Weak Amenability of C ∗-algebras by Kristian Knudsen Olesen

June 11th 2010 Bachelor Thesis in Mathematics. Department of Mathematical Sciences, University of Copenhagen Advisor: Ue Haagerup

Contents

Contents

1 Introduction 2 Weak amenability of C -algebras 3 Bounded Jordan derivations from C -algebras A Topologies on a von Neumann algebra B The universal enveloping von Neumann algebra C Amenable Groups ∗

∗

1

3 4 19 25 30 33

Abstract

Abstract This thesis is based on the article Weak amenability af C ∗ -algebras and a Theorem of Goldstein by Ue Haagerup and Niels J. Laustsen. The thesis deals with some results about derivations from a C ∗ -algebra to its conjugate space. Derivations are maps from a Banach algebra A to a Banach A -module, satisfying δ(ab) = a · δ(b) + δ(a) · b for all a, b ∈ A . The main result is that all C ∗ -algebras are weakly amenable, in the sence that all derivations from the C ∗ -algebra to its conjugate space are inner. Except the main result, which is done as in the article, a simpler version of a result on Jordan derivations from the article is proved. The proof of this result diers only a little from the article, and where it does, the proof is entirely based on methods from the rest of the article. To reach these results some theory is covered in form of appendicis. This theory is not done in the most general cases, but only general enough to ll in the needs for the thesis. The topics are some result on the interaction of the dierent topologies on a von Neumann algebra, a little on the universal enveloping von Neumann algebra and a few results on discrete amenable groups.

Resumé Dette projekt er baseret på artiklen Weak amenability af C ∗ -algebras and a Theorem of Goldstein af Ue Haagerup og Niels J. Laustsen. Projektet indeholder nogle resultater vedrørende derivationer fra en C ∗ -algebra til dens duale rum. Derivationer er afbildninger fra en Banach algebra A til et Banach A -bimodul, der opfylder at δ(ab) = a·δ(b)+δ(a)·b for alle a, b ∈ A . Hovedresultatet er at alle C ∗ -algebrarer er svagt amenable, i den forstand at alle derivationer fra C ∗ -algebraren til dens duale rum er indre. Ud over hovedresultatet, som er gennemgået som i artiklen, er en simplere version af et resultat om Jordan derivationer fra artiklen bevist. Beviset afviger ikke væsentligt fra artiklen, og hvor den gør er beviset udelukkende baseret på metoder fre resten af artiklen. For at nå disse resultater er en del teori gennemgået i form af appendiks. Meget af denne teori er ikke gennemgået i dens mest generelle form, men kun i tilstrækelig generel form til at projektets behov er opfyldt. De emner, der er berørt, er nogle resultater om sammenspil af de forskellige topologier på en von Neumann algebra, en lille smule om den universelle von Neumann algebra og et par resultater om diskrete amenable grupper.

Author's note  August 2010 After handing in the thesis, some corrections where made, none of which where very serious. This corrected version is thus ridded of a huge amount of typos, bad spelling and bad formulation. The original can be found through the author. 2

Introduction

1

Introduction

This thesis is written on the article Weak amenability af C ∗ -algebras and a Theorem of Goldstein by Ue Haagerup and Niels J. Laustsen.The article is divided in to three sections. The rst section cocerns weak amenability of C ∗ -algebras, proving that every bounded derivation from a C ∗ -algebra to its Banach dual is inner, and hence all C ∗ -algebras are wealy amenable. The second part concerns Bounded Jordan derivations, proving that every bounded Jordan derivations from a C ∗ -algebra A to a Banach A -bimodule is actually a derivation. The third and last part of the article contains the proof of a Theorem by S. Goldstein concerning bounded bilinear functional. The structure of this thesis is arranged such, that a large part is placed in appendicis. The reason for this is to avoid loosing track of the article on which the thesis is based. In the rst part it is proved that all C ∗ -algebras are weakly amenable. This is done, by rst proving that every derivation from a von Neumann algebra to its Banach dual is inner, and then extending the result to case of a general C ∗ -algebra. Both these steps uses the generalized Grothendieck inequality, [Haa], or consequences here of. This is a rather deep result that we will not go into details with. But we can state it here together with one of its consequences

The generalized Grothendieck inequality.

Let V : A ×B → C be a bounded bilinear form on a pair of A and B . Then there exist two states ϕ1 , ϕ2 on A and two states ψ1 and ψ2 on B such that

C ∗ -algebras

|V (x, y)| ≤ kV k(ϕ1 (x∗ x) + ϕ2 (xx∗ ))1/2 × (ψ1 (y ∗ y) + ψ2 (yy ∗ ))1/2 for all x ∈ A and all y ∈ B .

Consequence. Let V : A × B → C be a bounded bilinear form on a pair of C ∗ -algebras A and B . Then V can be extended to a jointly σ -strong∗ continuous bilinear functional V˜ : A ∗∗ × B ∗∗ → C. The second part of this thesis is, as in the article, about bounded Jordan derivation. Unlike the rst part of the thesis about weak amenability, this part only deals with some of the corresponding results from the article, only a less general result is proven. The section contains a proof that every bounded Jordan derivation from a C ∗ -algebra, to its Banach dual, is a derivation. This is done by proving that every bounded Jordan derivation from a von Neumann algebra M to a unital M -bimodule is a derivation, and then using techniques from the rst two sections of the article to prove that every bounded Jordan derivation from a C ∗ -algebra, to its Banach dual, is a derivation Kristian Knudsen Olesen June 2010

Acknowledgement. I would like to thank my advisor Ue, he has ben a great help and he has made it a plesant experience. I would also like to thank Amalie and Karen for reasons which need not be explanied here. 3

Weak amenability of

2

Weak amenability of

C ∗ -algebras

C ∗ -algebras

At rst we shall introduce some notation. Throughout this thesis the letters A and M will be used for C ∗ -algebras and von Neumann algebras respectively. If A is a C ∗ -algebra, then we denote by

• ProjA the set of projections in A , i.e. ProjA = {p ∈ A : p = p2 = p∗ }, • I (A ) the set of isometries in A , i.e. I (A ) = {v ∈ A : v ∗ v = 1A }, • U (A ) the set of unitaries in A , i.e. U (A ) = {u ∈ A : u∗ u = 1A = uu∗ } • Z (A ) the center of A , i.e. Z (A ) = {a ∈ A : ab = ba(b ∈ A )} and if a projection is in the center we say that its a central projection. If A is unital then the identity in A is denoted 1A . By X∗ we denote the Banach dual of a Banach space X, and we use h·, ·i for the duallity between X and X∗ , that is hϕ, xi := ϕ(x) for ϕ ∈ X∗ and x ∈ X. The weak topology on X we will denote as the σ(X, X∗ )-topology and the weak∗ topology on X∗ we will denote as the σ(X∗ , X)-topology. A linear functional, ϕ on a C ∗ -algebra A is said to be

•

self-adjoint if ϕ(a∗ ) = ϕ(a) for all a ∈ A ,

•

positive if ϕ(a∗ a) ≥ 0 for all a ∈ A ,

• a •

state if it is positive of norm 1,

central if ϕ(ab) = ϕ(ba) for all a, b ∈ A ,

• a

trace if it is positive and central.

We will denote the set of positive linear functionals on A by A+∗ .

Denition 2.1.

Let A be a Banach algebra. A A -bimodule, is a group X, together with maps (a, x) 7→ a · x, A × X → X and (x, a) 7→ x · a, X × A → X, linear in both variables and satisfying the following axioms

(ab) · x = a · (b · x),

a · (x · b) = (a · x) · b

x · (ab) = (x · a) · b

for all a, b ∈ A and x, y ∈ X. If A is unital we say that the X is a

1A · x = x

and

unital A -bimodule, if

x · 1A = x

for all x ∈ X.

Denition 2.2. Let A be a Banach algebra. A Banach space X A -bimodule if X is a A -bimodule, and the bimodule maps satisfy ka · xk ≤ kakkxk

is said to be a

Banach

kx · ak ≤ kxkkak

for all a ∈ A and x ∈ X. If in addition X is a Hilbert space, then X is said to be a A -bimodule.

4

Hilbert

Weak amenability of

C ∗ -algebras

Notice here that this makes the bimodule operations on a Banach A -bimodule jointly continuous. If X is a Banach A -bimodule, then there is a natural way to consider its dual X∗ as a Banach A -bimodule, namely if ϕ ∈ X∗ and a ∈ A , we dene a · ϕ and ϕ · a by

ha · ϕ, bi := hϕ, bai

hϕ · a, bi := hϕ, abi

and

for all b ∈ A . It's easy to check that this actually makes X∗ into a Banach A -bimodule. If A is a Banach algebra, then clearly A is a A -bimodule under the natural bimodule maps given by right and left multiplication. So this makes A ∗ a Banach A -bimodule as described above. Note that if A is unital then A ∗ becomes a unital Banach A -bimodule.

Note. In the rest of the thesis it will be implicit that the dual of a Banach A -bimodule is itself considered a Banach A -bimodule this way. Also A will only be considered a Banach A -bimodule the canonical way, with bimodule maps being left and right multiplication. Denition 2.3. Let A be a Banach algebra δ : A → X is a derivation provided that

and X a Banach A -bimodule. A linear map

δ(ab) = a · δ(b) + δ(a) · b for all a, b ∈ A . It's easy to check that the map a 7→ a · x0 − x0 · a, A → X for some x0 ∈ X denes a derivation, and this sort of derivation is called inner derivations.

