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

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 $T^{\mathrm{\prime }}:N^{\mathrm{\prime }}\to M^{\mathrm{\prime }}$ have trivial kernels and cokernels then $M\cong N^{\mathrm{\prime }}$. With the help of this result, for a monotone complete $C^{\ast }$-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.