DOCSLIB.ORG

Von Neumann Algebras and Ergodic Theory of Group Actions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Von Neumann Algebras and Ergodic Theory of Group Actions Von Neumann algebras and ergodic theory of group actions CEMPI Inaugural Conference Lille, September 2012 Stefaan Vaes∗ ∗ Supported by ERC Starting Grant VNALG-200749 1/21 Functional analysis Group theory I Hilbert space operators, I free groups, I Von Neumann algebras. I hyperbolic groups. Today's talk Ergodic theory I of group actions, I of equivalence relations. 2/21 Reminders : operators on a Hilbert space I Examples of bounded operators on a Hilbert space : • Operators on Cn $ n × n matrices. • Given f 2 L1(X ), we have the multiplication operator onL 2(X ) given by ξ 7! f ξ. 2 • The one-sided shift on ` (N) given by δn 7! δn+1. 2 Notation :( δn)n2N is the canonical orthonormal basis of ` (N). ∗ I Every bounded operator T has an adjoint T determined by hT ξ; ηi = hξ; T ∗ηi. ∗ I Self-adjoint operators : T = T , unitary operators : U∗U = 1 = UU∗, normal operators : T ∗T = TT ∗. 3/21 Von Neumann algebras The weak topology on B(H). Let Tn; T 2 B(H). We say that Tn ! T weakly if hTnξ; ηi ! hT ξ; ηi for all ξ; η 2 H. 2 n Example. If u 2 B(` (N)) is the one-sided shift, then u ! 0 weakly. Definition A von Neumann algebra is a weakly closed, self-adjoint algebra of operators on a Hilbert space. Examples : B(H), in particularM n(C), L1(X ) as multiplication operators onL 2(X ). Von Neumann's bicommutant theorem (1929) Let M ⊂ B(H) be a self-adjoint algebra of operators with1 2 M. TFAE I Analytic property : M is weakly closed. 00 0 I Algebraic property : M = M , where A is the commutant of A. 4/21 Group von Neumann algebras LetΓ be a countable group. I A unitary representation ofΓ on a Hilbert space H is a map π :Γ ! B(H): g 7! πg satisfying • all πg are unitary operators, • πg πh = πgh for all g; h 2 Γ. I The regular representation ofΓ is the unitary representation λ on 2 ` (Γ) given by λg δh = δgh, 2 where( δh)h2Γ is the natural orthonormal basis of ` (Γ). Definition (Murray - von Neumann, 1943) The group von Neumann algebra LΓ is defined as the weakly closed linear span of fλg j g 2 Γg. As we shall see, the relation between the groupΓ and its von Neumann algebraLΓ is extremely subtle. 5/21 The free groups The free group F2 is defined as \the group generated by a and b subject to no relations". −1 −1 I Elements of F2 are reduced words in the letters a; a ; b; b , like aba−1a−1b, or like bbbbbba−1bbbb. −1 −1 I Reduced means : no aa , no b b, ... in the word, because they \simplify". So bbaa−1a is not reduced. It reduces to bba. I Group operation : concatenation followed by reduction. Similarly : the free group Fn with n = 2; 3; ::: free generators. Big open problem (Murray - von Neumann, 1943) Are the free group factorsL Fn isomorphic for different n = 2; 3; ::: ? (Voiculescu 1990, Radulescu 1993) They are either all isomorphic, or all non-isomorphic. 6/21 II1 factors I A factor is a von Neumann algebra M with trivial center. Equivalently : M is not the direct sum of two von Neumann algebras. I A II1 factor is a factor M that admits a trace, i.e. a linear functional τ : M ! C such that τ(xy) = τ(yx) for all x; y 2 M. In a way, all von Neumann algebras can be assembled from II1 factors. The group von Neumann algebraLΓ is a factor iffΓ has infinite conjugacy classes : for every g 6= e, the set fhgh−1 j h 2 Γg is infinite. LΓ always has a trace, namely τ(x) = hxδe ; δe i, satisfying τ(λg ) = 0 for all g 6= e and τ(1) = τ(λe ) = 1. All icc groupsΓ give us II 1 factors LΓ, but their structure is largely non-understood. 