Note.

In Automatic continuity of derivations of operator algebras by J. R. Ringrose, [Rin], it was proved that every derivation from a C ∗ -algebra A to a Banach A -bimodule is automaticly bounded. This is used in the thesis without mentioning, since the argument would occur an emence amount of times.

Denition 2.4.

A Banach algebra A is said to be derivation from A to A ∗ is inner.

weakly amenable provided that every

Denition 2.5.

Let M be a von Neumann algebra, and let p, q ∈ M be projections. We say that p and q are Murray-von Neumann equuivalent or just equivalent, written p ∼ q , if there is a partial isometry v ∈ M such that p = v ∗ v and q = vv ∗ . In this case we call p the initial projection and q the nal projection. That ∼ is in fact a equivalence relation is shown in [KR2, Proposition 6.1.5.].

Denition 2.6. Let M be a von Neumann algebra, and let p ∈ M be a projection. We say that p is innite if there exist a projections p0 ∈ M such that p ∼ p0 < p. If p is innite and ep is either 0 or innite for all central projections e ∈ M then p is said to be properly innite. When the identity 1M in M is properly innite we say that M is itself properly innite. If a projection is not innite then we say that it is nite. Lemma 2.7. Let M be a von Neumann algebra, and let p be a properly innite projection in M . If q is a projection in M with q ∼ p then q is also properly innite. Proof. Suppose that p and q are projections in M with p ∼ q and p properly innite. Let p0 ∈ M be a projection such that p ∼ p0 < p. Choose partial isometries v and w such that q = v∗v

p = vv ∗

p = w∗ w 5

p0 = ww∗

Weak amenability of

C ∗ -algebras

Now let q0 := v ∗ p0 v , then we want to show that q ∼ q0 < q because then q will be innite. Note that v ∗ p0 v < v ∗ pv since p − p0 is a non zero projection with Ran(p − p0 ) ⊆ Ran v = (ker v ∗ )⊥ . But q0 = v ∗ p0 v and q = v ∗ pv so that is q0 < q and we need to show that q0 ∼ q . If we considder the partial isometry v0 := v ∗ w∗ p0 v then

v0 v0∗ = v ∗ w∗ p0 vv ∗ p0 wv

v0∗ v0 = v ∗ p0 wvv ∗ w∗ p0 v

and

= v ∗ w∗ p0 wv

= v ∗ p0 wpw∗ p0 v

= v ∗ pv

= v ∗ p0 v

=q

= q0

so q0 ∼ q , which proves that q is innite. That q is properly innite follows from the fact that if c is a central projection then cp ∼ cq .

Lemma 2.8. Let M be a properly innite von Neumann algebra. Then sequence of mutually orthorgonal projections which are all equivalent to 1M .

M

contains a

Proof. We construct the projections successive. Since the identity in M is properly innite, by [KR1, Lemma 6.3.3.] there is a projection p1 in M with p1 ≤ 1M and p1 ∼ 1M −p1 ∼ 1M , and by Lemma 2.7 both p1 and 1M − p1 are both properly innite. Again we can nd a projection p2 ≤ 1M − p1 and with p2 ∼ 1M − p1 − p2 ∼ 1M − p1 ∼ 1M Clearly p1 and p2 are orthorgonal since p2 ≤ 1M − p1 , and by Lemma 2.7 1M − (p1 + p2 ) is properly innite. Now suppose that we have constructed mutually orthorgonal projections P p1 , . . . , pk in M such that these are all equivalent to 1M and 1M − kn=1 pn is proporly innite and equivalent to 1M , then again we can nd a projection pk+1 in M such that Pk pk+1 ≤ 1M − n=1 pn and

pk+1 ∼ 1M −

k+1 X

pn ∼ 1M −

n=1

k X

pn ∼ 1M

n=1

P The projection pk+1 is orthorgonal to pn for n = 1, . . . , k since pk+1 ≤ 1M − kn=1 pn . So now p1P , . . . , pk+1 are mutually orthorgonal projections in M , all equivalent to 1M and both 1M − k+1 n=1 pn are properly innite and equivalent to 1M . Repeating the procedure we contruct a sequece p1 , p2 , . . . of mutually orthorgonal projections all Murray-von Neumann equivalent to 1M .

Lemma 2.9. Let M be a properly innite von Neumann algebra. For each element a ∈ M ,

0 ∈ conv{vav ∗ : v ∈ I (M )}

Proof. By Lemma 2.8 we can choose a sequence (pn )n≥1 of mutually orthorgonal projections which are all Murray-von Neumann equivalent to 1M . So there exist isometries (vn )n≥1 such that vn∗ vn = 1M and vn vn∗ = pn . Dene n

an :=

1X vi avi∗ ∈ conv{vav ∗ : v ∈ I (M )} n i=1

6

Weak amenability of

C ∗ -algebras

First we want to show that, since p1 , . . . , pn are mutually orthorgonal, vj∗ vi = 0 for i 6= j . So let i 6= j then we have

kvj∗ vi k4 = k(vj∗ vi )(vj∗ vi )∗ k2 = kvj∗ pi vj k2 = k(vj∗ pi vj )(vj∗ pi vj )∗ k = kvj∗ pi pj pi vj k = 0 and therefore we get that vj∗ vi = 0. By this we get that

a∗n an

n n 1 X ∗ ∗ 1 X ∗ ∗ ∗ = 2 vi a vi vi avi = 2 vi a avi n n i=1

i=1

Now since a∗ a ≤ kak2 1M we get that vi a∗ avi∗ ≤ kak2 vi vi∗ = kak2 pi so

a∗n an ≤

n kak2 X pi n2 i=1

P Since p1 , . . . , pn are pairwise orthorgonal (non-zero) projections we get that ni=1 pi is a nonzero projection and therefore have norm 1. Using this and the equation just above we get that 2

kan k =

ka∗n an k

n

kak2

X kak2 ≤ 2 pi = 2 n n i=1

which shows that an → 0 as n → ∞ and, hence 0 ∈ conv{vav ∗ : n ∈ I (M )}.

Lemma 2.10. Let M be a von Neumann algebra. There is a map T is bounded and linear and satises:

: M → Z (M )

which

(i) T (ab) = T (ba) for all a, b ∈ M ; (ii) T (a) ∈ conv{vav∗ : v ∈ I (M )} for each a ∈ M . Proof. By [Tak, Theorem 1.19] we can nd a nite central projection p, such that our von

Neumann algebra decomposes into the direct sum of a nite von Neumann algebra, M1 := pM , and a properly innite von Neumann algebra, M2 := (1M −p)M . From [KR2, Theorem 8.2.8] there exist a center-valued trace1 τ on M1 , which is a bounded linear map τ : M1 → Z (M ) and has the property that τ (ab) = τ (ba) for all a, b ∈ M1 . Now dene T : M → Z (M ) by

for a ∈ M

T (a) := τ (pa)

then T is well-dened since2 Z (M1 ) ⊆ Z (M ), and clearly T is linear and bounded since kT (a)k ≤ kpakkτ k ≤ kτ kkak. Since p is central (and idempotent), we get that

T (ab) = τ (pab) = τ (papb) = τ (pbpa) = τ (pba) = T (ba) where we use that τ (papb) = τ (pbpa) since pa, pb ∈ M1 and τ is central. This shows that T satises (i), so left is to show that T also satises (ii). Let a ∈ A . By [KR2, Theorem 8.3.6]

τ (pa) ∈ conv{upau∗ : u ∈ U (M1 )} ⊆ conv{vpav ∗ : v ∈ I (M1 )} 1 2

Note that it's important to remember that this is not a trace in the usual sence. this is a consequence of p being a central projection.

7

(2.1)

Weak amenability of

C ∗ -algebras

so Pmfor ε > 0 we can nd m ∈ N, w1 , . . . , wm ∈ I (M1 ) and r1 , . . . , rm ∈ (0, ∞) with j=1 rj = 1 such that m

ε X

rj wj pawj∗
c2 ≥ Re ρ(ϕ) for all ϕ ∈ σ(M ∗ , M )-conv{v ∗ · ψ · v : v ∈ I (M )} especially

Re ρ(τ ) ≥ c1 > c2 ≥ sup{Re ρ(v ∗ · ψ · v) : v ∈ I (M )}

(2.5)

By [KR1, Proporsition 1.3.5] the σ(M ∗ , M )-continuous functionals on M are exactly the point evaluations, so there exist a ∈ M such that ρ(ϕ) = ϕ(a) for all ϕ ∈ M ∗ . Now with this and (2.5)

Re τ (a) > sup{Re(v ∗ · ψ · v)(a) : v ∈ I (M )}

(2.6)

and from this it follows that τ (a) 6∈ conv{(v ∗ · ψ · v)(a) : v ∈ I (M )} since

Re z ≤ sup{Re(v ∗ · ψ · v)(a) : v ∈ I (M )} < Re τ (a) for all z ∈ conv{(v ∗ · ψ · v)(a) : v ∈ I (M )} by (2.6), but this is a contradiction since   τ (a) ∈ ψ conv{vav ∗ : v ∈ I (M )} ⊆ conv{(v ∗ · ψ · v)(a) : v ∈ I (M )} by property (ii) of T . The inclusion above follows from the fact that   ψ conv{vav ∗ : v ∈ I (M )} = conv{(v ∗ · ψ · v)(a) : v ∈ I (M )} since ψ is linear and the fact ψ(A) ⊆ ψ(A) for all A ⊆ M ∗ , since ψ is continuous. Since kv ∗ · ψ · vk ≤ kv ∗ kkψkkvk = kψk for all v ∈ I (M ) we get that3 kτ k ≤ kψk. If ψ is positive, then for a ∈ M

(v ∗ · ψ · v)(a∗ a) = ψ(va∗ av ∗ ) = ψ((av ∗ )∗ (av ∗ )) ≥ 0 which shows that v ∗ · ψ · v is also positive. Since τ then is the limit of convex combinations of positive linear functionals τ is itself positive, but then τ is a trace, since we saw earlier that τ was central. 3

This follows from the fact that the σ(M ∗ , M )-limit of a bounded net is again bounded, and the same bound applies.

