Non-commutative functional calculus in finite type I von Neumann algebras

Abstract

A certain class of matrix-valued Borel matrix functions is introduced. It is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f1, f2,… of functions that converge pointwise to 0 transform into sequences f1[T], f2[T],… of operators in T that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T that acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all functions from the aforementioned class. Some conclusions are drawn concerning so-called operator-spectra of such operators and a new variation of the spectral theorem is formulated.