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

Abstract

In this chapter let us show that the irreducible parts ℬ_ν, ℬ’_ν (see VII § 5.4) permit a representation by Hilbert space operators, i.e. for ℬ_ν, ℬ’_ν there is a Hilbert space ℋ_ν over the field R of real numbers or over the C of complex numbers or over the Q of quaternions where ℬ_ν 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 ∈ ℬ_ν with w ≧ 0, tr(w)= 1, while L_ν is the set of all operators ɡ ∈ ℬ’_ν with 0 ≦ɡ≦ 1