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  • AN INTRODUCTION TO VON NEUMANN ALGEBRAS

    ADRIAN IOANA

    1. von Neumann algebras basics

    Notation. Throughout these notes, H denotes a Hilbert space that we will typically assumeseparable (e.g. `2(N), L2([0, 1], ). We denote by B(H) the algebra of linear operators T : H Hthat are bounded, in the sense that the norm ||T || = supH,61 T () is finite. We also denoteby U(H) the group of unitary operators U : H H. The adjoint of T B(H) is the uniqueoperator T B(H) such that T (), = , T (), for all , H.There are three topologies on B(H) that we will consider:

    the norm topology: Ti T iff Ti T 0. the strong operator topology (SOT): Ti T iff Ti() T () 0, for all H. the weak operator topology (WOT): Ti T iff Ti(), T (), , for all , H.

    Proposition 1.1. If C B(H) is a convex set, then CSOT = CWOT .

    The proof of Proposition 1.1 relies on the following lemma:

    Lemma 1.2. If C H is a convex set, then the weak and norm closures of C are equal.

    Proof. Recall that i weakly iff i, , , for all H. Denote D = C.

    . It is clear

    that D Cweak. To prove the reverse inclusion, let Cweak. Since D is a norm closed andconvex subset of the Hilbert space H, we can find 0 D such that 0 = infD .Let C. Then the function [0, 1] 3 t (1 t)0 t2 = ( 0) t( 0)2 has aminimum at t = 0, hence its derivative at t = 0 is positive. It follows that

  • 2 ADRIAN IOANA

    Remarks. (1) Any von Neumann algebra is a C-algebra.

    (2) By Lemma 1.1, any SOT-closed -subalgebra is WOT-closed and hence a von Neumann algebra.(3) Let B B(H) be a set such that T B, for every T B. Then the commutant of B,defined as B = {T B(H)| TS = ST, for every S B} is a von Neumann algebra. Indeed, B isa -algebra. To see that B is WOT-closed, let Ti B be a net such that Ti T (WOT). Thenfor every S B and all , H we have that

    TS, = limiTiS, = lim

    iSTi, = lim

    iTi, S = T, S = ST, .

    Conversely, the next theorem shows that every von Neumann algebra arises in this way.

    Theorem 1.5 (von Neumanns bicommutant theorem). If M B(H) is a unital -subalgebra,then the following three conditions are equivalent:

    (1) M is WOT-closed.(2) M is SOT-closed.(3) M = M .

    This is a beautiful result which asserts that for -algebras, the analytic condition of being closedin the WOT is equivalent to the algebraic condition of being equal to the double commutant.

    Proof. It is clear that (3) (1) (2). We prove that (2) (3). It suffices to show that if x M , > 0, and 1, ..., n H, then there exists y M such that xi yi < , for all i = 1, ..., n. Westart with the following claim:

    Claim 1. Let K H be an M -invariant closed subspace. Let p be the orthogonal projection fromH onto K. Then p M .To see this, let x M . Then (1 p)xp (1 p)(K) = {0}, for all H. Hence (1 p)xp = 0and so xp = pxp. By taking adjoints, we get that px = pxp and hence px = pxp, for all x M .This shows that p commutes with x, as claimed.

    Assume first that n = 1 and let p be the orthogonal projection onto M1 = {xi|x M}. SinceM1 is M -invariant, Claim 1 gives that p M . Thus xp = px and so x1 = xp1 = px1 M1.Therefore, there is y M such that x1 y1 < .Now, for arbitrary n, we use a matrix trick. Let Hn = ni=1H be the direct sum of n copies ofH and identify B(Hn) = Mn(B(H)). Let : M B(Hn) be the diagonal -representation givenby (a) = (i,ja)16i,j6n.

    Exercise 1.6. Prove that the following holds:

    Claim 2. (M) = (M ).

    Finally, if x M , then Claim 2 gives that (x) (M). Let = (1, ..., n) Hn. Byapplying the case n = 1 we conclude that there is y M such that (x) (y) < . Since(x) (y)2 =

    ni=1 xi yi2, we are done.

    Examples. Next, we provide some basic examples of C- and von Neumann algebras.

    (1) If H is a Hilbert space, then B(H) is a von Neumann algebra and K(H), the algebra ofcompact operators on H, is a C-algebra.

    (2) Let (X,) be a standard probability space. This means that X is a Polish space(i.e. a topological space which is metrizable, complete and separable) and is a Borelprobability measure on X (i.e. a -additive probability measure defined on all Borelsubsets of X). Then L(X,) B(L(X,)) is a von Neumann algebra.

  • AN INTRODUCTION TO VON NEUMANN ALGEBRAS 3

    (3) If X is a compact Hausdorff space, then C(X) and B(X) (the algebras of complex-valuedcontinuous and, respectively, bounded Borel functions on X) are abstract C-algebras.More precisely, they are Banach algebras that have an intrinsic adjoint operation givenby conjugation: f(x) = f(x).

    (4) If I is a set, then `(I) B(`2(I)) is a von Neumann algebra and c0(I) B(`2(I)) is aC-algebra.

    Proposition 1.7. Let (X,) be a standard probability space. Define : L(X,) B(L2(X,))by letting f () = f, for all f L(X) and L2(X).

    Then (L(X)) = (L(X)). Therefore, (L(X)) is a maximal abelian von Neumann subalge-bra of B(L2(X)).

