Diagonalization of compact operators in Hilbert modules over finite W ∗ - algebras

It is known that a continuous family of compact operators can be diagonalized pointwise. One can consider this fact as a possibility of di-agonalization of the compact operators in Hilbert modules over a com-mutative W *-algebra. The aim of the present paper is to generalize this fact for a finite W *-algebra A not necessarily commutative. We prove that for a compact operator K acting in the right Hilbert A-module H * A dual to H A under slight restrictions one can find a set of " eigenvec-tors " x i ∈ H * A and a non-increasing sequence of " eigenvalues " λ i ∈ A such that K x i = x i λ i and the autodual Hilbert A-module generated by these " eigenvectors " is the whole H * A. As an application we consider the Schrödinger operator in magnetic field with irrational magnetic flow as an operator acting in a Hilbert module over the irrational rotation algebra A θ and discuss the possibility of its diagonalization.