9

Weak amenability of

C ∗ -algebras

Lemma 2.12. For each derivation δ from a unital bimodule, the following identity holds:

C ∗ -algebra A

to a unital Banach A -

v ∗ · δ(vav ∗ ) · v − δ(a) = a · δ(v ∗ ) · v − δ(v ∗ ) · va

for all a ∈ A and v ∈ I (A ). Proof. Let a ∈ A and v ∈ I (A ). By the derivation identity and the fact that δ maps into

a unital Banach A -bimodule we get that δ(1A ) = δ(12A ) = 2δ(1A ), hence δ(1A ) = 0. Since v is an isometry, it follows that

0 = δ(1A ) = δ(v ∗ v) = v ∗ · δ(v) + δ(v ∗ ) · v that is v ∗ · δ(v) = −δ(v ∗ ) · v . Using this we get   v ∗ · δ(vav ∗ ) · v = v ∗ v · δ(av ∗ ) + δ(v) · av ∗ · v

= δ(av ∗ ) · v + v ∗ · δ(v) · a = a · δ(v ∗ ) · v + δ(a) + v ∗ · δ(v) · a = a · δ(v ∗ ) · v + δ(a) − δ(v) · va

Lemma 2.13. Let M be a von Neumann algebra, and let δ : M → There is a trace τ ∈ M ∗ af norm at most one and a derivation δ0 : M

M ∗ be → M∗

a derivation. for which

(i) δ − δ0 is an inner derivation; √

(ii) |hδ0 (a), bi| ≤ 2 2kδkkak τ (b∗ b) p

for a, b ∈ M . Proof. The map (a, b) 7→ hδ(a), bi from M × M → C, is clearly a bilinear form and since |hδ(a), bi| ≤ kδ(a)kkbk ≤ kδkkakkbk we get that it is bounded with norm less than kδk. From the generalized Grothendieck inequality [Haa, Theorem 1.1] we get that there exist states ϕ1 , ϕ2 , ψ1 , ψ2 on M such that p p |hδ(a), bi| ≤ kδk ϕ1 (a∗ a) + ϕ2 (aa∗ ) ψ1 (b∗ b) + ψ2 (bb∗ ) for all a, b ∈ A . Using the C ∗ -identity and that states have unit norm we get that p p |hδ(a), bi| ≤ kδk 2ka∗ ak + kaa∗ k ψ1 (b∗ b) + ψ2 (bb∗ ) p √ = 2kδkkak ψ1 (b∗ b) + ψ2 (bb∗ ) Dene ψ := (ψ1 + ψ2 )/2, then clearly ψ ∈ M ∗ and ψi ≤ 2ψ for i = 1, 2. Using this we get that p √ |hδ(a), bi| ≤ 2kδkkak 2ψ(b∗ b) + 2ψ(bb∗ ) p = 2kδkkak ψ(b∗ b) + ψ(bb∗ ) (2.7)

10

Weak amenability of

C ∗ -algebras

By Corollary 2.11 there exists a trace τ in the σ(M ∗ , M )-closed convex hull of {v ∗ · ψ · v : v ∈ I (M )} with kτ k ≤ kψk = 1. Let (θλ )λ∈Λ be a net in the convex hull that σ(M ∗ , M )converges P λ to τ . For λ ∈ Λ choose nλ ∈ N, vλ,1 , . . . , vλ,nλ ∈ I (M ) and tλ,1 , . . . , tλ,nλ ∈ (0, ∞) with ni=1 ti = 1 and

θλ =

nλ X

∗ tλ,i vλ,i · ψ · vλ,i

i=1

Now dene a net (ωλ )λ∈Λ in M ∗ by

ωλ :=

nλ X

∗ tλ,i δ(vλ,i ) · vλ,i

i=1

It is easy to see that kωλ k ≤ kδk for all λ ∈ Λ, so by Banach-Alaoglu's Theorem the net (ωλ )λ∈Λ has a σ(M ∗ , M )-convergent subnet (ωµ )µ∈M with limit ω ∈ M ∗ . Now for a, b ∈ M we have

ha · ω − ω · a, bi = hω, ba − abi = limhωµ , ba − abi µ

= lim µ

∗  tµ,i δ(vµ,i ) · vµ,i , ba − ab

i=1

= lim µ

nµ X

nµ X

 ∗ ∗ tµ,i a · δ(vµ,i ) · vµ,i − δ(vµ,i ) · vµ,i a, b

i=1

and if we apply Lemma 2.12 we get that

ha · ω − ω · a, bi = lim µ

∗  ∗ tµ,i vµ,i · δ(vµ,i avµ,i ) · vµ,i − δ(a), b

i=1

= lim µ

nµ  X

∗   ∗ tµ,i vµ,i · δ(vµ,i avµ,i ) · vµ,i , b − tµ,i hδ(a), b

i=1 nµ

= lim µ

nµ X

X

∗   ∗ tµ,i vµ,i · δ(vµ,i avµ,i ) · vµ,i , b − hδ(a), b

(2.8)

i=1

Now dene δ0 : M → M ∗ by δ0 (a) := δ(a) + a · ω − ω · a, then clearly δ0 − δ is a inner, hence (i) is satised. To prove (ii) we use the calculations above in (2.8), nµ X

 ∗ ∗ |hδ0 (a), bi| = lim tµ,i δ(vµ,i avµ,i ), vµ,i bvµ,i µ

i=1

≤ lim sup µ

nµ X



 ∗ ∗ tµ,i δ(vµ,i avµ,i ), vµ,i bvµ,i

i=1

Here we used the result we got from the generalized Grothendieck inequality in (2.7) to get

11

Weak amenability of

C ∗ -algebras

that

|hδ0 (a), bi| ≤ lim sup µ

nµ X



 ∗ ∗ tµ,i δ(vµ,i avµ,i ), vµ,i bvµ,i

i=1 nµ

≤ lim sup µ

X

q ∗ ∗ + v bb∗ v ∗ ) tµ,i 2kδkkvµ,i avµ,i k ψ(vµ,i b∗ bvµ,i µ,i µ,i

i=1

≤ lim sup µ

nµ X

q ∗ + v bb∗ v ∗ ) tµ,i 2kδkkak ψ(vµ,i b∗ bvµ,i µ,i µ,i

i=1

Now using Caushy-Schwarz inequality for the usual inner product in Rnµ gives that

|hδ0 (a), bi| ≤ lim sup µ

nµ X

q ∗ + v bb∗ v ∗ ) tµ,i 2kδkkak ψ(vµ,i b∗ bvµ,i µ,i µ,i

i=1

v u nµ uX ∗ + v bb∗ v ∗ ) tµ,i ψ(vµ,i b∗ bvµ,i ≤ lim sup 2kδkkakt µ,i µ,i µ

i=1

v u nµ  u X ∗ ·ψ·v ∗ ∗ = lim sup 2kδkkakt tµ,i vµ,i µ,i (b b + bb ) µ

i=1

q = lim sup 2kδkkak θµ (b∗ b + bb∗ ) µ

The net (θµ )µ∈M was chosen as a net σ(M ∗ , M )-converging to τ , so q |hδ0 (a), bi| ≤ lim sup 2kδkkak θµ (b∗ b + bb∗ ) µ p = 2kδkkak τ (b∗ b + bb∗ ) p √ = 2 2kδkkak τ (b∗ b)

Note that by now we can easily prove the fact that every properly innite von Neumann algebra is weakly amenable.