7/21 Amenable groups and Connes's theorem Definition (von Neumann, 1929) A groupΓ is called amenable if there exists a finitely additive probability m on all the subsets ofΓ that is translation invariant : m(gU) = m(U) for all g 2 Γ and U ⊂ Γ. Non-amenable groups $ Banach-Tarski paradox. Theorem (Connes, 1976) All amenable groupsΓ with infinite conjugacy classes have isomorphic group von Neumann algebras LΓ. Actually : there is a unique \amenable" II1 factor. The following groups are amenable. • Abelian groups, solvable groups. • Stable under subgroups, extensions, direct limits. 8/21 Connes's rigidity conjecture I Open problem : are the group von Neumann algebras L(SL(n; Z)) for n = 3; 4; :::, non-isomorphic ? I Conjecturally, they are all non-isomorphic. I The groups SL(n; Z) with n = 3; 4; :::, have Kazhdan's property (T). Conjecture (Connes, 1980) LetΓ ; Λ be icc groups with Kazhdan's property (T). ThenLΓ =∼ LΛ if and only ifΓ =∼ Λ. 9/21 W∗-superrigidity for groups Theorem (Ioana-Popa-V, 2010) There are countable groups G such thatL G entirely remembers G : ifΛ is an arbitrary countable group withL G =∼ LΛ, then G =∼ Λ. (I ) These groups are of the form G = (Z=2Z) o Γ. Theorem (Berbec-V, 2012) (Γ) The same is true for G = (Z=2Z) o (Γ × Γ), for many groupsΓ, including the free groups and arbitrary free product groupsΓ=Γ 1 ∗ Γ2. 10/21 Group measure space version of the free group factor problem Recall : it is wide open whether theL Fn with n = 2; 3; :::, are isomorphic. Theorem (Popa-V, 2011) 1 The group measure space II1 factorsL (X ) o Fn, arising from free ergodic probability measure preserving actions Fn y X , are non-isomorphic for different values of n = 2; 3; :::, independently of the actions. GroupΓ: von Neumann algebraLΓ. 1 Group actionΓ y X : von Neumann algebraL (X ) o Γ. We will explain all these concepts, and some ideas behind the theorem. 11/21 Probability measure preserving group actions We study actions of countable groupsΓ on probability spaces( X ; µ) by • measurable transformations, • that preserve the measure µ. We callΓ y (X ; µ)a pmp action. Favorite examples : I Irrational rotation Z y T given by n · z = exp(2πiαn) z for a fixed α 2 R n Q. Γ I Bernoulli action Γ y (X0; µ0) given by( g · x)h = xhg . n n n I The action SL(n; Z) y T = R =Z . I The actionΓ y G=Λ for lattices Γ; Λ < G in a Lie group G. 12/21 Group measure space construction (MvN, 1943) Data : a pmp actionΓ y (X ; µ). 1 Output : von Neumann algebra M = L (X ) o Γ with trace τ : M ! C. I Notations : g 2 Γ acts on x 2 X as g · x. −1 Then, g acts on a function ξ : X ! C as( g · ξ)(x) = ξ(g · x). 1 1 I L (X ) o Γ is generated by unitaries( ug )g2Γ and a copy ofL (X ). I We have the algebraic relations (cf. semidirect products of groups) ∗ • ug uh = ugh and ug = ug −1 , ∗ • ug F ug = g · F . R I Trace : τ(F ) = F dµ and τ(F ug ) = 0 for g 6= e. 13/21 Essential freeness and ergodicity We do not want trivial actions. I Γ y (X ; µ) is called (essentially) free if for all g 6= e and almost every x 2 X , we have that g · x 6= x. 1 1 I Equivalently,L (X ) is a maximal abelian subalgebra ofL (X ) o Γ. We do not want the union of two actions. I Γ y (X ; µ) is called ergodic ifΓ-invariant subsets of X have measure 0 or 1. 