With τ a state of A, let πτ denote the corresponding GNS representation on the Hilbert space Hτ.
Using the notation defined here, τ is ωx ∘ πτ for a suitable unit vector x(=xτ) in Hτ.
Each bounded linear functional ρ on Φ(A) is weak-operator continuous and extends uniquely preserving norm, to a weak-operator continuous linear functional ρ on the von Neumann algebra Φ(A)−.
Each ρ in A* can be uniquely expressed in the form ρ=ρu+ρs, with ρu ultraweakly continuous and ρs singular.
Let f and g be continuous, real-valued functions on C4m and C4n, respectively, σ1, σ2, ..., σm be ultraweakly continuous, linear functionals on a von Neumann algebra R acting on the Hilbert space H, and ρ1, ρ2, ..., ρn be bounded linear functionals on R such that, for each a in R, Then the above inequality holds if each ρj is replaced by its ultraweakly continuous component (ρj)u.