Corollary 2.14. Every properly innite von Neumann algebra, M , is weakly amenable. Proof. In view of the the Lemma above, it is enough to show that the only trace on a properly

innite von Neumann algebra is the zero-functional. For, if δ is a derivation from M to M ∗ , by Lemma 2.13 (i) we get that there is a derivation δ0 such that δ − δ0 is inner, but it follows from (ii) the δ0 = 0 if the only trace on M is the zero-functional. So let τ be a trace on M . By [KR2, Lemma 6.3.3] there is a projection p in M such that p ∼ 1M − p ∼ 1M . Now there exist an isometry v in M such that v ∗ v = 1M and vv ∗ = p, and since τ is a trace, we get that

τ (1M ) = τ (v ∗ v) = τ (vv ∗ ) = τ (p) The same calculation shows that τ (1M ) = τ (1M − p), but then

τ (1M ) = τ (p) + τ (1M − p) = 2τ (1M ) hence τ (1M ) = 0, but τ (1M ) = kτ k (cf. [Zhu, Theorem 13.5]) so τ is the zero-functional. 12

Weak amenability of

C ∗ -algebras

Lemma 2.15. Let A be a unital C ∗ -algebra, and let H be a unital Hilbert A -bimodule. For each derivation δ : A → H, the closed convex hull of {δ(u) · u∗ : u ∈ U (A )} contains a vector x0 for which δ(a) = x0 · a − a · x0 for each a ∈ A . In particular, δ is inner. Proof. Let K = conv{δ(u) · u∗ : u ∈ U (A )}, and for u ∈ U (A ) dene αu : H → H by αu (x) = u · x · u∗ + δ(u) · u∗ Clearly αu continuous for all u ∈ U (A ) since the bimodule operations are continuous. The computation

kv − wk = ku∗ u · (v − w) · u∗ uk ≤ ku · (v − w) · u∗ k ≤ kv − wk shows that αu is in fact an isometry. Note that with these maps introduces we get that K = conv{αu (0) : u ∈ U (A )}. It's easy to realise that αu ◦ αv = αuv . Using this we get that

αu (K) = αu (conv{αv (0) : v ∈ U (A )}) = conv{αuv (0) : v ∈ U (A )} =K where the rst inequality follows from the fact that αu is an isometry and the second inequality follows from the fact that αu is ane. This shows that K is αu invariant for all u ∈ U (A ), that is αu (K) = K for all u ∈ U (A ). We proceed to show that the maps αu , u ∈ U (A ), has a common xed point in K . Since kδ(u) · u∗ k ≤ kδk for all u ∈ U (A ) we have that K is norm bounded by kδk, and therefore we may dene d : K → R by

d(x) := sup{kx − yk : y ∈ K} since then d(x) ≤ 2kδk for all x ∈ K . By the parallelogram law we get that

2kx − yk2 + 2kw − yk2 = kx − wk2 + kx + w − 2yk2 for all x, y, w ∈ H. From this we get that for x, w, y ∈ K

x + w

2

4 − y = 2kx − yk2 + 2kw − yk2 − kx − wk2 2 ≤ 2d(x)2 + 2d(w)2 − kx − wk2 and if we note that

x+w 2

∈ K we get that  x + w 2 4d ≤ 2d(x) + 2d(w) − kx − wk2 2

(2.9)

Now set d0 := inf{d(x) : x ∈ K} and choose a sequence (xn )n≥1 in K such that d(xn ) → d0 for n → ∞. We want to show that the sequence (xn )n≥1 is convergent with a limit x ∈ K satisfying d(x) = d0 . Let ε > 0 and dene ε0 := min{1, ε2 (8d0 + 4)−1 }. Find N ∈ N such that d(xn ) < d0 + ε0 for all n ≥ N . By (2.9) we get that  x + x 2 n m kxn − xm k2 ≤ 2d(xn )2 + 2d(xm )2 − 4d 2 < 2(d0 + ε0 )2 + 2(d0 + ε0 )2 − 4d0

= 8d0 ε0 + 4ε02 ≤ 8d0 ε0 + 4ε0 ≤ ε2 13

Weak amenability of

C ∗ -algebras

whenever n, m ≥ N , hence (xn )n≥1 is a Cauchy sequence. Let x ∈ K denote the limit of (xn )n≥1 in K , then

d(x) ≤ d(xn ) + kx − xn k −→ d0

for n → ∞

which shows that d(x) = d0 . Now we show that x is the only point in K having the property d(x) = d0 . So assume that y ∈ K with d(y) = d0 , then by (2.9) we get x + y  kx − yk ≤ 4d0 − 4d ≤0 2 hence x = y . This x is the candidate for a common xed point of αu , u ∈ U (A ). Let u ∈ U (A ), using that αu (K) = K we get that

d(αu (x)) = sup{kαu (x) − yk : y ∈ K} = sup{kαu (x) − αu (y)k : y ∈ K} = sup{kx − yk : y ∈ K} = d0 which by the uniqueness of x shows that αu (x) = x. Since u ∈ U (A ) was arbitrary x must be a xed point for αu for all u ∈ U (A ), that is

δ(u) = δ(u) · u∗ u = (αu (x) − u · x · u∗ ) · u = (x − u · x · u∗ ) · u =x·u−u·x for all u ∈ U (A ), and since A = span U (A ) (cf. [Zhu, Theorem 10.6]) we conclude that δ(a) = x · a − a · x for all a ∈ A , in particular δ is inner.

Lemma 2.16. Let A be a unital C ∗ -algebra and ϕ a positive linear functional on A , then it holds that (i) |ϕ(a∗ b)|2 ≤ ϕ(a∗ a)ϕ(b∗ b) (ii) ϕ(b∗ a∗ ab) ≤ kak2 ϕ(b∗ b) (iii) ϕ(a∗ a) = 0 if and only if ϕ(a∗ c) = 0 for all c ∈ A for all a, b ∈ A . If also ϕ is central (i.e. if ϕ is a trace) then it also holds that (iv) ϕ(b∗ a∗ ab) ≤ kbk2 ϕ(a∗ a) for all a, b ∈ A . Proof. The inequality in (i) is just the Cauchy-Schwarz inequality for the semi-inner product (b, a) 7→ ϕ(a∗ b), A × A → C. Now for (ii). Clearly a∗ a ≤ kak2 1A and therefore b∗ (1A kak2 − a∗ a)b ≥ 0, so since ϕ is positive we have 0 ≤ ϕ(b∗ (1A kak2 − a∗ a)b) = kak2 ϕ(b∗ b) − ϕ(b∗ a∗ ab) and (ii) follows. In (iii) the if part is just the case c = a and the only if part follows from (i). Suppose now that ϕ is central. Find c ∈ A such that cc∗ = 1A kak2 − aa∗ , then

0 ≤ ϕ(c∗ b∗ bc) = ϕ(cc∗ b∗ b) = kak2 ϕ(b∗ b) − ϕ(aa∗ b∗ b) = kak2 ϕ(b∗ b) − ϕ(a∗ b∗ ba) which shows (iv). 14

Weak amenability of

C ∗ -algebras

Lemma 2.17. Let A be a C ∗ -algebra and let δ : A → A ∗ be a derivation. Assume that there exist a trace τ on A and for each a ∈ A , exist a constant ca ≥ 0 such that p |hδ(a), bi| ≤ ca kak τ (b∗ b)

for all b ∈ A . Then δ is an inner derivation Proof. Let A be a C ∗ -algebra and let τ be a trace on A satisfying the inequality above with

some constants ca (a ∈ A ). The proof goes by constructing a Hilbert A -bimodule inside A ∗ , containing the image of δ , and then use the previous Lemma about Hilbert A -bimodules. We start by constructing a Hilber space. This will actually be the Hilbert space from the GNS-construction corresponding to τ . Set Nτ = {a ∈ A : τ (a∗ a) = 0}, then we will show that Nτ is a closed twosided ideal in A . Let a ∈ Nτ and b ∈ A , using Lemma 2.16 (iii) we get that

τ ((ba)∗ (ba)) = τ (a∗ b∗ ba) = 0 which shows that ba ∈ Nτ . In similar way one can show that ab ∈ Nτ using that τ is central. Now if a, b ∈ Nτ then

τ ((a + λb)∗ (a + λb)) = τ (a∗ a) + |λ|τ (b∗ b) + 2 Re λτ (a∗ b) = 0 by Lemma 2.16 (iii). This shows that Nτ is a twosided ideal in A . That Nτ is closed follows from the fact that the map b 7→ τ (b∗ b) is continuous. It's easy to show that the quotient algebra A /Nτ becomes a pre-Hilbert space with the inner product given by
a + Nτ | b + Nτ := τ (b∗ a) (2.10) for a, b ∈ A , and by completion we obtain a Hilbert space which we denote L2 (A , τ ), the inner product and the norm we denote by (·|·)2 and k · k2 respectively ˜ a and R ˜ a denote the linear operators on A /Nτ given by For a ∈ A let L

˜ a (b + Nτ ) = ab + Nτ L

˜ a (b + Nτ ) = ba + Nτ R

˜ a and R ˜ a are left and right multiplication by a respectively. By Lemma for all b ∈ A . So L 2.16 (ii) and (iv) we get that ˜ a (b + Nτ )k2 = τ ((ab)∗ ab) ≤ kak2 τ (b∗ b) = kak2 kb + Nτ k2 kL ˜ a (b + Nτ )k2 = τ ((ba)∗ ba) ≤ kak2 τ (b∗ b) = kak2 kb + Nτ k2 kR ˜ a and R ˜ a dene bounded linear operators on A /Nτ of norm at most kak and hence can so L be extended to bounded linear operators La ,
Ra on L2 (A , τ ) of norm at most kak. Now for ξ ∈ A the map a 7→ a + Nτ | ξ 2 , A → C denes a bounded linear functional on A since
a + Nτ | ξ ≤ ka + Nτ k2 kξk2 ≤ kakkτ k1/2 kξk2 2 If we denote this linear functional by T (ξ), then the map T : L2 (A , τ ) → A ∗ , is well-dened and clearly conjugate linear. If T (ξ) = T (η) then

a + Nτ | ξ 2 = a + Nτ | η 2 15

Weak amenability of

C ∗ -algebras

for all a ∈ A , and since A /Nτ is dense in L2 (A , τ ), this means that ξ = η , so T is injective. As a consequence4 of this, the image, Im T , of T is a Hilbert space under the inner product given by

∗ T (ξ) | T (η) 2 := ξ | η 2 Now we show that Im(T ) is a Hilbert A -bimodule under the bimodule operations inherited from A ∗ . Let ξ ∈ L2 (A , τ ) and a ∈ A , then for all b ∈ A
ha · T (ξ), bi = hT (ξ), bai = ba + Nτ | ξ 2