1 IfΓ y (X ; µ) is free and ergodic, thenL (X ) o Γ is a II1 factor. 14/21 Actions of free groups Theorem (Popa-V, 2011) If Fn y X and Fm y Y are free ergodic pmp actions with n 6= m, then 1 ∼ 1 L (X ) o Fn =6 L (Y ) o Fm. Approach to this theorem : 1 1 I Special role ofL (X ) ⊂ L (X ) o Γ: a Cartan subalgebra. I Uniqueness problem of Cartan subalgebras. I Countable pmp equivalence relations. I Gaboriau's work on orbit equivalence relations for free groups. 15/21 Cartan subalgebras Definition A Cartan subalgebra A of a II1 factor M is 0 I a maximal abelian subalgebra : A \ M = A, ∗ I such that the normalizer NM (A) := fu 2 U(M) j uAu = Ag spans a weakly dense subalgebra of M. 1 1 Typical example : L (X ) is a Cartan subalgebra ofL (X ) o Γ whenever Γ y X is a free, ergodic, pmp action. Main questions, given a II1 factor M : I Does M have a Cartan subalgebra ? I If yes, is it unique ? • up to unitary conjugacy : A = uBu∗ for some u 2 U(M), • up to automorphic conjugacy : A = α(B) for some α 2 Aut(M). 16/21 Non-uniqueness and non-existene of Cartan subalgebras Connes-Jones, 1981 : II1 factors with at least two Cartan subalgebras that are non-conjugate by an automorphism. Uniqueness of Cartan subalgebras seemed hopeless. Voiculescu, 1995 : LFn,2 ≤ n ≤ 1, has no Cartan subalgebra. 1 2 2 Ozawa-Popa, 2007 : the II1 factor M = L (Zp) o (Z o SL(2; Z)) has 1 2 2 two non-conjugate Cartan subalgebras, namelyL (Zp) andL( Z ). Speelman-V, 2011 : II1 factors with many Cartan subalgebras (where \many" has a descriptive set theory meaning). 17/21 Uniqueness of Cartan subalgebras Theorem (Popa-V, 2011) 1 Let Fn y X be an arbitrary free ergodic pmp action.
Load more

Recommended publications
  • Quasi-Expectations and Amenable Von Neumann
    PROCEEDINGS OI THE „ „ _„ AMERICAN MATHEMATICAL SOCIETY Volume 71, Number 2, September 1978 QUASI-EXPECTATIONSAND AMENABLEVON NEUMANN ALGEBRAS JOHN W. BUNCE AND WILLIAM L. PASCHKE1 Abstract. A quasi-expectation of a C*-algebra A on a C*-subalgebra B is a bounded linear projection Q: A -> B such that Q(xay) = xQ(a)y Vx,y £ B, a e A. It is shown that if M is a von Neumann algebra of operators on Hubert space H for which there exists a quasi-expectation of B(H) on M, then there exists a projection of norm one of B(H) on M, i.e. M is injective. Further, if M is an amenable von Neumann subalgebra of a von Neumann algebra N, then there exists a quasi-expectation of N on M' n N. These two facts yield as an immediate corollary the recent result of A. Cormes that all amenable von Neumann algebras are injective. Let A be a unital C*-algebra and B a unital C*-subalgebra of A. By a quasi-expectation of A on B, we mean a bounded linear projection Q: A -» B such that Q(xay) = xQ(a)y Vx,y G B, a G A. A classical result of J. Tomiyama [13] says that any projection of norm one of A onto F is a quasi-expectation. In what follows, we will show that if M is a von Neumann algebra of operators on Hubert space H, then the existence of a quasi-expec- tation of B(H) on M implies the existence of a projection of norm one of B(H) on M.
