$C^*$-Fredholm operators and polar decomposition type properties

We plan to recall the main properties of $A$-Fredholm operators (in Hilbert $C^*$-modules) and discuss the possibility of obtaining their index (for algebras close to $W^*$) in the classical form (as the difference of kernel and cokernel), that is necessary for non-commutative Hodge theory. Some new advances in this and related questions will be considered, based on a joint work with V.M.Manuilov https://arxiv.org/abs/2302.03760