= Ra (b + Nτ ) | ξ 2 = b + Nτ | Ra∗ (ξ) 2

= hT (Ra∗ (ξ)), bi So a · T (ξ) = T (Ra∗ (ξ)) which is in the image of T . Similarly it can be shown that T (ξ) · a = T (L∗a (ξ)) which is also in the image of T , so Im(T ) ia a Hilbert A -bimodule under the bimodule operations inherited from A ∗ . Now we proceed to show that the image of δ is contained in Im(T ). Let a ∈ A . Dene ϕa ∈ (A/Nτ )∗ by

hϕa , b + Nτ i := hδ(a), bi for b ∈ A , then ϕ is well-dened since Nτ ⊆ ker ϕa and

|hϕa , b + Nτ i| ≤ ca kakkb + Nτ k by our assumption on τ . Since A /Nτ is dense in L2 (A , τ ) we can extend ϕa to a linear functional ψa on L2 (A , τ ), and by Riesz' Representation Theorem [KR1, Theorem 2.3.1]
there exist ξa ∈ L2 (A , τ ) such that hψa , ηi = η | ξa 2 for all η ∈ L2 (A , τ ), but now

hδ(a), bi = hϕa (b + Nτ ) = ψ(b + Nτ ) = b + Nτ | ξa


2

= hT (ξa ), bi

for all b ∈ A which shows that δ(a) = T (ξa ), hence δ(a) ∈ Im(T ). Now we have shown that δ is a derivation from A to the Hilbert A -bimodule Im(T ) so we conclude by Lemma 2.15 that δ is inner.

Corollary 2.18. Let M be a von Neumann algebra. For each derivation δ : M → M ∗ there exists a functional ω in the σ(M ∗ , M )-closed convex hull of {δ(v∗ ) · v : v ∈ I (M )} for which δ(a) = ω · a − a · ω for all a ∈ M . In particular δ is inner, and M is weakly amenable. Proof. By Lemma 2.13 there exists a trace τ on M and a derivation δ0 such that δ − δ0 = x · a − a · x for some x ∈ M ∗ with

p √ |hδ0 (a), bi| ≤ 2 2kδkkak τ (b∗ b) for all a, b ∈ M , so by Lemma 2.17 δ0 is inner, clearly then δ is also inner. Now choose a functional θ ∈ A ∗ such that δ(a) = θ · a − a · θ for all a ∈ M . For v ∈ I (M ) we get by the derivation identity that

v ∗ · θ · v = θ − δ(v ∗ ) · v 4

The injectivity of T is needed so that the inner product is well dened.

16

Weak amenability of

C ∗ -algebras

Then by Lemma 2.11 the σ(M ∗ , M )-closed convex hull of {θ−δ(v ∗ )·v : v ∈ I (M )} contains a central functional τ . Now set ω = θ − τ . Since τ is central we get that

ha · τ, bi = hτ, bai = hτ, abi = hτ · a, bi for all b ∈ M , so τ · a − a · τ is the zero functional on A , hence δ(a) = ω · a − a · ω . Clearly by construction ω is in the σ(M ∗ , M )-closed convex hull of {δ(v ∗ ) · v : v ∈ I (M )} and we have proved what we wanted.

Theorem 2.19. Let A be a C ∗ -algebra. For every derivation δ : A → A ∗ there exist a functional ω ∈ A ∗ of norm at most kδk such that δ(a) = ω · a − a · ω for all a ∈ A . In particular δ is inner, and A is weakly amenable. Proof. The idea in the proof is to extend δ to a derivation on the universal enveloping von

Neumann algebra A ∗∗ , and then use Corollary 2.18 above to show that δ delta is inner. Dene a bilinear form V : A × A → C by

V (a, b) := hδ(a), bi Clearly V is bounded with kV k ≤ kδk. By the derivation identity we get that

V (ab, c) = V (a, bc) + V (b, ca)

(2.11)

for all a, b, c ∈ A . By [Haa, Corollary 2.4] we can extend V to a jointly σ -SO∗ -continuous bilinear form on V˜ : A ∗∗ × A ∗∗ → C. Note that since A is σ -SO∗ -dense in A ∗∗ by Lemma B.5 and V˜ is σ -SO∗ -continuous we must have kV˜ k = kV k. Now let a, b, c ∈ A ∗∗ . By Lemma B.5, there exist norm-bounded nets (aλ )λ∈Λ , (bµ )µ∈M and (cν )ν∈N in A , which σ -SO∗ -converges to a, b and c respectively. Using that the product is jointly σ -SO∗ -continuous on bounded subsets by Lemma A.3 (iii) and that V˜ is jointly σ -SO∗ -continuous we get that

V˜ (ab, c) = σ -SO∗ - lim V˜ (aα bµ , cν ) (α,µ,ν)

∗

= σ -SO - lim V (aα , bµ cν ) + V (bµ , cν aα ) (α,µ,ν)

∗

= σ -SO - lim V˜ (aα , bµ cν ) + V˜ (bµ , cν aα ) (α,µ,ν)

Where we applied continuity at the rst equality, (2.11) at the second equality and that V˜ extends V at the third equality. Again using that both the product and V˜ are jointly σ -SO∗ -continuous we get that

V˜ (ab, c) = V˜ (a, bc) + V˜ (b, ca)

(2.12)

If we consider A ∗∗ as a A ∗ -bimodule the natural way describet in the beginning of section 2, that is the bimodule maps dened by

ha · b, ϕi := hb, ϕ · ai

and

hb · a, ϕi := hb, a · ϕi

for a ∈ A , b ∈ A ∗∗ and ϕ ∈ A ∗ , then this clearly extends the bimodule structure on A identied with its cannonical image in A ∗∗ . If we then in the same manner consider A ∗∗∗ as an A ∗∗ -bimodule under the natural bimodule maps, then this bimodule structure extendt 17

Weak amenability of

C ∗ -algebras

the bimodule structure on A ∗ , when identied with its cannonical image in A ∗∗∗ . Dene a linear function δ˜ : A ∗∗ → A ∗∗∗ by

˜ hδ(a), bi := V˜ (a, b) This is bounded of course by kV˜ k, and that

˜ hδ(a), bi = V˜ (a, b) = V (a, b) = hδ(a), bi for all a, b ∈ A shows that δ˜ extends δ . Moreover by (2.12) we get that

˜ hδ(ab), ci = V˜ (a, bc) + V˜ (b, ca) ˜ ˜ = hδ(a), bci + hδ(b), cai ˜ · b + a · δ(b), ˜ = hδ(a) ci for all a, b, c ∈ A ∗∗ , which shows that δ˜ is a derivation. Now Corollary 2.18 states that ˜ ∗ ) · v : v ∈ I (A ∗∗ )} contains a functional θ (so the σ(A ∗∗∗ , A ∗∗ )-closed convex hull of {δ(v ∗∗∗ θ ∈ A ), such that

˜ δ(a) =θ·a−a·θ for all a ∈ A ∗∗ . Now if we let ω := θ|A then ω ∈ A ∗ and for a, b ∈ A

˜ hδ(a), bi = hδ(a), bi = hθ · a − a · θ, bi = hθ, abi − hθ, bai = hω, abi − hω, bai = hω · a − a · ω, bi ˜ ∗ ) · vk ≤ kδk ˜ So δ = ω · a − a · ω . In particular δ is inner. For the last part, note that since kδ(v ∗∗ ˜ for all v ∈ I (A ) we get that kθk ≤ kδk, thus ˜ = kV˜ k ≤ kδk kωk ≤ kθk ≤ kδk as desired.

18

Bounded Jordan derivations from

3

Bounded Jordan derivations from

C ∗ -algebras

C ∗ -algebras

In this section we introduce the notion of a Jordan derivation and prove that every bounded Jordan derivation from a C ∗ -algebra to its dual, is actually a derivation, and thus inner by the previous section.

Denition 3.1. A linear Jordan derivation

called a

map, δ , from a Banach algebra A to a Banach A -bimodule is provided that δ(a2 ) = δ(a) · a + a · δ(a) for all a ∈ A .

Clearly derivations are also Jordan derivations. It's easy to obtain that, being a Jordan derivation is equivalent to satisfying (3.1)

δ(ab + ba) = δ(a) · b + a · δ(b) + δ(b) · a + b · δ(a)

for all a, b ∈ A . This can be done, by using this Jordan derivation identity on ab + ba = (a + b)2 − a2 + b2 for one direction, and the case b = a for the other.

Lemma 3.2. Let δ be a bounded Jordan derivation from a Banach algebra A -bimodule X. Then

A

to a Banach

(i) For all a, b ∈ A , δ(aba) = δ(a) · ba + a · δ(b) · a + ab · δ(a). (ii) For each N ∈ {3, 4, . . .} and each a ∈ A , N

N −1

δ(a ) = δ(a) · a

+

N −2 X

aj · δ(a) · aN −1−j + aN −1 · δ(a)

j=1

If in addition both A and X is unital then (iii) δ(1A ) = 0 (iv) If a1 , . . . , an ∈ A satisfy a21 = · · · = a2n = 1A , then the map δˆ : A → X given by ˆ = an · · · a1 · δ(a1 · · · an ban · · · a1 ) · a1 · · · an − δ(b) is an inner derivation. δ(b) Proof. (i) By the alternative Jordan derivation identity (3.1) we get that 2δ(aba) = δ(a(ab + ba) + (ab + ba)a) − δ(ba2 + a2 b) = δ(a) · (ab + ba) + a · δ(ab + ba) + δ(ab + ba) · a + (ab + ba) · δ(a) − δ(b) · a2 − b · δ(a2 ) − δ(a2 ) · b − a2 · δ(b) Now expanding this even more using the Jordan derivation identity and (3.1) will eventually give the desired expression. To prove (ii) use induction. Clearly (ii) is satised in the case N = 3 by (i), and assuming that the identity holds for N = 3, 4, . . . , k , where5 k > 3, we get using (i) that

δ(ak+1 ) = δ(aak−1 a) = δ(a) · ak + a · δ(ak−1 ) · a + ak · δ(a) and inserting the expression for δ(ak−1 ) gives the desired result, so by induction (ii) holds. 5

In the case N = 4 it follows by rst applying (i) and then the Jordan derivation identity.

