DOCSLIB.ORG

Algebras and Noncommutative Integration Ryszard Paweł Kostecki

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebras and Noncommutative Integration Ryszard Paweł Kostecki 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.
Load more

Recommended publications
  • Map Composition Generalized to Coherent Collections of Maps
    Front. Math. China 2015, 10(3): 547–565 DOI 10.1007/s11464-015-0435-5 Map composition generalized to coherent collections of maps Herng Yi CHENG1, Kang Hao CHEONG1,2 1 National University of Singapore High School of Mathematics and Science, Singapore 129957, Singapore 2 Tanglin Secondary School, Singapore 127391, Singapore c Higher Education Press and Springer-Verlag Berlin Heidelberg 2015 Abstract Relation algebras give rise to partial algebras on maps, which are generalized to partial algebras on polymaps while preserving the properties of relation union and composition. A polymap is defined as a map with every point in the domain associated with a special set of maps. Polymaps can be represented as small subcategories of Set∗, the category of pointed sets. Map composition and the counterpart of relation union for maps are generalized to polymap composition and sum. Algebraic structures and categories of polymaps are investigated. Polymaps present the unique perspective of an algebra that can retain many of its properties when its elements (maps) are augmented with collections of other elements. Keywords Relation algebra, partial algebra, composition MSC 08A02 1 Introduction This paper considers partial algebras on maps (similar to those defined in [15]), constructed in analogy to relation algebras [5,12,16]. The sum of two maps f0 : X0 → Y0 and f1 : X1 → Y1 is defined if the union of their graphs is a functional binary relation R ⊆ (X0 ∪ X1) × (Y0 ∪ Y1), in which case the sum is the map (f0 + f1): X0 ∪ X1 → Y0 ∪ Y1 with graph R. This is defined similarly to the union operation + from [10].
    [Show full text]
  • LOW-DIMENSIONAL UNITARY REPRESENTATIONS of B3 1. Introduction Unitary Braid Representations Have Been Constructed in Several
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 9, Pages 2597{2606 S 0002-9939(01)05903-2 Article electronically published on March 15, 2001 LOW-DIMENSIONAL UNITARY REPRESENTATIONS OF B3 IMRE TUBA (Communicated by Stephen D. Smith) Abstract. We characterize all simple unitarizable representations of the braid group B3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ1 and σ2 denote the two generating twists of B3,thenasimple representation ρ : B3 ! GL(V )(fordimV ≤ 5) is unitarizable if and only if the eigenvalues λ1,λ2;::: ;λd of ρ(σ1) are distinct, satisfy jλij =1and (d) ≤ ≤ (d) µ1i > 0for2 i d,wheretheµ1i are functions of the eigenvalues, explicitly described in this paper. 1. Introduction Unitary braid representations have been constructed in several ways using the representation theory of Kac-Moody algebras and quantum groups (see e.g. [1], [2], and [4]), and specializations of the reduced Burau and Gassner representations in [5]. Such representations easily lead to representations of PSL(2; Z)=B3=Z ,where Z is the center of B3,andPSL(2; Z)=SL(2; Z)={1g,where{1g is the center of SL(2; Z). We give a complete classification of simple unitary representations of B3 of dimension d ≤ 5 in this paper. In particular, the unitarizability of a braid rep- resentation depends only on the the eigenvalues λ1,λ2;::: ,λd of the images of the two generating twists of B3. The condition for unitarizability is a set of linear in- equalities in the logarithms of these eigenvalues. In other words, the representation is unitarizable if and only if the (arg λ1; arg λ2;::: ;arg λd) is a point inside a poly- hedron in (R=2π)d, where we give the equations of the hyperplanes that bound this polyhedron.
