Troitsky. 2023-02-07. On Hilbert C*-modules with Hilbert dual and C*-Fredholm operators. https://arxiv.org/abs/2302.03760

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