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Projections from a von Neumann algebra

arXiv:math/9302215v1 [math.FA] 18 Feb 1993Projections from a von Neumann algebraonto a subalgebrabyGilles Pisier*Texas A&M UniversityandUniversit´e Paris VIIntroduction.This paper is mainly devoted to the following question: Let M, N bevon Neumann algebras with M ⊂N, if there is a completely bounded (c.b. in short)projection P: N →M, is there automatically a contractive projection eP: N →M?We give an affirmative answer with the only restriction that M is assumed semi-finite.At the time of this writing, the case when the subalgebra M is a type III factor seemsunclear, although this might be not too hard to deduce from our results using crossedproduct techniques from the Tomita-Takesaki theory with which we are not familiar.If N = B(H), a positive answer (without any restriction on M) was given in [P1,P2] (and independently in [CS]).

I am grateful to Eberhard Kirchberg for mentioning tome that a more general statement might be true. It should be mentioned that the abovequestion seems open if “completely bounded” is replaced by “bounded” in the assumptionon the projection P. For more results in this direction, see [P3] and [HP2].

We shouldrecall that, by a classical result of Tomiyama [T], every norm one projection P from Nonto M necessarily is a conditional expectation and in particular is completely positive. Inthe second part of the paper we give an interpolation theorem which generalizes a resultin [P1], as follows.

Let N be a von Neumann algebra equipped with a normal semi-finitefaithful trace ϕ. Let us denote by Lp(ϕ) the noncommutative Lp-space associated to (N, ϕ)in the usual way.

Fix n ≥1. Let us denote by A0 (resp.

A1) the space N n equipped with* Supported in part by the NSF1

the norms∥(x1, . .

., xn)∥A0 =Xxix∗i1/2N∥(x1, . .

., xn)∥A1 =Xx∗i xi1/2N.We prove in section 2 that the complex interpolation space (A0, A1)θ is the space N nequipped with the norm∥(x1, . .

. , xn)∥θ =XLxiRx∗i1/2B(Lp(ϕ))where we have denoted by Lx (resp.

Rx) the operator of left (resp. right) multiplication byx on Lp(ϕ), and where p = θ−1.

Note that the case θ = 0 corresponds to L∞(ϕ) identifiedwith N and θ = 1 corresponds to L1(ϕ) identified with N∗in the usual way. Again in theparticular case N = B(H) this result was proved in [P1].We refer to [Ta1] for background on von Neumann algebras and to [Pa] for completeboundedness.We will use several times the following elementary fact.Lemma 0.1.

Let M ⊂N be von Neumann algebras. Let (pi)i∈I be a directed net ofprojections in M such that, for all x in M, pixpi tends to x in the σ(M, M∗) topology.Assume that for each i there is a norm one projection Pi: N →piMpi.

Then there is anorm one projection P from N onto M.Proof.Let U be a nontrivial ultrafilter refining the net.For any x in N, we defineP(x) = limU Pi(pixpi) where the limit is in the σ(M, M∗) sense.Then P(x) ∈M and∥P(x)∥≤∥x∥. Moreover, for any x in M we havePi(pixpi) = pixpi.Hence P(x) = x for all x in M, and we conclude that P is a projection from N to M.2

§1. Projections.The main result of this section is the following.Theorem 1.1.

Let M ⊂N ⊂B(H) be von Neumann algebras with M semi-finite. Ifthere is a completely bounded (c.b.

in short) projection P: N →M, then there is a normone projection eP: N →M.Actually, we use less than complete boundedness, we only need to assume that thereis a constant C such that for all x1, . .

., xn in N we have(1.1)XP(xi)∗P(xi) ≤C2 Xx∗i xiandXP(xi)P(xi)∗ ≤C2 Xxix∗i .The proof is given at the end of this section.Notation: Let ϕ be a normal faithful semi-finite trace on a von Neumann algebra N. Wedenote by L2(ϕ) the usual associated Hilbert space. For any a in N, we denote by La(resp.