19

Bounded Jordan derivations from

C ∗ -algebras

That (iii) holds is an easy consequence the fact that X is unital and the Jordan derivation identity δ(1A ) = δ(12A ) = 2δ(1A ). To prove (vi) we also use induction. First the case n = 1, by (i) we have

ˆ = a1 · δ(a1 ba1 ) · a1 − δ(b) δ(b) = a1 · δ(a1 ) · ba21 + a21 · δ(b) · a21 + a21 b · δ(a1 ) · a1 − δ(b) = a1 · δ(a1 ) · b + b · δ(a1 ) · a1 = (a1 · δ(a1 )) · b − b · (a1 · δ(a1 )) for all b ∈ A . Where the last equality follows from the fact that

δ(a1 ) · a1 = δ(a21 ) − a1 · δ(a1 ) = δ(1A ) − a1 · δ(a1 ) = −a1 · δ(a1 ) Now to prove the induction step. Let n ≥ 2 and assume that the assertion holds for n − 1. Let a := a1 · · · an−1 , then by the assumption there is some x0 ∈ X such that

a−1 · δ(aba−1 ) · a − δ(b) = x0 · a − a · x0

(3.2)

for each b ∈ A . Using what we proved in the case n = 1 we get that with y0 = an · δ(an ) we have that

an · δ(an ban ) · an − δ(b) = y0 · b − b · y0

(3.3)

for each b ∈ A . Now by using (3.2) and (3.3) we get

ˆ = an a−1 · δ(aan ban a−1 ) · aan − δ(b) δ(b)
= an · x0 · an ban − an ban · x0 + δ(an ban ) · an − δ(b) = an · x0 · an b − ban · x0 · an + an · δ(an ban ) · an − δ(b) = an · x0 · an b − ban · x0 · an + y0 · b − b · y0 = (an · x0 · an + y0 ) · b − b · (an · x0 · an + y0 ) for all b ∈ A which shows that δˆ is inner, hence the assertion is proved.

Proposition 3.3. Let M be a von Neumann algebra. Every bounded Jordan derivation from to a unital Banach M -bimodule is automaticly a derivation.

M

Proof. Let M be a von Neumann algebra. Let δ : M → X be a bounded Jordan derivation from M to a unital M -bimodule X. Then the claim is that δ is in fact a derivation, that is δ satises the derivation identity δ(ab) = δ(a) · b + a · δ(b) for all a, b ∈ M . We start the proof by reducing to the case where a and b are selfadjoint unitaries. Since δ is continuous and M is the norm-closed linear span of its projections (cf. [Zhu, Theorem 20.3]) it suces to check the derivation identity for projections p, q ∈ ProjM . Putting u := 2p − 1M and v := 2q − 1M , using Lemma 3.2 (iii) and that X is unital, we see that δ(pq) = δ(p) · q + p · δ(q) ⇐⇒ δ((u + 1M )(v + 1M )) = δ(u + 1M ) · (v + 1M ) + (u + 1M ) · δ(v + 1M ) ⇐⇒ δ(u) + δ(v) + δ(uv) = δ(u) · (v + 1M ) + (u + 1M ) · δ(v) ⇐⇒ δ(uv) = δ(u) · v + u · δ(v) 20

Bounded Jordan derivations from

C ∗ -algebras

It's easy to check that u and v are self-adjoint unitaries, and all self-adjoint unitaries arise in this fashion, since 12 (w + 1M ) will be a projection if w is a self-adjoint unitary. Because of this, the equivalence above shows that the derivation identity holds projections if and only if it holds for self-adjoint unitaries. Now since it was sucient to prove the derivation identity for projections it is also sucient to prove the derivation identity for self-adjoint unitaries. Suppose that u, v ∈ M are dierent6 self-adjoint unitaries. If either u or v are the identity in M , then by Lemma 3.2 (iii) the derivation identity is trivially satised, so we may therefore assume that u 6= 1M and v 6= 1M . Natually U (M ) is a group with the multiplication inherited from M . Let G denote the subgroup of U (M ) gennerated by u and v . This means that G consists of 1M and all reduced words in u and v . The claim now is that the subgroup generated by uv has index 2. Let H denote the subgroup generated by uv . Then since (uv)−1 = vu, H will consists of the identity and all words with dierent starting and ending letter (in the reduced word notation). Because of this it is easy to check that this makes H a normal subgroup of index 2. Now since H has index 2 in G, this clearly makes G/H abelian, and since H is generated by one element, H is also abelian. By Theorem C.4 both G/H and H are amenable in the discrete topology, and by Theorem C.5 this makes G amenable in the discrete topology. So let m be a right invariant mean on `∞ (G). Dene a map δ0 : M → X∗∗ by Z hδ0 (a), ϕi := hϕ, w−1 · δ(waw−1 ) · widm(w) G

for all a ∈ M and ϕ ∈ X∗ , then it is easy to check that δ0 is linear. If we consider X∗∗ as a M -bimodule with the natural bimodule maps

ha · x, ϕi := hx, ϕ · ai

hx · a, ϕi := hx, a · ϕi

for all a ∈ M , ϕ ∈ X∗ and x ∈ X∗∗ , then this extends the bimodule structure on X if X is identied with its canonical image in X∗∗ . Now for a ∈ M , z ∈ G and ϕ ∈ X ∗ we get that Z −1 hδ0 (zaz ), ϕi = hϕ, w−1 · δ(wzaz −1 w−1 ) · widm(w) ZG = hϕ, zz −1 w−1 · δ(wzaz −1 w−1 ) · wzz −1 idm(w) ZG = hz −1 · ϕ · z, z −1 w−1 · δ(wzaz −1 w−1 ) · wzidm(w) ZG = hz −1 · ϕ · z, (wz)−1 · δ((wz)a(wz)−1 ) · (wz)idm(w) G

Here we use that right invariance of m and get that Z −1 hδ0 (zaz ), ϕi = hz −1 · ϕ · z, (wz)−1 · δ((wz)a(wz)−1 ) · (wz)idm(w) ZG = hz −1 · ϕ · z, w−1 · δ(waw−1 ) · widm(w) G

= hδ0 (a), z −1 · ϕ · zi = hz · δ0 (a) · z −1 , ϕi 6 The case where u = v the derivation identity reduces to the Jordan derivation identity which is satised by obvious reasons.

21

Bounded Jordan derivations from

C ∗ -algebras

Since this holds for all ϕ ∈ X ∗ , it show that

δ0 (zaz −1 ) = z · δ0 (a) · z −1

(3.4)

for all a ∈ M and z ∈ G. Now suppose thatPµ is a nite mean on `∞ (G), so there is n ∈ N, w1 , . . . , wn ∈ G and s1 , . . . , sn ∈ (0, 1) with ni=1 si = 1, such that

Z f dµ = G

n X

si f (wi )

i=1

for all f ∈ `∞ (G). For a ninte mean µ let δµ : M → X be the map

δµ (a) :=

n X

si wi−1 · δ(wi awi−1 ) · wi

(3.5)

i=1

Now since wi ∈ G for all i ∈ {1, . . . , n}, wi must be a word in u and v , so by Lemma 3.2 (iv) δµ − δ is a sum of inner derivations, hence a inner derivation. By Theorem C.7 there is a net (mλ )λ∈Λ of nite means on `∞ (G) which converges to m in the σ(`∞ (G)∗ , `∞ (G))-topology on `∞ (G). That is Z Z lim f dmλ = f dm λ

G

G

for all f ∈ `∞ (G). Let a ∈ M . Now for λ ∈ Λ let δmλ denote the construction from (3.5), then Z hδ0 (a), ϕi = hϕ, w−1 · δ(waw−1 ) · widm(w) G Z = lim hϕ, w−1 · δ(waw−1 ) · widmmλ (w) λ

G

= limhδmλ (a), ϕi λ

for all ϕ ∈ X∗∗ . Since δmλ − δ is a derivation for all λ ∈ Λ it satises the derivation identity and we get that

h(δ0 − δ)(ab), ϕi = limh(δmλ − δ)(ab), ϕi λ

= limh(δmλ − δ)(b), ϕ · ai + limh(δmλ − δ)(a), b · ϕi λ

λ

= h(δ0 − δ)(b), ϕ · ai + h(δ0 − δ)(a), b · ϕi

for all a, b ∈ M , which shows that δ0 − δ is a derivation. As mentioned in the start of the proof we only ned to check that

δ(uv) = u · δ(v) + δ(u) · v for this particular set of unitaries. So since δ0 − δ is a derivation it suces to show that δ0 satises

δ0 (uv) = u · δ0 (v) + δ0 (u) · v 22

Bounded Jordan derivations from

C ∗ -algebras

Now since δ0 is the sum of two Jordan derivations, namely δ and (δ0 − δ), it is itself a Jordan derivation. By Lemma 3.2 (ii) we get that