    [Show full text]
  • Von Neumann Algebras Form a Model for the Quantum Lambda Calculus∗
    Von Neumann Algebras form a Model for the Quantum Lambda Calculus∗ Kenta Cho and Abraham Westerbaan Institute for Computing and Information Sciences Radboud University, Nijmegen, the Netherlands {K.Cho,awesterb}@cs.ru.nl Abstract We present a model of Selinger and Valiron’s quantum lambda calculus based on von Neumann algebras, and show that the model is adequate with respect to the operational semantics. 1998 ACM Subject Classification F.3.2 Semantics of Programming Language Keywords and phrases quantum lambda calculus, von Neumann algebras 1 Introduction In 1925, Heisenberg realised, pondering upon the problem of the spectral lines of the hydrogen atom, that a physical quantity such as the x-position of an electron orbiting a proton is best described not by a real number but by an infinite array of complex numbers [12]. Soon afterwards, Born and Jordan noted that these arrays should be multiplied as matrices are [3]. Of course, multiplying such infinite matrices may lead to mathematically dubious situations, which spurred von Neumann to replace the infinite matrices by operators on a Hilbert space [44]. He organised these into rings of operators [25], now called von Neumann algebras, and thereby set off an explosion of research (also into related structures such as Jordan algebras [13], orthomodular lattices [2], C∗-algebras [34], AW ∗-algebras [17], order unit spaces [14], Hilbert C∗-modules [28], operator spaces [31], effect algebras [8], . ), which continues even to this day. One current line of research (with old roots [6, 7, 9, 19]) is the study of von Neumann algebras from a categorical perspective (see e.g.
    [Show full text]
  • Algebra of Operators Affiliated with a Finite Type I Von Neumann Algebra
    UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: 10.4467/20843828AM.16.005.5377 53(2016), 39{57 ALGEBRA OF OPERATORS AFFILIATED WITH A FINITE TYPE I VON NEUMANN ALGEBRA by Piotr Niemiec and Adam Wegert Abstract. The aim of the paper is to prove that the ∗-algebra of all (closed densely defined linear) operators affiliated with a finite type I von Neumann algebra admits a unique center-valued trace, which turns out to be, in a sense, normal. It is also demonstrated that for no other von Neumann algebras similar constructions can be performed. 1. Introduction. With every von Neumann algebra A one can associate the set Aff(A) of operators (unbounded, in general) which are affiliated with A. In [11] Murray and von Neumann discovered that, surprisingly, Aff(A) turns out to be a unital ∗-algebra when A is finite. This was in fact the first example of a rich set of unbounded operators in which one can define algebraic binary op- erations in a natural manner. This concept was later adapted by Segal [17,18], who distinguished a certain class of unbounded operators (namely, measurable with respect to a fixed normal faithful semi-finite trace) affiliated with an ar- bitrary semi-finite von Neumann algebra and equipped it with a structure of a ∗-algebra (for an alternative proof see e.g. [12] or x2 in Chapter IX of [21]). A more detailed investigations in algebras of the form Aff(A) were initiated by a work of Stone [19], who described their models for commutative A in terms of unbounded continuous functions densely defined on the Gelfand spectrum X of A.
    [Show full text]
  • Set Theory and Von Neumann Algebras
    SET THEORY AND VON NEUMANN ALGEBRAS ASGER TORNQUIST¨ AND MARTINO LUPINI Introduction The aim of the lectures is to give a brief introduction to the area of von Neumann algebras to a typical set theorist. The ideal intended reader is a person in the field of (descriptive) set theory, who works with group actions and equivalence relations, and who is familiar with the rudiments of ergodic theory, and perhaps also orbit equivalence. This should not intimidate readers with a different background: Most notions we use in these notes will be defined. The reader is assumed to know a small amount of functional analysis. For those who feel a need to brush up on this, we recommend consulting [Ped89]. What is the motivation for giving these lectures, you ask. The answer is two-fold: On the one hand, there is a strong connection between (non-singular) group actions, countable Borel equiva- lence relations and von Neumann algebras, as we will see in Lecture 3 below. In the past decade, the knowledge about this connection has exploded, in large part due to the work of Sorin Popa and his many collaborators. Von Neumann algebraic techniques have lead to many discoveries that are also of significance for the actions and equivalence relations themselves, for instance, of new cocycle superrigidity theorems. On the other hand, the increased understanding of the connection between objects of ergodic theory, and their related von Neumann algebras, has also made it pos- sible to construct large families of non-isomorphic von Neumann algebras, which in turn has made it possible to prove non-classification type results for the isomorphism relation for various types of von Neumann algebras (and in particular, factors).
