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Algebras and Noncommutative Integration Ryszard Paweł Kostecki

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W *-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 [email protected] October 27, 2014 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, 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 C*-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 W *-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 M/&, 4M/輝く星Íというのがù用d環.ã界です. 6õ正., ù用d環8.qJL1 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 W *-algebras.2 Our exposition is intended to fill this gap.3 The theory of W *-algebras is presented in an algebraic fashion, with elimination of dependence on the Hilbert spaces, and downplaying the usual noncommutative topo- logical (C*-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 Lϒ(N ) canonically associated with arbitrary W -algebra N and arbitrary Orlicz function ϒ, the construction of the family of commutative Lp(A) spaces canonically associated with arbitrary mcb-algebra4 A for p 2 [1; 1[, as well as Propositions 5.1-5.9. The category theoretic formulation of equivalences (541)-(543), (552), (597), (598), (603), (609), and the diagram (540) are also new. These results allow us to establish an explicit relationship between noncommutative and commutative integration theories without passing to topological representations (see Section 5.8), and to provide the category theoretic description of the main structures of integration theory in terms of the diagram (616). Aknowledgments: I am very indebted to Professor Stanisław L. Woronowicz for numerous discussions that introduced me to the worlds of modular theory and noncommutative integration theory. I would like also to thank Jan Dereziński, Wojciech Kamiński, Jerzy Kijowski, Władysław Adam Majewski, Mustafa Muratov, Włodzimierz Natorf, Dmitri˘ıPavlov, Aleksander Pełczyński, David Sherman, and Piotr Sołtan for valuable comments, discussions, xerocopies, and/or correspondence. Naturally, all eventual drawbacks of this text are my own responsibility. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. Conventions: i, e, π, R, C, Z are standard; R+ ≡ [0; 1[, N ≡ f1; 2;:::g; id denotes identity morphism in the category of objects of a given kind, dom denotes domain, cod denotes codomain, ran denotes image, span K denotes linear span in the vector space over K 2 fR; Cg, int denotes interior of subspace of a topological space, Set denotes the category of sets and functions; given a small category C , Ob(C ) denotes the set of all objects in C , Mor(C ) denotes the set of all arrows in C , while HomC (X; Y ) denotes the set of all arrows between X; Y 2 Ob(C ); 훿ij denotes Kronecker’s delta; sp denotes spectrum of an operator on a Hilbert space. All Cyrillic titles and names were transliterated from the original papers and books. For the Latin transliteration of the Cyrillic script (in references and surnames) we use the following modification of the system GOST 7.79- 2000B: c = c, q = ch, h = kh, = zh, x = sh, w = shh, = yu, = ya, e = ë, = ‘, ~ = ’, = è, $i = ˘ı, with an exception that names beginning with H are transliterated to H.5 For Russian texts: y = y, i = i; for Ukrainian: i = y, i = i, ï = ï. 1«It is in the world of operator algebras that the forefront is dark and behind is a twinkling constellation.» Masamichi Takesaki, Entrance to operator algebras [561]. 2The canonical references for the theory of W *-algebras are [465, 535, 559], and we definitely recommend them. 3On this occasion, we fill some gaps in the literature, clarifying the functorial character of some constructions. The category theoretic formulation of equivalences (187), (502), and the diagrams (466), (503) seem to be new. 4That is, the Dedekind–MacNeille complete boolean algebras admitting a strictly positive semi-finite countably additive measure (see Section 5.6 for definitions of these terms). 5This is required for agreement with the widespread practice to transliterate Holevo as Holevo, etc. 2 2 Algebras and functionals The theory of abstract C*- and W *- algebras emerged from the works of von Neumann and Murray [603, 358, 610, 359, 614, 360, 615] on von Neumann algebras, and the works of Gel’fand, Na˘ımark, Ra˘ıkov and Shilov on Banach algebras and C*-algebras [167, 169, 168, 174, 170, 172, 426, 173, 171]. The most important results obtained at that early stage (up to 1950) were von Neumann’s reduc- tion theorem [615] allowing to represent separable all von Neumann algebras in terms of factor von Neumann algebras, Gel’fand–Na˘ımark[170] representation theory for commutative C*-algebras, and its generalisation to noncommutative C*-algebras by Segal [487]. Among key results obtained later (up to early 1970s) within the frames of a general theory, it is necessary to mention Sakai’s charac- terisation of W *-algebras in purely algebraic terms [454], development of the theory of weights by Combes and Pedersen [87, 88, 89, 91, 90, 92, 394, 395, 396, 397, 398], and development of the the- ory of integral decomposition of functionals on C*-algebras by Tomita [567, 568], Ruelle [447, 448], Sakai [459, 465], Wils [634, 635, 636] and others [490, 269, 516] using the Choquet theory of boundary integrals [83, 80, 81, 82, 44, 84]. 2.1 C*-algebras * A C -algebra [170] is defined as an algebra C over the field C with unit I, equipped with: (i) an operation * : C!C satisfying * * * * * * * * * * * (xy) = y x ; (x + y) = x + y ; (x ) = x; (휆푥) = 휆 x 8x; y 2 C 8휆 2 C; (1) * with the action of on 휆 2 C given by standard complex conjugation, * * : C 3 휆 = (re (휆) + i · im(휆)) 7! 휆 := (re (휆) − i · im(휆)) 2 C; (2) (ii) a norm map j|·|j : C! [0; +1[ such that C is a Banach space [29] over C with norm j|·|j, and jjx*xjj = jjxjj2 8x 2 C; (3) or, equivalently [383, 177], jjx*xjj = jjx*jjjjxjj 8x 2 C: (4) * * The properties jjIjj = 1 and jjx yjj ≤ jjxjjjjy jj 8x; y 2 C follow from (3)[19] (see also [326]), ensuring * continuity of and of multiplication in the topology generated by j|·|j. Algebra over C that is equipped only with the map * defined as above (but with no norm) is called a *-algebra, while the algebra over * C that is equipped only with the norm j|·|j such that it becomes a Banach space (but with no map) is called a Banach algebra [363]. A *-algebra equipped with a structure of a topological vector space such that the operations · and * are continuous will be called topological [595]. (It is possible to * develop a theory of C -algebras without unit I, as well as the theory of Banach *-algebras, defined as *-algebras that are also Banach spaces and satisfy jjx*jj = jjxjj and jjxyjj ≤ jjxjjjjyjj instead of (3). Such theories, however, will not be useful for our purposes.) The most remarkable property of C*-algebras follows from the condition (3): their norm structure is uniquely determined by their algebraic structure [170], which follows from the property6 p jjxjj = sup fj휆j j 휆 2 C; (x*x − 휆I) is not invertibleg 8x 2 C: (5) Let 휆1; 휆2 2 C be arbitrary.
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  • On the Lefschetz Problem in Non Commutative Geometry