δ0 (u) = δ0 (u3 ) = δ0 (u) + u · δ0 (u) · u + δ0 (u) But by (3.4), with a = z = u we know that δ0 (u) = u · δ0 (u) · u, and consequently δ0 (u) = 0. The same argument applied to v will show that δ0 (v) = 0. Let N ∈ N. If we again apply (3.4) bu now to uv we get that

δ0 (uv) = uv · δ0 (uv) · vu which gives that δ0 (uv) · uv = uv · δ0 (uv). Together with Lemma 3.2 (ii) we get that

δ0 ((uv)N ) = δ0 (uv) · (uv)N −1 +

N −2 X

(uv)j · δ(a)0 · (uv)N −1−j + (uv)N −1 · δ0 (uv)

j=1

= 2δ0 (uv) · (uv)N −1 +

N −2 X

δ0 (uv) · (uv)N −1

j=1 N −1

= N δ0 (uv) · (uv) or equivalently that δ0 (uv) = get that

1 N N δ0 ((uv) )

· (vu)N −1 . By taking the norm on both sides we

1 kδ0 ((uv)N ) · (vu)N −1 k N 1 ≤ kδ0 kk(uv)N kk(vu)N −1 k N 1 ≤ kδ0 k N

kδ0 (uv)k =

hence δ0 (uv) = 0 since N ∈ N was arbitrary. Now we have proved that δ0 (u) = δ0 (v) = δ0 (uv) = 0 so clearly

δ0 (uv) = u · δ0 (v) + δ0 (u) · v is satised, and thus the theorem is proved.

Theorem 3.4. Let A be a C ∗ -algebra. Every bounded Jordan derivation from A to A ∗ is automaticly a derivation. Proof. We start by doing the unital case, and for this we proceed as in the proof of Theorem

2.19 by extending δ to the universal enveloping von Neumann algebra. Let A be a unital C ∗ -algebra, then A ∗ becomes a unital Banach A -bimodule the natural way. Let δ : A → A ∗ be a Jordan derivation, and let V : A × A → C be the bilinear form given by

V (a, b) := hδ(a), bi for a, b ∈ A . This is clearly bounded since δ is bounded and by [Haa, Corollary 2.4] we can extend V to a bilinear form V˜ : A ∗∗ × A ∗∗ → C that is jointly σ -SO∗ -continuous. Since δ is a Jordan derivation we get, by the Jordan derivation identity, that

V (a2 , b) = V (a, ba) + V (a, ab) 23

Bounded Jordan derivations from

C ∗ -algebras

for all a, b ∈ A . Let a, b ∈ A ∗∗ . By B.5 there exist norm-bounded nets (aα )α∈Λ and (bµ )µ∈M in A (identifyed with its canonical image in A ∗∗ ) that σ -SO∗ -converges to a and b respectively. Now using that V˜ is jointly σ -SO∗ -continuous, and that the product in A ∗∗ is also jointly σ -SO∗ -continuous on norm bounded sets, by Lemma A.3 (iii), we get that

V˜ (a2 , b) = σ -SO∗ - lim V (a2α , bµ ) (α,µ)

∗

= σ -SO - lim V (aα , bµ aα ) + V (aα , aα bµ ) (α,µ)

= V˜ (a, ba) + V˜ (a, ab)

(3.6)

where continuity of V˜ and the product where used at the rst and last equality. Now dene a linear map δ˜ : A ∗∗ → A ∗∗∗ by

˜ hδ(a), bi := V˜ (a, b) for all a, b ∈ A ∗∗ . If we considder A ∗∗∗ as a A ∗∗ -bimodule (cf. proof of Theorem 2.19), we get by (3.6) that

˜ 2 ), bi = V˜ (a, ba) + V˜ (a, ab) hδ(a ˜ ˜ = hδ(a), bai + hδ(a), abi ˜ + δ(a) ˜ · a, bi = ha · δ(a) for all a, b ∈ A ∗∗ , which shows that δ˜ is a Jordan derivation from the von Neumann algebra A ∗∗ to its unital Banach A ∗∗ -bimodule A ∗∗∗ , and hence by Proposition 3.3, δ˜ is a derivation. In particular since δ˜ extends δ we get that δ is a derivation. Now we proceed to the non-unital case. Suppose that A is a non-unital C ∗ -algebra and that δ : A → A ∗ is a bounded jordan derivation. Let A ∼ denote A with identity adjoint. We identify A ∗ with the subset (C1A ∼ )⊥ , the annihilator of C1A ∼ in (A ∼ )∗ , that is

(C1A ∼ )⊥ = {ϕ ∈ (A ∼ )∗ : ϕ(1A ∼ ) = 0} by to ϕ ∈ A ∗ assosiating the linear functional on A ∼ given by

a + λ1A ∼ 7→ ϕ(a) for a ∈ A , λ ∈ C. Then the A ∼ -bimodule structure on (A ∼ )∗ extendst the A -bimodule structure on A ∗ , in the sence that bimodule maps from A ∼ × (A ∼ )∗ to (A ∼ )∗ and from (A ∼ )∗ × A ∼ to (A ∼ )∗ extends the bimodule maps form A × A ∗ to A ∗ and from A ∗ × A to A ∗ respectively. Dene a map

δ˜ : a + λ1A ∼ 7→ δ(a) ,

A ∼ → (A ∼ )∗

where δ(a) here is considered a linear functional on A ∼ as described above. Clearly δ˜ is bounded and extends δ . Now using that δ is a Jordan derivation we get that

˜ + λ1A ∼ )2 ) = δ(a ˜ 2 + 2λa + λ2 1A ∼ ) δ((a = δ(a2 ) + 2λδ(a) = (a + λ1A ∼ ) · δ(a) + δ(a) · (a + λ1A ∼ ) = (a + λ1A ∼ ) · δ(a + λ1A ∼ ) + δ(a + λ1A ∼ ) · (a + λ1A ∼ ) which shows that δ˜ is a Jordan derivation. By the rst part of the proof, δ˜ is a derivation, so since δ˜ extends δ , we conclude that δ is a drivation.

24

Topologies on a von Neumann algebra

A

Topologies on a von Neumann algebra

In this section we prove some results about the interaction between some of the dierent topologies on a von Neumann algebra. First we recall how the topologies are dened.

Denition A.1.

Let M be a von Neumann algebra acting on a Hilbers space H, and let (· | ·) denote the inner product on H. (i) The weak operator topology (or W O-topology) on M is the locally convex topology generated by the family of seminorms

a ∈ M 7→ |(aξ | η)|,

ξ, η ∈ H.

(ii) The strong operator topology (or SO-topology) on M is the locally convex topology generated by the family of seminorms

a ∈ M 7→ kaξk,

ξ ∈ H.

(iii) The strong∗ operator topology (or SO∗ -topology) on M is the locally convex topology generated by the family of seminorms
1/2 a ∈ M 7→ kaξk2 + ka∗ ξk2 , ξ ∈ H. (iv) The σ -weak operator topology (or σ -W O-topology) on M is the locally convex topology generated by the family of seminorms ∞ X a ∈ M 7→ (aξj | ηj ) ,

ξi , ηi ∈ H,

∞ X

j=1

kξi kkηi k < ∞.

i=1

(v) The σ -strong operator topology (or σ -SO-topology) on M is the locally convex topology generated by the family of seminorms

a ∈ M 7→

∞ X

kaξj k2

1/2

ξi ∈ H,

,

j=1

∞ X

kξi k2 < ∞.

i=1

(vi) The σ -strong∗ operator topology (or σ -SO∗ -topology) on M is the locally convex topology generated by the family of seminorms

a ∈ M 7→

∞ X

kaξj k2 + ka∗ ξj k

1/2

ξi ∈ H,

,

∞ X

j=1

kξi k2 < ∞.

i=1

There are some obvious inclusions between the topologies, namely Norm-topology

⊇

σ -SO∗ -topology

⊇

SO-topology

σ -W O-topology

⊇

W O-topology

⊆

SO∗ -topology

⊇

⊆

σ -SO-topology

⊆

⊇

where inclusion means actual inclusion of the topologies.