    [Show full text]
  • Approximation Properties for Group C*-Algebras and Group Von Neumann Algebras
    transactions of the american mathematical society Volume 344, Number 2, August 1994 APPROXIMATION PROPERTIES FOR GROUP C*-ALGEBRAS AND GROUP VON NEUMANN ALGEBRAS UFFE HAAGERUP AND JON KRAUS Abstract. Let G be a locally compact group, let C*(G) (resp. VN(G)) be the C*-algebra (resp. the von Neumann algebra) associated with the left reg- ular representation / of G, let A(G) be the Fourier algebra of G, and let MqA(G) be the set of completely bounded multipliers of A(G). With the com- pletely bounded norm, MqA(G) is a dual space, and we say that G has the approximation property (AP) if there is a net {ua} of functions in A(G) (with compact support) such that ua —»1 in the associated weak '-topology. In par- ticular, G has the AP if G is weakly amenable (•» A(G) has an approximate identity that is bounded in the completely bounded norm). For a discrete group T, we show that T has the AP •» C* (r) has the slice map property for sub- spaces of any C*-algebra -<=>VN(r) has the slice map property for a-weakly closed subspaces of any von Neumann algebra (Property Sa). The semidirect product of weakly amenable groups need not be weakly amenable. We show that the larger class of groups with the AP is stable with respect to semidirect products, and more generally, this class is stable with respect to group exten- sions. We also obtain some results concerning crossed products. For example, we show that the crossed product M®aG of a von Neumann algebra M with Property S„ by a group G with the AP also has Property Sa .
    [Show full text]
  • Categorical Characterizations of Operator-Valued Measures
    Categorical characterizations of operator-valued measures Frank Roumen Inst. for Mathematics, Astrophysics and Particle Physics (IMAPP) Radboud University Nijmegen [email protected] The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras. 1 Introduction The logic governing quantum measurements differs from classical logic, and it is still unknown which mathematical structure is the best description of quantum logic. The first attempt for such a logic was discussed in the famous paper [2], in which Birkhoff and von Neumann propose to use the orthomod- ular lattice of projections on a Hilbert space. However, this approach has been criticized for its lack of generality, see for instance [22] for an overview of experiments that do not fit in the Birkhoff-von Neumann scheme. The operational approach to quantum physics generalizes the approach based on pro- jective measurements. In this approach, all measurements should be formulated in terms of the outcome statistics of experiments. Thus the logical and probabilistic aspects of quantum mechanics are combined into a unified description. The basic concept of operational quantum mechanics is an effect on a Hilbert space, which is a positive operator lying below the identity.
    [Show full text]
  • Von Neumann Algebras
    1 VON NEUMANN ALGEBRAS ADRIAN IOANA These are lecture notes from a topics graduate class taught at UCSD in Winter 2019. 1. Review of functional analysis In this section we state the results that we will need from functional analysis. All of these are stated and proved in [Fo99, Chapters 4-7]. Convention. All vector spaces considered below are over C. 1.1. Normed vector spaces. Definition 1.1. A normed vector space is a vector space X over C together with a map k · k : X ! [0; 1) which is a norm, i.e., it satisfies that • kx + yk ≤ kxk + kyk, for al x; y 2 X, • kαxk = jαj kxk, for all x 2 X and α 2 C, and • kxk = 0 , x = 0, for all x 2 X. Definition 1.2. Let X be a normed vector space. (1) A map ' : X ! C is called a linear functional if it satisfies '(αx + βy) = α'(x) + β'(y), for all α; β 2 C and x; y 2 X. A linear functional ' : X ! C is called bounded if ∗ k'k := supkxk≤1 j'(x)j < 1. The dual of X, denoted X , is the normed vector space of all bounded linear functionals ' : X ! C. (2) A map T : X ! X is called linear if it satisfies T (αx+βy) = αT (x)+βT (y), for all α; β 2 C and x; y 2 X. A linear map T : X ! X is called bounded if kT k := supkxk≤1 kT (x)k < 1. A linear bounded map T is usually called a linear bounded operator, or simply a bounded operator.