Ra) the operator of left (resp. right) multiplication by a in L2(ϕ), i.e.

we set for allx in L2(ϕ)Lax = ax,Rax = xa.The key lemma in the proof of Theorem 1.1 is the next statement.Lemma 1.2. Let N be a semi-finite von Neumann algebra with a normal faithful semi-finite trace ϕ as above.

Consider a finite set x1, . .

. , xn in N and assume(1.2)Xn1 LxiRx∗iB(L2(ϕ)) ≤1,then there is a decomposition xi = ai + bi with ai ∈N, bi ∈N such that(1.3)Xa∗i ai1/2 +Xbib∗i1/2≤1.More generally, the main idea of this paper seems to be the identification of theexpression∥(x1, .

. .

, xn)∥=Xn1 LxiRx∗i1/2B(L2(ϕ))with the norm of a simple interpolation space obtained by the complex interpolationmethod. See section 2 for further details.3

Corollary 1.3. Let N be as in Lemma 1.2 and let M be a finite von Neumann algebraequipped with a normalized finite trace τ.

Let P: N →M be any linear map satisfying(1.1). Then for all finite sequences x1, .

. ., xn in N we havenX1τ(P(xi)P(xi)∗) =nX1τ(P(xi)∗P(xi)) ≤C2nX1LxiRx∗iB(L2(ϕ)).Proof.

Assume P LxiRx∗i ≤1. Let ai, bi be as in Lemma 1.2.

Let us denote ∥x∥2 =(τ(x∗x))1/2 for all x in M. Then we haveX∥P(xi)∥221/2≤X∥P(ai)∥221/2+X∥P(bi)∥221/2≤XP(ai)∗P(ai)1/2+XP(bi)P(bi)∗1/2≤C.Lemma 1.4. Let N be as in Lemma 1.2 and let M ⊂N be a finite von Neumannsubalgebra.

Assume that there is a projection P: N →M satisfying (1.1). Then for allnonzero projection p in the center of M and for all unitary operators u1, .

. ., un in M wehave(1.4)n =nX1LpuiR(pui)∗B(L2(ϕ)).Proof.

Fix p as in Lemma 1.4. By [Ta1, p. 311 ] there is a finite trace τ on M withτ(p) ̸= 0.

By Corollary 1.3 applied to the normalized trace x →τ(p)−1τ(x) on pMp = pMwe haven =X∥pui∥22 ≤C2nX1LpuiR(pui)∗B(L2(ϕ)).To replace C2 by 1 in this inequality, we use the same trick as Haagerup in [H1]. LetTn =nX1LpuiR(pui)∗.We have for each kT kn =X1≤m≤nkLxmRx∗m4

where each xm is of the form pu with u unitary in M. It follows thatnk ≤C2∥T kn∥≤C2∥Tn∥khence n ≤C2/k∥Tn∥. Letting k tend to infinity we obtain (1.4) (since the other directionis trivial by the triangle inequality.

)Proof of Lemma 1.2. We will use the duality between N n and N n∗.Let C be the set of elements (xi)i≤n in N n which admit a decomposition xi = ai +bi in Nsatisfying (1.3).

We will show that if (1.2) holds, then necessarily (xi) lies in the bipolarCoo of C in the duality between N n and N n∗. This is enough to conclude.

Indeed sincethe set C is clearly convex and σ(N n, N n∗) closed we have C = Coo, so we obtain that (xi)is in C if (xi) satisfies (1.2).Hence assume given (xi) satisfying (1.2).Consider (ξi)i≤n in N n∗and assume(ξi)i≤n ∈Co. This means that for any ai in N such that(1.5)eithernX1aia∗i1/2≤1orXa∗i ai1/2≤1,we haveXξi(ai) ≤1.We use the classical identification N∗= L1(ϕ) and we use the density of N ∩L1(ϕ) inL1(ϕ).

By these well known properties of L1(ϕ) for each ε > 0 we can find a projection pin N with ϕ(p) < ∞and elements b1, . .