    [Show full text]
  • LOCAL-GLOBAL PRINCIPLE for TRANSVECTION GROUPS 1. Introduction in 1956, J-P. Serre Asked If a Finitely Generated Projective Modu
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 4, April 2010, Pages 1191–1204 S 0002-9939(09)10198-3 Article electronically published on November 20, 2009 LOCAL-GLOBAL PRINCIPLE FOR TRANSVECTION GROUPS A. BAK, RABEYA BASU, AND RAVI A. RAO (Communicated by Martin Lorenz) Abstract. In this article we extend the validity of Suslin’s Local-Global Prin- ciple for the elementary transvection subgroup of the general linear group GLn(R), the symplectic group Sp2n(R), and the orthogonal group O2n(R), where n>2, to a Local-Global Principle for the elementary transvection sub- group of the automorphism group Aut(P ) of either a projective module P of global rank > 0 and constant local rank > 2, or of a nonsingular symplec- tic or orthogonal module P of global hyperbolic rank > 0 and constant local hyperbolic rank > 2. In Suslin’s results, the local and global ranks are the same, because he is concerned only with free modules. Our assumption that the global (hyperbolic) rank > 0 is used to define the elementary transvec- tion subgroups. We show further that the elementary transvection subgroup ET(P )isnormalinAut(P ), that ET(P )=T(P ), where the latter denotes the full transvection subgroup of Aut(P ), and that the unstable K1-group K1(Aut(P )) = Aut(P )/ET(P )=Aut(P )/T(P ) is nilpotent by abelian, pro- vided R has finite stable dimension. The last result extends previous ones of Bak and Hazrat for GLn(R), Sp2n(R), and O2n(R). An important application to the results in the current paper can be found in a preprint of Basu and Rao in which the last two named authors studied the decrease in the injective stabilization of classical modules over a nonsingular affine algebra over perfect C1-fields.
    [Show full text]
  • Duals of Quantum Semigroups with Involution
    DUALS OF QUANTUM SEMIGROUPS WITH INVOLUTION YULIA KUZNETSOVA Abstract. We define a category QSI of quantum semigroups with involution which carries a corep- c resentation-based duality map M 7→ M. Objects in QSI are von Neumann algebras with comultipli- cation and coinvolution, we do not suppose the existence of a Haar weight or of a distinguished spatial realisation. In the case of a locally compact quantum group G, the duality b in QSI recovers the uni- \∞ u b ∗∗ u\∗∗ \∞ b u ∗∗ u\b ∗∗ versal duality of Kustermans: L (G) = C0 (G) = C0 (G) , and L (G) = C0 (G) = C0 (G) . Other various examples are given. 1. Introduction In the theory of locally compact groups, a crucial role is played by the Haar measure; in the theory of locally compact quantum groups, its place it taken by a pair of left and right Haar weights. It is known that in the classical case, the existence of a Haar measure is a theorem, whereas in the quantum case this is a part of axioms. The dual of a locally compact quantum group in the setting of Kustermans and Vaes [8] is defined via an explicit construction involving the Haar weight; in the setting of Woronowicz [17], the spatial realisation of the multiplicative unitary gives readily the corresponding pair of quantum group algebras in duality. This should be compared with the classical Pontryagin duality: we consider first the group G of continuous characters of G, which is meaningful for any topological group, and then prove that if G is locally compact abelian, then so is G, and G is isomorphic to G.