    [Show full text]
  • A GENTLE INTRODUCTION to VON NEUMANN ALGEBRAS for MODEL THEORISTS Contents 1. Topologies on B(H) and the Double Commutant Theore
    A GENTLE INTRODUCTION TO VON NEUMANN ALGEBRAS FOR MODEL THEORISTS ISAAC GOLDBRING Contents 1. Topologies on B(H) and the double commutant theorem 2 2. Examples of von Neumann algebras 5 2.1. Abelian von Neumann algebras 6 2.2. Group von Neumann algebras and representation theory 6 2.3. The Hyperfinite II1 factor R 9 3. Projections, Type Classification, and Traces 10 3.1. Projections and the spectral theorem 10 3.2. Type classification of factors 12 3.3. Traces 13 4. Tracial ultrapowers and the Connes Embedding Problem 15 4.1. The 2-norm 15 4.2. Tracial ultraproducts 16 4.3. The Connes Embedding Problem 17 5. Some Model Theory of tracial von Neumann algebras 17 5.1. Axiomatizability 18 5.2. The model theoretic version of CEP 20 5.3. Model companions and connection to CEP 20 5.4. Stability 22 References 23 These notes were part of my course on Continuous Model Theory at UIC in the fall of 2012. We spent several weeks on the model theory of von Neumann algebras and these notes served as an introduction to von Neumann algebras with the intended audience being model theorists who know a little bit of functional analysis. These notes borrow in great part from excellent notes of Vaughn Jones [10], Martino Lupini and Asger Tornquist [11], and Jacob Lurie [12]. Date: May 3, 2013. 1 2 ISAAC GOLDBRING 1. Topologies on B(H) and the double commutant theorem In this section, H denotes a (complex) Hilbert space and B(H) denotes the set of bounded operators on H: recall that the linear operator T : H ! H is bounded if T (B1(H)) is bounded, where B1(H) := fx 2 H : kxk ≤ 1g is the closed unit ball in H.
    [Show full text]
  • Basic Von Neumann Algebra Theory
    BASIC VON NEUMANN ALGEBRA THEORY FARBOD SHOKRIEH Contents 1. Introduction1 2. von Neumann algebras and factors1 3. von Neumann trace2 4. von Neumann dimension2 5. Tensor products3 6. von Neumann algebras associated with a discrete group3 References 5 1. Introduction The theory of von Neumann algebras and von Neumann dimensions allows one to measure some infinite-dimensional subspaces in a Hilbert space, by assigning to them a notion of \dimension" (not necessarily an integer). We quickly review some very basic notions in the theory. We follow the presentation in [Shu93]. See also [L¨uc02] and [Pan96] for proofs and a more thorough treatment. 2. von Neumann algebras and factors Let H be a Hilbert space over C with the (Hermitian) inner product h·; ·i. Let B(H) be the algebra of all bounded linear operators on H. This is Banach space with respect to the operator norm. Moreover, this is a C∗-algebra: (i) B(H) is a Banach algebra, meaning kABk ≤ kAkkBk for all A; B 2 B(H). (ii) B(H) is a ∗-algebra, i.e. it is closed under the operation of taking adjoints of operators, where A 7! A∗ is defined by the usual identity hAx; yi = hx; A∗yi for all x; y 2 H. (iii)( AB)∗ = B∗A∗ for all A; B 2 B(H). (iv) kA∗Ak = kAk2 for all A 2 B(H). For any subset M ⊆ B(H), its commutant is the subalgebra M0 = fA 2 B(H): AB = BA; 8B 2 Mg in B(H). Clearly, the identity operator I is in M0.
    [Show full text]
  • Notes on Von Neumann Algebras
    Notes on von Neumann algebras Jesse Peterson April 5, 2013 2 Chapter 1 Spectral theory If A is a complex unital algebra then we denote by G(A) the set of elements which have a two sided inverse. If x 2 A, the spectrum of x is σA(x) = fλ 2 C j x − λ 62 G(A)g: The complement of the spectrum is called the resolvent and denoted ρA(x). Proposition 1.0.1. Let A be a unital algebra over C, and consider x; y 2 A. Then σA(xy) [ f0g = σA(yx) [ f0g. Proof. If 1 − xy 2 G(A) then we have (1 − yx)(1 + y(1 − xy)−1x) = 1 − yx + y(1 − xy)−1x − yxy(1 − xy)−1x = 1 − yx + y(1 − xy)(1 − xy)−1x = 1: Similarly, we have (1 + y(1 − xy)−1x)(1 − yx) = 1; and hence 1 − yx 2 G(A). Knowing the formula for the inverse beforehand of course made the proof of the previous proposition quite a bit easier. But this formula is quite natural to consider. Indeed, if we just consider formal power series then we have 1 1 X X (1 − yx)−1 = (yx)k = 1 + y( (xy)k)x = 1 + y(1 − xy)−1x: k=0 k=0 1.1 Banach and C∗-algebras A Banach algebra is a Banach space A, which is also an algebra such that kxyk ≤ kxkkyk: 3 4 CHAPTER 1. SPECTRAL THEORY A Banach algebra A is involutive if it possesses an anti-linear involution ∗, such that kx∗k = kxk, for all x 2 A.
