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

TL;DR

This paper investigates Hilbert C*-modules with Hilbert duals over monotone complete C*-algebras, establishing isomorphisms and index calculations for A-Fredholm operators that generalize Hilbert space results.

Contribution

It proves that under certain conditions, the Banach A-dual of a Hilbert A-module is itself a Hilbert A-module and establishes an index formula for A-Fredholm operators.

Findings

Banach A-dual modules can carry a Hilbert A-module structure.

Isomorphism between a self-dual module and the dual of another module.

Index of A-Fredholm operators equals the difference of kernel and cokernel.

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.