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Let u: A →B be a bounded linear operator between two C∗-algebras A, B. The

arXiv:math/9302214v1 [math.FA] 18 Feb 1993Bounded linear operatorsbetween C∗-algebrasbyUffe HaagerupandGilles Pisier*PlanIntroduction§1. Operators between C∗-algebras.§2.

Description of Ekn.§3. Random series in non-commutative L1-spaces.§4.

Complements.IntroductionLet u: A →B be a bounded linear operator between two C∗-algebras A, B. Thefollowing result was proved in [P1].Theorem 0.1. There is a numerical constant K1 such that for all finite sequences x1, .

. ., xnin A we havemaxXu(xi)∗u(xi)1/2B,Xu(xi)u(xi)∗1/2B(0.1)1≤K1∥u∥maxXx∗i xi1/2A,Xxix∗i1/2A.A simpler proof was given in [H1].

More recently an other alternate proof appearedin [LPP]. In this paper we give a sequence of generalizations of this inequality.The above inequality (0.1)1 appears as the case of “degree one” in this sequence.

Thenext case of degree 2 seems particularly interesting, we now formulate it explicitly. * Partially supported by the N.S.F.

Let us assume that A ⊂B(H) (embedded as a C∗-subalgebra) for some Hilbert spaceH, and similarly that B ⊂B(K). Let (aij) be an n ×n matrix of elements of A.

We define[(aij)](2) = max((aij)Mn(A),(a∗ij)Mn(A),Pij a∗ijaij1/2A,Pij aija∗ij1/2A).Then we haveTheorem 0.2. There is a numerical constant K2 such that for all n and for all (aij) inMn(A) we have(0.1)2[(u(aij))](2) ≤K2∥u∥[(aij)](2).We recall in passing the following identities for aij ∈A and ai ∈A∥(aij)∥Mn(A) = supXij⟨yi, aijxj⟩,xj, yi ∈HX∥xj∥2 ≤1,X∥yi∥2 ≤1o,andXa∗i ai1/2A= supnX⟨yi, aix0⟩ , x0 ∈H, yi ∈H∥x0∥≤1,X∥yi∥2 ≤1o.We will denote(0.2)[(ai)](1) = maxXa∗i ai1/2A,Xaia∗i1/2A.More generally, let us explain the general case of ”degree k” of our main result.

Let k ≥1.Let n be a fixed integer. We will denote [n] = {1, 2, .

. ., n}.

Let {aJ | J ∈[n]k} be afamily of elements of A indexed by [n]k. Let us denote by Pk the set of all the 2k subsets(including the void set) of {1, 2, . .

., k}.For any α ⊂{1, . .

., k} we denote by αc the complement of α and byπα: [n]k →[n]αthe canonical projection, i.e.∀J = (j1, . .

., jk) ∈[n]kπ(J) = (ji)i∈α.2

For any α with α ̸= ∅and αc ̸= ∅we define(0.3)∥(aJ)∥α = supXJ∈[n]k⟨aJxπα(J), yπαc(J)⟩where the supremum runs over all families{xℓ| ℓ∈[n]α}and{ym | m ∈[n]αc}of elements of H such that P ∥xℓ∥2 ≤1 and P ∥ym∥2 ≤1. There is an alternate descrip-tion, we can identify [n]k with [n]αc × [n]α so that J ∈[n]k is identified with (i, j) withi = παc(J), j = πα(J).

Then ∥(aJ)∥α is nothing but the norm of the matrix (aij) actingfrom ℓ2([n]α, H) into ℓ2([n]c, H). For α = ∅, this definition extends naturally to∥(aJ)∥∅= supXJ∈[n]k⟨aJx0, yJ⟩=XJ∈[n]ka∗JaJ1/2Awhere the supremum runs over all x0 ∈H, yJ ∈H such that ∥x0∥≤1 and P ∥yJ∥2 ≤1.Similarly, for α = {1, .

. ., k} we set∥(aJ)∥α =XaJa∗J1/2A.We then define(0.4)[(aJ)](k) = maxα∈Pk{∥(aJ)∥α}.We can now state one of our main results.Theorem 0.k.

For each k ≥1, there is a constant Kk such that for any bounded linearoperator u: A →B, for any n ≥1 and for any family {aJ | J ∈[n]k} in A we have(0.1)k[(u(aJ))](k) ≤Kk∥u∥[(aJ)](k).Moreover, we have Kk ≤2(3k/2)−1.The proof is essentially in section 1 (it is completed in section 2).3

We now reformulate this result in a fashion which emphasizes the connection with thenotion of complete boundedness for which we refer to [Pa].Let A ⊂B(H) be a C∗-algebra embedded as a C∗-subalgebra. (H a Hilbert space.

)We denote as usual by Mn the set of all n × n complex matrices (equipped with the normof the space B(ℓn2)) and by Mn(A) the space Mn ⊗A equipped with its natural C∗-norm,induced by B(ℓn2(H)). More generally, let S ⊂B(H) be any closed linear subspace of B(H)(H is a Hilbert space).