    [Show full text]
  • C*-Algebras of Matricially Ordered *-Semigroups
    C*-algebras of Matricially Ordered *-Semigroups Berndt Brenken COSy 2014 A map of a C*-algebra defined via an implementing partial isometry does not preserve algebra structure. It is, however, a completely positive *-linear map. We consider *-semigroups S, matricial partial order orders on S; along with a universal C*-algebra associated with S and a matricial ordering on S. Preface Universal C*-algebras involving an automorphism realized via an implementing unitary, or an endomorphism via an isometry, have played a fundamental role in operator algebras. Such maps preserve algebraic structure. We consider *-semigroups S, matricial partial order orders on S; along with a universal C*-algebra associated with S and a matricial ordering on S. Preface Universal C*-algebras involving an automorphism realized via an implementing unitary, or an endomorphism via an isometry, have played a fundamental role in operator algebras. Such maps preserve algebraic structure. A map of a C*-algebra defined via an implementing partial isometry does not preserve algebra structure. It is, however, a completely positive *-linear map. Preface Universal C*-algebras involving an automorphism realized via an implementing unitary, or an endomorphism via an isometry, have played a fundamental role in operator algebras. Such maps preserve algebraic structure. A map of a C*-algebra defined via an implementing partial isometry does not preserve algebra structure. It is, however, a completely positive *-linear map. We consider *-semigroups S, matricial partial order orders on S; along with a universal C*-algebra associated with S and a matricial ordering on S. The resulting Cuntz-Pimsner C*-algebra for this example is the universal C*-algebra P generated by a partial isometry.
    [Show full text]
  • A Characterization of the Galois Subbialgebras L-&(K/F)
    JOURNAL OF ALGEBRA 42, 31.5-362 (1976) A Characterization of the Galois Subbialgebras l-&(K/F) MITSUHIRO TAKEUCHI Liniversity of Tsukuba, Iboraki, 300-31 Japan Communicated by N. Jacobson Received April 30, 1975 Let K/k be an extension of fields of characteristic p > 0. Sweedler [6] defines a pointed cocommutative k-bialgebra H,(K) and a k-linear map W: H,(K) @ K - K such that (i) w meaSures K to K, i.e., w(a @ I) = e(a)1 and da 0 W L C 4ah) 0 4 4w 0 4 (a) for a E H,(K), h, p E K, (ii) K is a left H,(K)-module through w. The pair (H,(K), W) is characterized by some universality. We call H,(K) the GaZoi> k-bialgebra of K/k. When H is a subbialgebra of H,(K) put KH = {A E K / w(a @ A) = E(a)& Va E H}, which is a subfield of K containing k. Conversely if K 3 F 3 k are fields, there is a unique maximal subbialgebra H = H,(K/F) such that F C KH. We call H,(K/F) the Galois subbialgebra associated with F. In order to establish the Galois correspondence between the set of inter- mediate fields K r) F 3 k and the set of subbialgebras H of H,(K): F * H,(W) and KHclH we must solve the following two kinds of problems: (i) to characterize the subfields F of the form KH, (ii) to characterize the Galois subbialgebras H = H,(K/F).
    [Show full text]
  • Introduction to Representation Theory by Pavel Etingof, Oleg Golberg
    Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina with historical interludes by Slava Gerovitch Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Contents Chapter 1. Introduction 1 Chapter 2. Basic notions of representation theory 5 x2.1. What is representation theory? 5 x2.2. Algebras 8 x2.3. Representations 9 x2.4. Ideals 15 x2.5. Quotients 15 x2.6. Algebras defined by generators and relations 16 x2.7. Examples of algebras 17 x2.8. Quivers 19 x2.9. Lie algebras 22 x2.10. Historical interlude: Sophus Lie's trials and transformations 26 x2.11. Tensor products 30 x2.12. The tensor algebra 35 x2.13. Hilbert's third problem 36 x2.14. Tensor products and duals of representations of Lie algebras 36 x2.15. Representations of sl(2) 37 iii Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms iv Contents x2.16. Problems on Lie algebras 39 Chapter 3. General results of representation theory 41 x3.1. Subrepresentations in semisimple representations 41 x3.2. The density theorem 43 x3.3. Representations of direct sums of matrix algebras 44 x3.4. Filtrations 45 x3.5. Finite dimensional algebras 46 x3.6. Characters of representations 48 x3.7. The Jordan-H¨oldertheorem 50 x3.8. The Krull-Schmidt theorem 51 x3.9.