., bn in pNp such that(1.6)∥ξi −bi∥N∗< ε.It follows that for any (ai) satisfying (1.5) we have P⟨bi, ai⟩ ≤1 + nε. So that replacingbi bybi1+nε we may as well assume (since ε > 0 is arbitrary) that, for any (ai) satisfying(1.5) we have(1.7)Xϕ(biai) ≤1.5

We first claim that this implies(1.8)ϕXb∗i bi1/2≤1andϕXbib∗i1/2≤1.Indeed, let r (resp. c) be the element of Mn(N)∗corresponding to the n × n matrix whichhas coefficients equal to b1, .

. ., bn on the first row (resp.

column) and zero elsewhere.Then by (1.7) r and c are in the unit ball of Mn(N)∗. From this (1.8) immediately followsby the identification between Mn(N)∗and L1(eϕ) where eϕ is the semi-finite trace definedon Mn(N) byeϕ((aij)) =Xϕ(aii).Secondly, we claim that, for any δ > 0, bi can be written as bi = αyiβ with α, yi, β in pNpsuch thatϕ(|a|4) ≤1 + δϕ(p),ϕ(|β|4) ≤1 + δϕ(p)andXϕ(|yi|2) ≤1.Letα =Xbib∗i1/2+ δp1/4andβ =Xb∗i bi1/2+ δp1/4.Note that we clearly have(1.9)α−2 Xbib∗iα−2 ≤Xbib∗i1/2andβ−2 Xb∗i biβ−2 ≤Xb∗i bi1/2.We also note that(1.10)ϕ(β4) ≤1 + δϕ(p)andϕ(α4) ≤1 + δϕ(p).Now in the von Neumann algebra pNp (with unit p) we introduce the analytic pNp valuedfunctions fk (k = 1, ..., n) defined on the strip S = {z ∈C | 0 < Re(z) < 1} byfk(z) = α−2(1−z)bkβ−2z.We havefk(it) = α2itα−2bkβ−2itandfk(1 + it) = α2itbkβ−2−2it.6

Since α2it and β−2it are unitary in pNp, it follows that for all real tX∥fk(it)∥2L2(ϕ) = ϕα−2 Xbkb∗kα−2,hence by (1.9) and (1.8)≤ϕXbkb∗k1/2≤1.Note that fk is bounded on S since α, β are bounded below in pNp. We now invoke thethree lines lemma (cf.

[BL, p. 4]). Note that, as is well known, this lemma remains valid forbounded analytic functions on S, not necessarily continuous on S, using the nontangentialboundary values to extend the functions to S. Using this, we conclude that, for all z inthe strip S, we haveX∥fk(z)∥2L2(ϕ) ≤1.In particular this holds for z = 1/2 and we can define yk = fk(1/2).Then we havebk = αykβ and all the announced properties hold.

We now return to our original n-tuplex1, . .

., xn in N.We have by (1.6)X⟨ξk, xk⟩ ≤X⟨bk, xk⟩ + nε≤Xϕ(αykβxk) + nεhence by Cauchy-Schwarz and by (1.10)≤X∥βxkα∥2L2(ϕ)1/2+ nε=ϕXβxkαα∗x∗kβ∗1/2+ nε=Dβ∗β,XLxkαα∗Rx∗kE1/2L2(ϕ) + nε≤XLxkRx∗k1/2B(L2(ϕ)) (1 + δϕ(p))1/2 + nε.Since ε, δ > 0 are arbitrary, we conclude that if (1.2) holds we have P⟨ξk, xk⟩ ≤1 forall (ξk) in Co. Hence we have (xk) ∈Coo and the proof is complete.To prove Theorem 1.1, we will combine Lemma 1.2 with a rather straightforward ex-tension of some results of Haagerup in [H1] on injective von Neumann algebras.

Haagerup’swork is based on Connes’ ideas on injective factors [Co].7

Definition 1.5. Let M ⊂N be von Neumann algebras.

We will say that a state ω on Nis an M-hypertrace on N if we have∀a ∈M∀x ∈Nω(ax) = ω(xa).Theorem 1.6. Let M ⊂N be von Neumann algebras with N semi-finite and ϕ a faithfulnormal semi-finite trace on N. The following are equivalent(i) M is finite and there is a norm one projection P from N onto M.(ii) For any finite set u1, .