    On the Lefschetz Problem in Non Commutative Geometry

    EXPOSITIONES Expo. Math. 21 (2003): 1-31 © Urban & Fischer Verlag MATHEMATICAE www.u rba nfischer.de/journals/expomath On the Lefschetz Problem in Non Commutative Geometry Moulay-Tahar Benameur Institut Girard Desargues, Universit6 Claude Bernard, Lyon I, 43, Bd du 11-novembre- 1918, 69622 Villeurbanne Cedex, France Abstract: In this paper, we state a Lefschetz problem in Noncommutative geometry. The main tool is a new notion of equivariant cyclic cohomology and its well defined index pairing with equivariant K-theory. We also inves- tigate invariant Connes-von Neumann spectral triples involving some new type II geometric situations. Keywords: C*-algebras, K-theory, Lefschetz, equivariant, cyclic homology. 1. INTRODUCTION In the classical work of Atiyah, Bott and Segal on Lefschetz formu- lae for isometries, the equivariant index theorem plays a fundamental E-mail address: [email protected] 0732-0869/03/21/1-01 $15.00/0 2 M.-T. Benameur part, see [2, 4]. On the other hand, index theory has benefited from Connes' Noncommutative Geometry point of view and local index theo- rems have been proven in many new singular situations, see for instance [22, 23, 11, 32]. The aim of the present paper is to show how equivari- ant index theory can be extended to yield higher Lefschetz formulae, in such singular situations [6, 7]. In the process, we also give a defini- tion of the equivariant Chern-Connes character which is appropriate for Lefschetz formulae and which is closely related with previously known definitions, see for instance [30] or [33]. Moreover, one can include in the present discussion some new results based on Segal's abstract in- tegration theory and the continuous Murray-von Neumann dimension theory [10].