The Tomita-Takesaki Theory

Abstract

Section 2.1 discusses the following question (which, in the case M = L^∞(X, F,μ) with μ finite, is answered affirmatively by the existence of the Lebesgue integral): if m: P(M) → [0,1] is countably additive in the sense that

$$ m\left[ {\mathop{V}\limits_{{n = 1}}^{\infty } {e_n}} \right] = \sum\limits_{{n = 1}}^{\infty } {m({e_n})} $$

for any countable collection of pairwise orthogonal projections in M, does m extend to a linear functional on M which is well-behaved under monotone convergence?