. .

, un of unitaries in M and any nonzero central projection p inM we have(1.11)n =nX1LpuiR(pui)∗B(L2(ϕ)). (iii) For every nonzero central projection p in M there is an M-hypertrace ω on N, suchthat ω(1 −p) = 0.

(iv) For every state ω0 on the center of M there is an M-hypertrace ω on N extending ω0.Proof. The proof of Lemma 2.2 in [H1] extends word for word.

We simply replace thereB(H) by N and we denote by M the subalgebra.Remark. For the convenience of the reader, we recall the key idea which is behind thepreceding statement.

This is best described in the case when M is a factor. In that casethe implication (ii) ⇒(i) is proved as follows: using the uniform convexity of L2(ϕ) oneshows that (ii) implies the existence of a net (zα) in the unit sphere of L2(ϕ) such that∥uzαu∗−zα∥L2(ϕ) →0 for all u unitary in M. Then if we define on Nω(x) = limU ⟨xzα, zα⟩L2(ϕ) = limU ϕ(xz∗αzα)we find that ω is an M-hypertrace on N. Moreover since M is a factor, ω restricted to M isthe trace of M. It is then easy to conclude that there is a norm one projection P: N →Mwhich is built exactly like a conditional expectation.8

Proof of Theorem 1.1. By a well known crossed product argument, (cf.

[Ta2]) there isa semi-finite algebra eN with N ⊂eN and a completely contractive projection Q: eN →N.Hence, replacing N by eN we may assume that N is semi-finite.Let P be a projection satisfying (1.1). We first assume M finite.

Then by Lemma 1.4, thesecond assertion in Theorem 1.6 holds. Therefore, by (ii) ⇒(i) in Theorem 1.6 there is anorm one projection from N onto M.Now if M is semi-finite, we can write M = ∪piMpi (weak-∗closure) where pi is anincreasing net of finite projections in M such that pixpi →x in the σ(M, M∗)-sense forall x in M. Clearly x →piP(x)pi is a projection from piNpi onto piMpi which satisfies(1.1) hence by the first part of the proof, there is a norm one projection from piNpi ontopiMpi.

A fortiori there is a norm one projection from N onto piMpi, hence we concludeby Lemma 0.1 that there is a norm one projection eP from N onto M.9

§2. An interpolation theorem.Let N be a semi-finite von Neumann algebra, let 1 ≤p < ∞and let Lp(ϕ) be theclassical non-commutative Lp-space associated to a faithful normal semi-finite trace ϕ onN.

For the construction and the basic properties of Lp(ϕ), the classical references are[D,S,Ku,Sti]. For a more concise and recent exposition, see [N].The following statement extends a result proved in [P1] in the particular case N =B(H).Theorem 2.1.

Fix an integer n ≥1. Let A0 (resp.

A1) be the space N n equipped withthe norm∥(x1, . .

., xn)∥A0 =nX1xix∗i1/2 resp. ∥(x1, .

. ., xn)∥A1 =Xx∗i xi1/2.Then for 0 < θ < 1, the complex interpolation space (A0, A1)θ is the space N n equippedwith the norm∥(x1, .

. .

, xn)∥θ =XLxiRx∗i1/2B(Lp(ϕ))where θ = 1/p.Remark. Note that Theorem 2.1 implies Lemma 1.2 by a well known property of theinterpolation spaces, namely the (norm one) inclusion (A0, A1)θ ⊂A0 + A1, (see [BL] formore details).

But actually, the proof of Theorem 2.1 is quite similar to that of Lemma 1.2,although slightly more technical.We will use Szeg¨o’s classical factorization theorem which says that under a nonva-nishing condition, a positive function W in L1(T) can always be written as W = |F|2(W = FF is more suggestive in view of the non-commutative case) for some F in H2.Moreover, this can be done with F “outer”, so that z →1/F(z) is analytic inside the disc,and if we additionally require F(0) > 0 then F is unique. Actually, we will need an ex-tension of this theorem (due to Devinatz) valid for B(H)-valued functions.

