Noncommutative Integration and Operator Theory

This chapter presents the basic theory of measurable and \(\tau \)-measurable operators affiliated with a semi-finite von Neumann algebra. Particular attention is given to the properties of the measure topology and the order structure in spaces of \(\tau \)-measurable operators. Properties of operator functions on these spaces are studied in some detail.

This chapter is a detailed study of properties of the generalized singular value function of a \(\tau \)-measurable operator. Particular attention is given to the notion of submajorization (in the sense of Hardy–Littlewood–Polya), and several basic submajorization inequalities are obtained. Noncommutative \(L_1\) and \(L_2\)-spaces are introduced, together with fundamental convergence theorems. In particular, noncommutative versions of the dominated convergence theorem and Fatou’s lemma are presented. The chapter concludes with a discussion of contractions in the noncommutative pair \((L_1, L_\infty )\).

This chapter exhibits the theory of strongly symmetric spaces of \(\tau \)-measurable operators (without any additional assumptions on the underlying semi-finite von Neumann algebra). This includes, in particular, the class of fully symmetric spaces. Duality theory is discussed in detail and a version of the fundamental Hewitt–Yosida decomposition is presented. Several characterizations of order continuity of the norm are given.

This chapter presents several interpolation theorems in the setting of noncommutative spaces introduced in previous chapters. In particular, several classical interpolation results are obtained in the noncommutative setting, including noncommutative versions of theorems of Marcinkiewicz, Calderón, and Boyd. Applications are given of the complex method to the geometry of noncommutative \(L_p\)-spaces.