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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. We shall denote by kxk, or sometimes kxk1, the operator norm of x 2 B(H). Throughout this text, we shall consider the two following weaker topologies on B(H): • the strong operator topology (s.o. topology), that is, the locally con- vex topology on B(H) generated by the seminorms pξ(x) = kxξk; ξ 2 H; • the weak operator topology (w.o. topology), that is, the locally con- vex topology on B(H) generated by the seminorms pξ,η(x) = j!ξ,η(x)j; ξ; η 2 H; where !ξ,η is the linear functional x 7! hξ; xηi on B(H). This latter topology is weaker than the s.o. topology. One of its important properties is that the unit ball of B(H) is w.o. compact. This is an immediate consequence of Tychonoff's theorem. This unit ball, endowed with the uniform structure associated with the s.o. topology is a complete space. In case H is separable, both topologies on the unit ball are metrizable and second-countable. On the other hand, when H is infinite dimensional, this unit ball is not separable with respect to the operator norm (Exercise 1.1). A von Neumann algebra M on H is a ∗-subalgebra of B(H) which is closed 2 in the s.o. topology and contains IdH. We shall sometimes write (M; H) to specify the Hilbert space on which M acts. The unit IdH of M will also be denoted 1M or simply 1. Given a subset S of B(H), we denote by S0 its commutant in B(H): S0 = fx 2 B(H): xy = yx for all y 2 Sg: 1In this text, H is always assumed to be separable. 2We shall see in Theorem 1.3 that we may require, equivalently, that M is closed in the w.o. topology. ERGODIC THEORY AND VON NEUMANN ALGEBRAS 3 Then S00 is the commutant of S0, that is, the bicommutant of S. Note that S0 is a s.o. closed unital subalgebra of B(H); if S = S∗, then S0 = (S0)∗ and therefore S0 is a von Neumann algebra on H. The first example of von Neumann algebra coming to mind is of course 0 n M = B(H). Then, M = C IdH. When H = C , we get the algebra Mn(C) of n × n matrices with complex entries, the simplest example of von Neumann algebra. We recall that a C∗-algebra on H is a ∗-subalgebra of B(H) which is closed in the norm topology. Hence a von Neumann algebra is a C∗-algebra, but the converse is not true. For instance the C∗-algebra K(H) of compact operators on an infinite dimensional Hilbert space H is not a von Neumann algebra on H : its s.o. closure is B(H). For us, a homomorphism between two C∗-algebras preserves the algebraic operations and the involution. We recall that it is automatically a contrac- tion and a positive map, i.e. it preserves the positive cones. For a concise introduction to the theory of C∗-algebras, we refer to [16]. 1.1.1. Von Neumann's bicommutant theorem. [25] We begin by showing that, although different (for infinite dimensional Hilbert spaces), the s.o. and w.o. topologies introduced in the first chapter have the same continuous linear functionals. For ξ; η in a Hilbert space H we denote by !ξ,η the linear functional x 7! hξ; xηi on B(H). We set !ξ = !ξ,ξ. Proposition 1.1. Let ! be a linear functional on B(H). The following conditions are equivalent : Pn (i) there exist ξ1; : : : ; ξn; η1; : : : ηn 2 H such that !(x) = i=1 !ηi,ξi (x) for all x 2 B(H) ; (ii) ! is w.o. continuous ; (iii) ! is s.o. continuous. Proof. (i) ) (ii) ) (iii) is obvious. It remains to show that (iii) ) (i). Let ! be a s.o. continuous linear functional. There exist vectors ξ1; : : : ; ξn 2 H and c > 0 such that, for all x 2 B(H), n X 21=2 j!(x)j ≤ c kxξik : i=1 n times z }| { Let H⊕n = H ⊕ · · · ⊕ H be the Hilbert direct sum of n copies of H. We ⊕n set ξ = (ξ1; : : : ; ξn) 2 H and for x 2 B(H), θ(x)ξ = (xξ1; : : : ; xξn): The linear functional : θ(x)ξ 7! !(x) is continuous on the vector sub- space θ(B(H))ξ of H⊕n. Therefore it extends to a linear continuous func- tional on the norm closure K of θ(B(H))ξ. It follows that there exists 4 CLAIRE ANANTHARAMAN-DELAROCHE η = (η1; : : : ; ηn) 2 K such that, for x 2 B(H), n X !(x) = (θ(x)ξ) = hη; θ(x)ξiH⊕n = hηi; xξii: i=1 Corollary 1.2. The above proposition remains true when B(H) is replaced by any von Neumann subalgebra M. Proof. Immediate, since by the Hahn-Banach theorem, continuous w.o. (resp. s.o.) linear functionals on M extend to linear functionals on B(H) with the same continuity property. In the sequel, the restrictions of the functionals !ξ,η and !ξ = !ξ,ξ to any von Neumann subalgebra of B(H) will be denoted by the same expressions. Recall that two locally convex topologies for which the continuous linear functionals are the same have the same closed convex subsets. Therefore, the s.o. and w.o. closures of any convex subset of B(H) coincide. The following fundamental theorem shows that a von Neumann algebra may also be defined by purely algebraic conditions. Theorem 1.3 (von Neumann's bicommutant theorem). Let M be a unital self-adjoint subalgebra of B(H). The following conditions are equiva- lent : (i) M = M 00. (ii) M is weakly closed. (iii) M is strongly closed. Proof. (i) ) (ii) ) (iii) is obvious. Let us show that (iii) ) (i). Since the inclusion M ⊂ M 00 is trivial, we only have to prove that every x 2 M 00 belongs to the s.o. closure of M (which is M, by assumption (iii)). More precisely, given " > 0 and ξ1; : : : ; ξn 2 H, we have to show the existence of y 2 M such that for 1 ≤ i ≤ n; kxξi − yξik ≤ ": We consider first the case n = 1. Given ξ 2 H, we denote by [Mξ] the orthogonal projection from H onto the norm closure Mξ of Mξ. Since this vector space is invariant under M, the projection [Mξ] is in the commutant M 0. Hence xξ = x[Mξ]ξ = [Mξ]xξ; and so we have xξ 2 Mξ. Therefore, given " > 0, there exists y 2 M such that kxξ − yξk ≤ ". We now reduce the general case to the case n = 1 thanks to the following very useful and basic matrix trick. We identify the algebra B(H⊕n) with the algebra Mn(B(H)) of n by n matrices with entries in B(H). We denote ERGODIC THEORY AND VON NEUMANN ALGEBRAS 5 by θ : B(H) !B(H⊕n) the diagonal map 0y ··· 01 B. .. .C y 7! @. .A : 0 ··· y We set N = θ(M). A straightforward computation shows that the commu- tant N 0 of N is the algebra of n × n matrices with entries in M 0. It follows that for every x 2 M 00, we have θ(x) 2 N 00. We apply the first part of the ⊕n proof to θ(x) and N. Given " > 0 and ξ = (ξ1; : : : ; ξn) 2 H , we get an element θ(y) 2 N such that kθ(x)ξ − θ(y)ξk ≤ ", that is, kxξi − yξik ≤ " for i = 1; : : : ; n. 1.2. Measure space von Neumann algebras. Every probability mea- sure space (X; µ) gives rise in a natural way to an abelian von Neumann algebra. Proposition 1.4. Let (X; µ) be a probability measure space. We set A = L1(X; µ). 1 (i) For f 2 L (X; µ), denote by Mf the multiplication operator by f 2 on L (X; µ).
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