The followingconsequence of Devinatz’s theorem will be enough for our purposes (cf. [D], [He]).Theorem 2.2.

Let H be a separable Hilbert space and let W: T →B(H) be a functionsuch that, for all x, y in H, the function t →⟨W(t)x, y⟩is in L1(T). Assume that there is10

δ > 0 such that W(t) ≥δI for all t. Then there is a unique analytic function F: D →B(H)such that(i) For all x in H, z →F(z)x is in H2(H) and its boundary values satisfy almosteverywhere on T⟨W(t)x, y⟩= ⟨F(t)x, F(t)y⟩,(ii) F(0) ≥0,(iii) z →F(z)−1 exists and is bounded analytic on D.The following corollary was pointed out to me by Uffe Haagerup during our collaborationon [HP1] (we ended up not using it in our paper).Corollary 2.3. Consider a von Neumann subalgebra N ⊂B(H).

Then in the situationof Theorem 2.2, if W is N-valued, F necessarily also is N-valued.Proof. Indeed, for any unitary u in the commutant N ′, the function z →u∗F(z)u stillsatisfies the conclusions of Theorem 2.2, hence (by uniqueness) we must have F = u∗Fu,which implies by the bicommutant theorem thatF(z) ∈N ′′ = N.Proof of Theorem 2.1.

By well known results, N can be written as a direct sum ofσ-finite semi-finite algebras. Hence we can assume that N is σ-finite and that H = L2(ϕ)is separable.Let θ = 1/p.

Let us denote L∞(ϕ) = N. Then it is well known that we have isometrically(L∞(ϕ), L1(ϕ))θ = Lp(ϕ).Clearly if x1, . .

., xn in Nare such that ∥(x1, . .

. , xn)∥0≤1,then we have P LxiRx∗iB(L∞(ϕ))≤1.Similarly, it is easy to check by transposition that if∥(x1, .

. ., xn)∥1 ≤1, then P LxiRx∗iB(L1(ϕ)) ≤1.

Hence, if (x1, . .

., xn) is in the unitball of (A0, A1)θ, we have necessarily by classical interpolation theoryXLxiRx∗iB(Lp(ϕ)) ≤111

where 1/p = θ.This is the easy direction. To prove the converse, we assume that(2.1)XLxiRx∗iB(Lp(ϕ)) ≤1.We will proceed by duality as in the proof of Lemma 1.2.

Let B denote the open unit ballin the space (A0∗, A1∗)θ. Note that A0∗(resp.

A1∗) coincides with N n∗equipped with thenorm ∥(ξ1, . .

., ξn)∥= ϕ[(P ξ∗i ξi)1/2] (resp. ∥(ξ1, .

. ., ξn)∥= ϕ[(P ξiξ∗i )1/2]).

Let Bo bethe polar of B in the duality between N n and N n∗. By a well known duality property ofinterpolation spaces (cf.

[BL,Be]) Bo coincides with the unit ball of (A0, A1)θ. Hence toconclude it suffices to show that (2.1) implies (x1, .

. ., xn) ∈Bo.

Equivalently, to completethe proof it suffices to show that, if (2.1) holds, then for any (ξ1, . .

. , ξn) in B we have P ξi(xi) ≤1.

The rest of the proof is devoted to the verification of this. By density, ifwe identify again N∗with L1(ϕ) in the usual way, we may assume that ξi is of the formξi(x) = ϕ(bix) for some bi in qMq where q is a finite projection in M, i.e.

a projectionwith ϕ(q) < ∞.In that case we have ξi(xi) = ξi(qxiq). Note that (2.1) remains true if we replace (xi)by (qxiq).

