W*-algebras and noncommutative integration

Abstract

This text is a detailed overview of the theories of W*-algebras and noncommutative integration, up to the Falcone-Takesaki theory of noncommutative Lp spaces over arbitrary W*-algebras, and its extension to noncommutative Orlicz spaces. The topics under consideration include the Tomita-Takesaki modular theory, the relative modular theory (featuring bimodules, spatial quotients, and canonical representation), the theory of W*-dynamical systems (featuring derivations, liouvilleans, and crossed products), noncommutative Radon-Nikodym type theorems, and operator valued weights. We pay special attention to abstract algebraic formulation of all properties (avoiding the dependence on Hilbert spaces wherever it is possible), to functoriality of canonical structures arising in the theory, and to the relationship between commutative and noncommutative integration theories. Several new results are proved.

111 pages, 671 references, contains index

References in corpus (9)

• Remote preparation of arbitrary ensembles and quantum bit commitment
• Noncommutative Lp modules
• Noncommutative Lp structure encodes exactly Jordan structure
• On the structure of isometries between noncommutative Lp spaces
• Orlicz Spaces associated with a Semi-Finite Von Neumann Algebra
• Quantum Orlicz spaces in information geometry
• A new proof of the noncommutative Banach-Stone theorem
• Quantum L_p and Orlicz spaces
• A classification for 2-isometries of noncommutative Lp-spaces

Cited by in corpus (4)

• Lüders' and quantum Jeffrey's rules as entropic projections
• Local quantum information dynamics
• Noncommutative Orlicz spaces over W*-algebras
• Contractively decomposable projections on noncommutative ${\mathrm{L}}^p$-spaces