Diagonalization of compact operators in Hilbert modules over finiteW*-algebras

It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH*_A dual toH_A under slight restrictions one can find a set of eigenvectorsx_i H*_A and a non-increasing sequence of eigenvalues _i A such thatK x_i=x_{i i} and the selfdual HilbertA-module generated by these eigenvectors is the wholeH*_A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA and discuss the possibility of its diagonalization.