Therefore, at this point we may as well replace N by the finite von Neumannalgebra qNq (with unit q) so that we are reduced to the finite case. Hence, for simplicity,we assume in the rest of the proof that N is finite with unit I and that ξi lies in N viewedas a subspace of L1(ϕ) (i.e.

that the elements bi above are in N and q = I). By definitionof (A0∗, A1∗)θ, since (ξi) is in B there are functions fi: S →L1(ϕ) which are bounded,continuous on S and analytic on S such that denoting∂0 = {z ∈C | Rez = 0},∂1 = {z ∈C | Rez = 1}we have ξi = fi(θ) for i = 1, ..., n, with(2.2)supz∈∂0ϕXfi(z)∗fi(z)1/2< 1andsupz∈∂1ϕXfi(z)fi(z)∗1/2< 1.Since ξi is in N ⊂L1(ϕ) and N n is dense in N n∗, we may as well assume by a well knownfact (cf.

[St]) that the functions f1, . .

., fn take their values into a fixed finite dimensionalsubspace of N ⊂L1(ϕ). We are then in a position to use Theorem 2.2 and its corollary.12

Let δ > 0 to be specified later. We define functions W1 and W2 on ∂S = ∂0 ∪∂1 by setting∀z ∈∂1W1(z) =Xfi(z)fi(z)∗1/2+ δI1/2∀z ∈∂0W1(z) = I∀z ∈∂1W2(z) = I∀z ∈∂0W2(z) =Xfi(z)∗fi(z)1/2+ δI1/2.By (2.2) we can choose δ small enough so that(2.3)supz∈∂1ϕ(W 21 ) < 1andsupz∈∂0ϕ(W 22 ) < 1.By Theorem 2.2 and Corollary 2.3, using a conformal mapping from S onto D, we findbounded N-valued analytic functions F and G on S with (nontangential) boundary valuessatisfying(2.4)FF ∗= W 21andG∗G = W 22 .Moreover, F −1 and G−1 are analytic and bounded on S. Therefore we can writefi(z) = F(z)gi(z)G(z)where(2.5)gi(z) = F(z)−1fi(z)G(z)−1.We claim that(2.6)∀z ∈SX∥gi(z)∥2L2(ϕ) ≤1.By the three lines lemma, to verify this it suffices to check it on the boundary of S. (Notethat we know a priori that supz∈S∥gi(z)∥L2(ϕ) < ∞since ∥F −1∥< δ−1/2 and ∥G−1∥≤δ−1/2,hence gi is an H∞function with values in L2(ϕ), and its nontangential boundary valuesstill satisfy (2.5) a.e.

on the boundary of S.) We have∀z ∈∂1X∥gi(z)∥2L2(ϕ) = ϕXgi(z)gi(z)∗= ϕF(z)−1 Xfi(z)fi(z)∗F(z)−1∗= ϕ((FF ∗)−1(W 21 −ε2I)2)hence by (2.4) and (2.3)13

≤ϕ(W 21 ) < 1.Similarly, we find∀z ∈∂0X∥gi(z)∥2L2(ϕ) ≤ϕ(W 22 ) < 1.This proves our claim (2.6). Finally, if θ = 1/p we haveL2p(ϕ) = (N, L2(ϕ))θandL2p′(ϕ) = (L2(ϕ), N)θ.Hence by definition of the latter complex interpolation spaces, since ∥F(z)∥N = ∥W1∥N ≤1on ∂0 and (by (2.3)) ∥F(z)∥L2(ϕ) < 1 on ∂1, we have∥F(θ)∥L2p(ϕ) ≤1and similarly ∥G(θ)∥L2p′(ϕ) ≤1.Therefore we can conclude as in section 1: we haveξi = fi(θ) = F(θ)gi(θ)G(θ), hence if (xi) satisfies (2.1) we have by (2.6) (and Cauchy-Schwarz)Xξi(xi) =Xϕ(F(θ)gi(θ)G(θ)xi)≤X∥G(θ)xiF(θ)∥2L2(ϕ)1/2≤XxiF(θ)F(θ)∗x∗i1/2Lp(ϕ)≤XLxiRx∗i1/2B(Lp(ϕ)) ≤1.Thus we have verified that (2.1) implies (xi) ∈Bo.

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