Hilbert C*-Modules with Hilbert Dual and C*-Fredholm Operators

We study Hilbert C*-modules over a C*-algebra A for which 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? N and its Banach A-dual T' : N'? M' have trivial kernels and cokernels then M ? 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.