    [Show full text]
  • Causal Symmetric Spaces Geometry and Harmonic Analysis Joachim Hilgert Gestur ´Olafsson
    Causal Symmetric Spaces Geometry and Harmonic Analysis Joachim Hilgert Gestur Olafsson´ Contents Preface viii Introduction x 1 Symmetric Spaces 1 1.1 BasicStructureTheory .. .. .. .. .. .. .. .. .. 1 1.2 DualSymmetricSpaces .. .. .. .. .. .. .. .. .. 7 1.2.1 The c-Dual Space ˜ c ................. 7 M 1.2.2 The Associated Dual Space a ........... 9 M 1.2.3 The Riemannian Dual Space r ........... 9 M 1.3 The Module Structure of To(G/H).............. 12 1.4 A-Subspaces........................... 22 1.5 TheHyperboloids........................ 24 2 Causal Orientations 29 2.1 ConvexConesandTheirAutomorphisms . 29 2.2 CausalOrientations . .. .. .. .. .. .. .. .. .. 39 2.3 Semigroups ........................... 43 2.4 TheOrderCompactification. 45 2.5 Examples ............................ 50 2.5.1 TheGroupCase .................... 50 2.5.2 TheHyperboloids . .. .. .. .. .. .. .. .. 51 2.6 Symmetric Spaces Related to Tube Domains . 52 2.6.1 BoundaryOrbits .. .. .. .. .. .. .. .. .. 56 2.6.2 The Functions Ψm ................... 58 2.6.3 The Causal Compactification of .......... 63 M 2.6.4 SU(n,n)......................... 65 2.6.5 Sp(n, R)......................... 68 v vi CONTENTS 3 Irreducible Causal Symmetric Spaces 71 3.1 ExistenceofCausalStructures . 71 3.2 The Classificationof Causal Symmetric Pairs . 83 4 Classification of Invariant Cones 91 4.1 Symmetric SL(2, R)Reduction ................ 91 4.2 TheMinimalandMaximalCones. 98 4.3 TheLinearConvexityTheorem . 105 4.4 TheClassification. 110 4.5 ExtensionofCones. 115 5 The Geometry 120 5.1 The Bounded Realization of H/H K ........... 121 ∩ 5.2 The Semigroup S(C)...................... 126 5.3 TheCausalIntervals . 130 5.4 CompressionSemigroups. 132 5.5 TheNonlinearConvexityTheorem . 143 5.6 The B]-Order.......................... 152 5.7 The Affine Closure of B] .................... 157 6 The Order Compactification 172 6.1 CausalGaloisConnections.
    [Show full text]
  • Group Von Neumann Algebras and Related Algebras
    Group von Neumann Algebras and Related Algebras Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fachbereiche der Georg-August-Universit¨at zu G¨ottingen vorgelegt von Holger Reich aus Uelzen G¨ottingen 1998 D7 Referent: Prof. Dr. W. L¨uck Korreferent: Prof. Dr. T. tom Dieck Tag der m¨undlichen Pr¨ufung: Contents 1 Introduction 1 1.1 NotationsandConventions . 4 2 The Algebra of Operators Affiliated to a Finite von Neumann Algebra 5 3 Dimensions 10 3.1 -modules, -modules and Hilbert -modules .......... 10 3.2 A Notion ofU Dimension for -modulesA ............... 15 3.3 The Passage from -modulesU to -modules ............ 19 A U 4 New Interpretation of L2-Invariants 24 5 The Atiyah Conjecture 26 5.1 TheAtiyahConjecture. 26 5.2 AStrategyfortheProof. 29 5.3 The Relation to the Isomorphism Conjecture in Algebraic K-theory 32 6 Atiyah’s Conjecture for the Free Group on Two Generators 37 6.1 FredholmModules .......................... 37 6.2 Some Geometric Propertiesofthe Free Group . 39 6.3 ConstructionofaFredholmModule . 40 7 Classes of Groups and Induction Principles 43 7.1 ElementaryAmenableGroups. 43 7.2 Linnell’s Class ........................... 45 C 8 Linnell’s Theorem 52 8.1 Induction Step: Extensions - The (A)-Part . 55 8.2 Induction Step: Extensions - The (B)-Part . 69 8.3 InductionStep: DirectedUnions . 71 8.4 Starting the Induction - Free Groups . 74 8.5 Starting the Induction - Finite Extensions of Free Groups .... 75 9 Homological Properties and Applications 77 10 K-theory 81 10.1 K0 and K1 of and ........................ 81 10.2 LocalizationSequences.A U . 85 11 Appendix I: Affiliated Operators 87 12AppendixII:vonNeumannRegularRings 95 i 13 Appendix III: Localization of Non-commutative Rings 98 13.1 OreLocalization ..........................