Lemma A.2. Let M be a von Neumann algebra acting on a Hilbert space H. (i) 1) The W O- and σ-W O-topologies agree on norm-bounded sets. 2) The SO- and σ-SO-topologies agree on norm-bounded sets. 25

Topologies on a von Neumann algebra

3) The SO∗ - and the σ-SO∗ -topologies agree on norm-bounded sets. (ii) The W O-continuous and the SO∗ -continuous linear functionals coincide. Proof. The proofs of (i1), (i2) and (i3) are essentially the same so we only do (i2). As

mentioned the SO-topology is weaker than the σ -SO-topology, so it is sucient to show that, on norm-bounded set, the σ -SO-topology is weaker than the SO-topology. To show this we show that if a net in a normbounded set converges to an element in SO-topology, then it converges to the same element in σ -SO-topology Let X ⊆ M be a subset, norm-bounded by some K > 0, and let (aα )α∈Λ a net in X Pbe ∞ which SO-converges to some a ∈ X . Let (ξn )n∈N be a sequence in M with i=1 kξi k2 < ∞ then what we want to show, is that ∞ X

k(aα − a)ξj k2

1/2

−→ 0

(A.1)

for α → ∞

j=1

because then since (ξn )n∈N was arbitrary we get that (aα )α∈Λ σ -SO-converges to a. P ε2 2 Let ε > 0 be given. Choose N ∈ N such that ∞ n=N +1 kξn k < 8K 2 . Since X is normbounded by K we get that kaα − ak ≤ 2K for all α ∈ Λ, and hence ∞ X

∞ X

k(aα − a)ξn k2 ≤ kaα − ak2

n=N +1

kξn k2
0, then for g ∈ B(f, 2c ), the ball in `∞ (G, R) with center f and radius 2c where c = inf x∈G f (x), we have that

inf g(x) = inf f (x) − (f (x) − g(x))

x∈G

x∈G

≥ inf |f (x)| − kf − gk x∈G

c >0 2 This shows that the set {g ∈ `∞ (G) : inf x∈G g(x) ≤ 0} is closed, and thus the claim is proved. Now suppose that9 >

φ=

N X

fn − fn .xn

n=1

for some f1 , . . . , fN ∈ `∞ (G, R) and x1 , . . . , xN ∈ G, then we want to show that inf{φ(x) : x ∈ G} ≤ 0. For p ∈ N let

Λp := {(λ1 , . . . , λN ) : λi ∈ {1, 2, . . . , p}, i = 1, . . . , N } Then clearly the cardinality |Λp | = pN . For λ = (λ1 , . . . , λN ) ∈ Λp let y(λ) := xλ1 1 · · · xλNN , then of course y(λ) ∈ G for all λ ∈ Λp . Consider now for i ∈ {1, . . . , N } the sum X fi (y(λ)) − fi (y(λ)xi ) λ∈Λp

An easy combinatorial argument (here using that G is abelian) will show that atmost pN −1 terms in this sum remains, namely those on the form fi (y(λ1 , . . . , λN )) where λi = 1 and those on the form fi (y(µ1 , . . . , µN )xi ) where µi = p, and the rest cancel out. Now it follows that

X

φ(y(λ)) =

λ∈Λp

N XX

fi (y(λ)) − fi (y(λ)xi ) ≤ p

N −1

N X

kfn k∞

n=1

λ∈Λp i=1

Using this we get that

|Λp | inf{φ(x) : x ∈ G} = pN inf{φ(x) : x ∈ G} X ≤ φ(y(λ)) λ∈Λp

≤ pN −1

N X

kfn k∞

n=1 9

We can ignore scalar multiplication since a scalar multiple of a function f − f.x is again on this form.

34

Amenable Groups

But now dividing by |Λp | = pN on both sides yealds N

1X inf{φ(g) : g ∈ G} ≤ kfn k∞ p n=1

and since this holds for all p ∈ N we conclude that inf{φ(g) : g ∈ G} ≤ 0. Now we have shown that inf{g(x) : x ∈ G} ≤ 0 for all g ∈ X so clearly 1G 6∈ X . Let

d := inf{k1G − f k : f ∈ X} then since 0 ∈ X we get that d ≤ 1, and for f ∈ X we also have k1G −f k ≥ 1−inf x∈G f (x) ≥ 1 so d = 1. Now by the Hahn-Banach Theorem [KR1, Corollary 1.6.3] there is a bounded linear functional m ˜ on `∞ (G, R) of norm 1 such that

hm, ˜ 1G i = d

hm, ˜ fi = 0

and

for all f ∈ X , that is m ˜ is a state on `∞ (G, R). If we dene m : `∞ (G) → C by

hm, f i := hm, ˜ Re f i + ihm, ˜ Im f i for all f ∈ `∞ (G), then this is a bounded linear functional on `∞ (G). Now since hm, f i = hm, ˜ f i for all f ≥ 0 and m ˜ is positive, we get that m is positive. Moreover

hm, 1G i = hm, ˜ 1G i = 1 which shows that m is a mean on `∞ (G). Left is to show that m is right invariant. Let f ∈ `∞ (G) and x ∈ G, then we need to show that hm, f − f.xi = 0. By construction m ˜

hm, ˜ Re f − Re f.xi = 0

and

hm, ˜ Im f − Im f.xi = 0

since Re f − Re f.x ∈ X and Im f − Im f.x ∈ X . But then clearly hm, f − f.xi = 0, hence m is a right invariant mean on `∞ (G), and G is amenable.

Theorem C.5. Let G be a group and N a normal subgroup of G. If both N and amenable in the discrete topology, then G is amenable in the discrete topology.

G/N

is

Proof. Let m1 and m2 be right invariant means on G/N and N respectively. If x, y ∈ G with

xN = yN then since N is normal there is an n ∈ N such that x = ny . Let f ∈ `∞ (G) and consider f.x and f.y as functions restricted10 to N , then hm2 , f.xi = hm2 , (f.ny)i = hm2 , (f.y).ni = hm2 , f.yi where we used that n ∈ N and that m2 is right invariant. This shows that

x 7→ hm2 , f.xi is constant on cosets of N for all f ∈ `∞ (G), and we may dene a function Ff ∈ `∞ (G/N ) by Ff (xN ) = hm2 , f.xi with the property that

(Ff .yN )(xN ) = Ff (xyN ) = hm2 , f.xyi = hm2 , (f.y).xi = Ff.y (xN ) 10

(C.2)

We will usually not write this restriction unless it is unclear whether the functions is considered as a function on G or on N .

35

Amenable Groups

Since m2 is linear we obtain that Fλf +g = λFf + Fg for all f, g ∈ `∞ (G), λ ∈ C, and hence we can dene a linear map m : `∞ (G) → C by

hm, f i := hm1 , Ff i for all f ∈ `∞ (G). Clearly m is a bounded linear functional and we want to show that m is a mean on `∞ (G), that is hm, 1G i = 1 = kmk. Let f ∈ `∞ (G), then

kFf k = sup{|hm2 , f.xi| : x ∈ G} ≤ sup{km2 kkf k : x ∈ G} = kf k This shows that |hm, f i| = |hm1 , Ff i| ≤ kf k, hence kmk ≤ 1. Obviously 1H .x = 1H for all x ∈ G, and since 1G restricted to H is 1H we get that

F1G (xN ) = hm2 , 1H .xi = hm2 , 1H i = 1 since m2 is a mean. This shows that F1G = 1G/N so we get that

hm, 1G i = hm1 , F1G i = hm1 , 1G/N i = 1 using that m1 is a mean. This also shows that kmk = 1 and therefore that m is a mean on `∞ (G). Last we need to check that m is right invariant. Using (C.2) we get

hm, f.xi = hm1 , Ff.x i = hm1 , Ff .xN i = hm1 , Ff i = hm, f i agian using the right invariance of m1 . This shows that m is a right invariant mean on `∞ (G), and therefore that G is amenable in the discrete topology.

Denition C.6. Let G be a group. A linear functional m on `∞ (G) mean if m is a convex combination of point evaluations, that is

is said to be a

nite

m ∈ conv{ϕx : x ∈ G} where ϕx ∈ `∞ (G)∗ is the function hϕx , f i := f (x) for all f ∈ `∞ (G). It is easy to verify P that nite means are in fact P mean. In fact if x1 , . . . , xn ∈ G and s1 , . . . , sn ∈ (0, 1) with ni=1 si = 1 then for ψ := ni=1 si ϕx we have

hψ, 1G i =

n X

si hϕx , 1G i = 1

and

i=1

kψk ≤

n X

si kϕx k = 1

i=1

which shows that ψ is a mean on G.

Theorem C.7. The set of nite means on means on `∞ (G).

`∞ (G)

is σ(`∞ (G)∗ , `∞ (G))-dense in the set of

Proof. Let X denote the σ(`∞ (G)∗ , `∞ (G))-closed convex hull of {ϕx : x ∈ G} and assume

towards a contradiction that there is a mean m ∈ `∞ (G)∗ with m 6∈ X . The set X is clearly closed and convex so by the Hahn-Banach seperation theorem [KR1, Theorem 1.2.10.] there exist a f ∈ `∞ (G) such that

Rehm, f i > sup{Rehψ, f i : ψ ∈ X} 36

Amenable Groups

In particular we get by only considering {ϕx : x ∈ G} that

Rehm, f i > sup{Re f (x) : x ∈ G}

(C.3)

Since m is positive it follows that Re m(f ) = m(Re f ). If we let C := sup{Re f (x) : x ∈ G} then f ≤ C1G . Now m is positive with m(1G ) = 1 so

Rehm, f i ≤ m(C1G ) = sup{Re f (x) : x ∈ G} which is in contradiction with (C.3). It follows that m ∈ X , which was what we wanted to show.

37

References

References

[Haa] Ue Haagerup. The Grothendieck inequality for bilinear forms on C ∗ -algebras. in Math., 56(2):93116, 1985.

Adv.

[KR1] Richard V. Kadison and John R. Ringrose. Fundamentals of the theory of operator algebras. Vol. I, volume 100 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. Elementary theory. [KR2] Richard V. Kadison and John R. Ringrose. Fundamentals of the theory of operator algebras. Vol. II, volume 100 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1986. Advanced theory. [Rin] J. R. Ringrose. Automatic continuity of derivations of operator algebras. Math. Soc. (2), 5:432438, 1972.

J. London

Theory of operator algebras. I, volume 124 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2002. Reprint of the rst (1979) edition,

[Tak] M. Takesaki.

Operator Algebras and Non-commutative Geometry, 5. [Zhu] Kehe Zhu. An introduction to operator CRC Press, Boca Raton, FL, 1993.

38

algebras. Studies in Advanced Mathematics.