푊 *-algebras and noncommutative integration Ryszard Paweł Kostecki
Perimeter Institute of Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada* Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warszawa, Poland
October 27, 2014
Abstract This text is a detailed overview of the theories of 푊 *-algebras and noncommutative integration, up * to the Falcone–Takesaki theory of noncommutative 퐿푝 spaces over arbitrary 푊 -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 푊 *-dynamical systems (featuring derivations, liou- villeans, and crossed products), noncommutative Radon–Nikodým type theorems, and operator valued weights. We pay special attention to abstract algebraic formulation of all properties (avoid- ing 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.
Contents
1 Introduction 2 2 Algebras and functionals 3 2.1 퐶*-algebras ...... 3 2.2 Functionals ...... 6 2.3 Representations ...... 9 2.4 von Neumann algebras ...... 11 2.5 Barycentric decompositions ...... 12 3 Modular theory 15 3.1 Kubo–Martin–Schwinger condition ...... 16 3.2 Tomita–Takesaki modular theory ...... 18 3.3 Relative modular theory ...... 20 3.4 Canonical representation and bimodules ...... 24 3.5 Classification of 푊 *-algebras ...... 29 4 Automorphisms and their representations 32 4.1 Derivations ...... 33 4.2 Standard liouvilleans ...... 35 4.3 Crossed products ...... 37 4.4 Canonical core algebra ...... 41 5 Noncommutative integration 44 arXiv:1307.4818v5 [math.OA] 27 Oct 2014 5.1 Noncommutative Radon–Nikodým type theorems ...... 46 5.2 Integration relative to a trace ...... 50 5.3 Operator valued weights ...... 52 5.4 Integration relative to a weight ...... 56 5.5 Canonical noncommutative integration ...... 60 5.6 Integration relative to a measure ...... 66 5.7 Canonical commutative integration ...... 76 5.8 Categories of integration theory ...... 80 5.9 Commutative and noncommutative Orlicz spaces ...... 85 References 91 Index 108 List of symbols 111
*Current affiliation.
1 前は闇, 後ろは輝く星座というのが作用素環の世界です. 竹崎正道, 作用素環への入り口1 1 Introduction
While there exists a wide range of detailed expositions of various aspects of the theory of noncommu- tative algebras of operators, to mention only [264, 373, 127, 430, 128, 485, 577, 465, 50, 190, 538, 25, 539, 594, 399, 535, 250, 546, 576, 150, 357, 466, 625, 391, 106, 141, 559, 45], there is no self-contained text covering the theory of noncommutative integration over arbitrary 푊 *-algebras.2 Our exposition is intended to fill this gap.3 The theory of 푊 *-algebras is presented in an algebraic fashion, with elimination of dependence on the Hilbert spaces, and downplaying the usual noncommutative topo- logical (퐶*-algebraic) point of view in favour of exposition of the role played by predualisation and relative modular theory. Our presentation of noncommutative integration covers large parts of the theory that are otherwise scattered among many papers and books. In order to provide an overview that is detailed but also has a reasonable size, we have omitted proofs. Yet, we compensate for this by discussion of notions and results under consideration and by providing quite detailed references to original papers. The new mathematical results of this text are: the construction of the family of noncommutative * Orlicz spaces 퐿ϒ(풩 ) canonically associated with arbitrary 푊 -algebra 풩 and arbitrary Orlicz