    [Show full text]
  • Unbounded Normal Operators in Octonion Hilbert Spaces and Their
    Unbounded normal operators in octonion Hilbert spaces and their spectra Ludkovsky S.V. 15 February 2012 Abstract Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investi- gated. 1 1 Introduction Unbounded normal operators over the complex field have found many-sided applications in functional analysis, differential and partial differential equa- tions and their applications in the sciences [4, 11, 12, 14, 31]. On the other hand, hypercomplex analysis is fast developing, particularly in relation with problems of theoretical and mathematical physics and of partial differential equations [2, 7, 9]. The octonion algebra is the largest division real algebra arXiv:1204.1554v1 [math.FA] 6 Apr 2012 in which the complex field has non-central embeddings [3, 1, 13]. It is inten- sively used especially in recent years not only in mathematics, but also in applications [5, 10, 8, 15, 16]. In previous works analysis over quaternion and octonions was developed and spectral theory of bounded normal operators and unbounded self-adjoint 1key words and phrases: non-commutative functional analysis, hypercomplex numbers, quaternion skew field, octonion algebra, operator, operator algebra, spectra, spectral mea- sure, non-commutative integration Mathematics Subject Classification 2010: 30G35, 17A05, 17A70, 47A10, 47L30, 47L60 1 operators was described [18, 19, 20, 21, 22]. Some results on their applications in partial differential equations were obtained [23, 24, 25, 26, 27]. This work continuous previous articles and uses their results. The present paper is devoted to unbounded normal operators and affiliated operators in octonion Hilbert spaces, that was not yet studied before.
    [Show full text]
  • Knot Symmetries and the Fundamental Quandle
    KNOT SYMMETRIES AND THE FUNDAMENTAL QUANDLE EVA HORVAT Abstract. We establish a relationship between the knot symmetries and the automorphisms of the knot quandle. We identify the homeomorphisms of the pair (S3;K) that induce the (anti)automorphisms of the fundamental quandle Q(K). We show that every quandle (anti)automorphism of Q(K) is induced by a homeomorphism of the pair (S3;K). As an application of those results, we are able to explore some symmetry properties of a knot based on the presentation of its fundamental quandle, which is easily derived from a knot diagram. 1. Introduction It is well known that the knot quandle is a complete knot invariant [8]. The knot quandle and various derived quandle invariants have been extensively used to study and distinguish nonequivalent knots. Our idea is to study knot equivalences from the viewpoint of the funda- mental quandle. In this paper, we establish a relationship between the knot symmetries and the automorphisms of the knot quandle. Our goal is to investigate the complex topological information, hidden in the group of knot symmetries, from a purely algebraical perspective of the knot quandle. Our main results are the following. 3 3 Proposition 1.1. Let f :(S ;K) ! (S ;K) be a homeomorphism for which [fj@NK ] = ±1 2 2 MCG(T ). Then f induces a map f∗ : Q(K) ! Q(K) that is either a quandle automorphism or a quandle antiautomorphism. Proposition 1.2. Let F : Q(K) ! Q(K) be an (anti)automorphism of the fundamental quan- dle of a nontrivial knot K.
    [Show full text]
  • 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].
    [Show full text]