    Proof. Note that is an isometric -homomorphism: f = f, for every f L(X). Weidentify L(X) with its image under .

    Now, let T L(X) and put g = T (1). Then fg = fT (1) = Tf (1) = T (f) and hence

    fg2 = T (f)2 6 T f2, for every f L(X).

    Let > 0 and f = 1{xX| |g(x)|>T+}. Then it is clear that fg2 > (T+)f2. In combinationwith the last inequality, we get that (T + )f2 6 Tf2, and so f = 0, almost everywhere.Thus, we conclude that g L(X). Since T (f) = fg = g(f), for all f L(X), and L(X) is.2-dense in L2(X), it follows that T = g L(X).

    2. Abelian group algebras and standard probability spaces

    The most studied classes of von Neumann algebras arise from groups and from group actions. Let

    be a countable group and consider `2 = {f : C| f2 := (

    g |f(g)|2)12 < }. An

    orthonormal basis for `2 is given by {g}g, where g(h) = g,h is the Dirac function at g. Weconsider the left regular representation : U(`2) given by

    (g)(h) = gh, for all g, h .

    It is easy to check that (g) is a unitary, (gh) = (g)(h) and (g) = (g1), for all g, h . LetA = {

    gF ag(g)| F is finite, ag C, for all g F}. Then A B(`2) is a -subalgebra

    which is isomorphic to the group algebra C(). The norm closure of A is denoted by Cr () andcalled the reduced C-algebra of .

    Definition 2.1. L() := AWOT is called the group von Neumann algebra of .

    A related construction that we will discuss later on associates to every measure preserving action y (X,) of a countable group on a standard probability space (X,) a crossed product vonNeumann algebra L(X) o .

    It is typically hard to understand the structure of L() is a simpler way. One situation when thisis possible is for abelian groups.

    Proposition 2.2. Let A be a countable abelian group. Denote by A the Pontryagin dual of A, andby the Haar measure of A.

    Then L(A) is -isomorphic to L(A, ).

  • 4 ADRIAN IOANA

    Recall that A is the group of characters of A, i.e. homomorphisms h : A T = {z C| |z| = 1}.Also, recall that A is a compact metrizable group. More precisely, if we enumerate A = {an}n>1,then d(h, h) =

    n=1 2

    n|h(an) h(an)| is a compatible metric.Before proving Proposition 2.2, let us establish the following useful fact:

    Lemma 2.3. Let X be a normal topological space (i.e. any two disjoint closed sets have disjointopen neighborhoods). Let f B(X).Then we can find a net fi C(X) such that fi 6 f, for all i, and

    X fi d

    X f d, for

    every Borel probability measure on X.

    Note that every compact Hausdorff space X is normal.

    Proof. Recall that a Borel probability measure on X is regular if for every -measurable setA X we have

    (A) = sup {(F )| F A,F closed} = inf {(G)| G A,G open}.

    Let f B(X), 1, ..., n be a finite collection of regular Borel probability measures on X, and > 0.Let 1, ..., m C and 1, ....,m be Borel subsets of X such that f

    mk=1 k1k < /2

    and |k| 6 f, for all k = 1, ...,m. Since 1, ..., n are regular measures, we can find closedsets Fk X and open sets Gk X such that we have Fk k Gk, for all k = 1, ...,m, andi(Gk \ Fk) < 2mf , for all k = 1, ...,m and every i = 1, ..., n.

    Since X is normal, Uryshons lemma provides g1, ..., gm C(X) such that 1Fk 6 gk 6 1Gk , for allk = 1, ...,m. Finally, let f =

    mk=1 kgk C(X). Note that |gk1k | 6 1Gk\Fk , for all k = 1, ...,m.

    Using this, for all i = 1, ..., n we get that

    |Xf di

    Xf di| 6

    X|f

    mk=1

    k1k | di +mk=1

    |k|X|gk 1k | di 6 .

    Moreover, one can arrange for f to satisfy f 6 f.Since > 0 and the probability measures 1, ..., n are arbitrary, it is easy to finish the proof.

    Proof of Proposition 2.2. We begin with the following claim:

    Claim. If a A \ {e}, then there exists h A such that h(a) 6= 1.Enumerate A = {an}n>1 with a1 = a. For every n, denote by An the subgroup of A generatedby {a1, ..., an}. Since a 6= e, there is a character h1 : A1 T such that h1(a) 6= 1. We prove byinduction that there is a character hn : An T such that hn+1|An = hn, for all n > 1. Once this isdone, it is clear how to define h. Thus, it suffices to show that a character hn of An extends to acharacter of An+1. Indeed, let l be the smallest integer such that a

    ln+1 An. Define hn+1(an+1) = z,

    where z T is such that zl = hn(aln+1). Then hn+1 has the desired property, proving the claim.

    Now, for a A, let a L2(A, ) be given by a(h) = h(a). If a A \ {e}, the claim gives h Asuch that h(a) 6= 1. Then

    A a(h) d(h) =

    A a(h

    h) d(h) = h(a)A a(h) d(h) and therefore

    A a(h) d(h) = 0. This implies that U : `2A L2(A, ) given by U(a) = a is an isometry.

    We claim that U is onto, i.e. it is a unitary. Denote by B the linear span of {a|a A}. Since Bseparates points in A, the Stone-Weierstrass theorem implies that B is