    [Show full text]
  • Notes on Von Neumann Algebras
    Von Neumann Algebras. Vaughan F.R. Jones 1 October 1, 2009 1Supported in part by NSF Grant DMS93–22675, the Marsden fund UOA520, and the Swiss National Science Foundation. 2 Chapter 1 Introduction. The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. In this sense it is opposite in spirit from the treatises of Dixmier [], Takesaki[], Pedersen[], Kadison-Ringrose[], Stratila-Zsido[]. The philosophy is to lavish attention on a few key results and examples, and we prefer to make simplifying assumptions rather than go for the most general case. Thus we do not hesitate to give several proofs of a single result, or repeat an argument with different hypotheses. The notes are built around semester- long courses given at UC Berkeley though they contain more material than could be taught in a single semester. The notes are informal and the exercises are an integral part of the ex- position. These exercises are vital and mostly intended to be easy. 3 4 Chapter 2 Background and Prerequisites 2.1 Hilbert Space A Hilbert Space is a complex vector space H with inner product h; i : HxH! C which is linear in the first variable, satisfies hξ; ηi = hη; ξi, is positive definite, i.e. hξ; ξi > 0 for ξ 6= 0, and is complete for the norm defined by test jjξjj = phξ; ξi. Exercise 2.1.1. Prove the parallelogram identity : jjξ − ηjj2 + jjξ + ηjj2 = 2(jjξjj2 + jjηjj2) and the Cauchy-Schwartz inequality: jhξ; ηij ≤ jjξjj jjηjj: Theorem 2.1.2.
    [Show full text]
  • Finite Von Neumann Algebras : Examples and Some Basics 2 1.1
    ERGODIC THEORY AND VON NEUMANN ALGEBRAS : AN INTRODUCTION CLAIRE ANANTHARAMAN-DELAROCHE Contents 1. Finite von Neumann algebras : examples and some basics 2 1.1. Notation and preliminaries 2 1.2. Measure space von Neumann algebras 5 1.3. Group von Neumann algebras 5 1.4. Standard form 9 1.5. Group measure space von Neumann algebras 11 1.6. Von Neumann algebras from equivalence relations 16 1.7. Two non-isomorphic II1 factors 20 Exercises 23 2. About factors arising from equivalence relations 25 2.1. Isomorphisms of equivalence relations vs isomorphisms of their von Neumann algebras 25 2.2. Cartan subalgebras 25 2.3. An application: computation of fundamental groups 26 Exercise 27 1 2 1 2 3. Study of the inclusion L (T ) ⊂ L (T ) o F2 29 3.1. The Haagerup property 30 3.2. Relative property (T) 31 References 35 1 2 CLAIRE ANANTHARAMAN-DELAROCHE 1. Finite von Neumann algebras : examples and some basics This section presents the basic constructions of von Neumann algebras coming from measure theory, group theory, group actions and equivalence relations. All this examples are naturally equipped with a faithful trace and are naturally represented on a Hilbert space. This provides a plentiful source of tracial von Neumann algebras to play with. 1.1. Notation and preliminaries. Let H be a complex Hilbert space1 with inner-product h·; ·i (always assumed to be antilinear in the first vari- able), and let B(H) be the algebra of all bounded linear operators from H to H. Equipped with the involution x 7! x∗ (adjoint of x) and with the operator norm, B(H) is a Banach ∗-algebra with unit IdH.
    [Show full text]