Representation of ℬ, ℬ’ by Banach Spaces of Operators in a Hilbert Space

Abstract

In this chapter let us show that the irreducible parts ℬν, ℬ’ν (see VII § 5.4) permit a representation by Hilbert space operators, i.e. for ℬν, ℬ’ν there is a Hilbert space ℋν over the field R of real numbers or over the C of complex numbers or over the Q of quaternions where ℬν can be identified with the set of self-adjoint operators of the trace class and ℬ’ν with the set. of all bounded, self-adjoint operators, so that µ (x, y) = tr (x y). Then Kν is the set of all operators w ∈ ℬν with w ≧ 0, tr(w)= 1, while Lν is the set of all operators ɡ ∈ ℬ’ν with 0 ≦ɡ≦ 1