We call S an “operator space”.We denote by S ⊗A the completion of the linear space S ⊗A equipped with the norminduced by B(H ⊗2 H) (here H ⊗2 H denotes the Hilbert space tensor product of H andH). We will repeatedly use the following fact (for a proof see Lemma 1.5 in [DCH]).

Let Kbe an arbitrary Hilbert space. Whenever u : S →B(K) is completely bounded, the mapIA ⊗u : A ⊗S →A ⊗B(K) is bounded and we have(0.5)∥IA ⊗u∥A⊗S→A⊗B(K) ≤∥u∥cb.Clearly S ⊗A is again an operator space embedded into B(H ⊗2 H).For example, we will need to consider a particular embedding of the Euclidean spaceℓn2 into Mn ⊕Mn as follows.

(We equip Mn ⊕Mn with the norm ∥(x, y)∥= max{∥x∥, ∥y∥},for which it clearly is an operator space embedded – say – into M2n in a block diagonalway.) We denote by En the subspace of Mn ⊕Mn formed by all the elements of the formx1...⃝xn⊕x1.

. .xn⃝with x1, .

. ., xn ∈C.

Let (eij) be the usual basis of Mn. We denote byδi = ei1 ⊕e1ithe natural basis of En, (so that the above element can be written as P xiδi.) As aBanach space, En is clearly isometric to ℓn2.

More precisely, for any C∗-algebra A and forany a1, . .

., an in A we have (this known fact is easy to check)Xδi ⊗aiEn⊗A = maxXa∗i ai1/2 ,Xaia∗i1/2(0.6)or equivalently4

= [(ai)](1)in the preceding notation.Let us denote by Ekn the tensor productEn ⊗· · · ⊗En(k times).Then, Theorem 0.k implies (and is actually equivalent to) the following.Proposition 0.k. For any u: A →B∥IEkn ⊗u∥Ekn⊗A→Ekn⊗B ≤2(3k/2)−1∥u∥.This proposition is proved in section 1.

In section 2 we extend (0.6) and compute thenorm of an element of Ekn ⊗A for k > 1 to deduce Theorem 0.k from Proposition 0.k.In section 3, we develop the viewpoint of [LPP] which dualizes inequalities such as(0.1)1 or (0.1)k to compute (an equivalent of) the norm of certain random series withcoefficients in a non-commutative L1-space. Let (εj)j∈N be an i.i.d.

sequence of randomvariables each distributed uniformly over the unimodular complex numbers. (Such vari-ables are sometimes called Steinhaus variables.) Let A∗be a non-commutative L1-space.Roughly, while [LPP] treats the case of A∗-valued random variables which depend linearlyon the sequence (εj), we can treat variables which depend bilinearly or multilinearly in thevariables (εj).

For a precise statement see Theorem 3.6 below.It might be useful for some readers to emphasize that the variables (εj) can be replacedby independent choices of signs or more importantly by i.i.d. Gaussian variables.

All ourresults remain true in this setting, but with different numerical constant, this follows fromthe fact (due to N. Tomczak-Jaegermann) that A∗is of cotype 2, see e.g. [P3] p. 36 for moredetails.

We also would like to draw the reader’s attention to Kwapie´n’s paper [K] whichcontains “decoupling inequalities” quite relevant to the situation considered in Theorem 3.6below. Using [K] one can deduce from (3.1) below some “non-decoupled” inequalities.

Forinstance, we can find an equivalent of integrals of the formRP1≤i

probability space (Ω, P), and similarly in the multilinear case. We will not spell out thedetails.The results of the first three sections of this paper rely heavily on the following fac-torization result proved in section 1: The identity map IEn on the operator space En hasa completely bounded factorization through the von Neumann algebra V N(Fn) associ-ated with the left regular representation of the free group with n generators, i.e.

there arewn : En →V N(Fn) and vn : V N(Fn) →En such thatIEn = vnwnand∥vn∥cb∥wn∥cb ≤2.In section 4, we show that for any sequence of factorizations IEn = vnwn (n = 1, 2, ...) ofthe identity maps IEn through injective von Neumann algebras we havelimn→∞∥vn∥cb∥wn∥cb = +∞.Combining these two facts about the factorization of IEn with Voiculescu’s recent result([V1]) that the algebra of all n×n matrices over V N(F∞) is isomorphic (as a von Neumannalgebra) to V N(F∞), we show at the end of section 4 that the von Neumann algebraV N(Fn) is not a complemented subspace of B(H) for any n ≥2. (For very recent resultson similar questions, see [P4,CS].) We also include several general remarks about therelation between the existence of a -completely bounded linear projection from B(H) ontoa subspace S and that of a bounded linear projection from B(ℓ2) ⊗B(H) onto B(ℓ2) ⊗S.For instance, if S is weak-∗closed and if B(ℓ2)⊗S denotes the weak-∗closure of B(ℓ2)⊗Sin B(ℓ2 ⊗H), we show that there is a bounded linear projection from B(ℓ2 ⊗H) ontoB(ℓ2)⊗S if and only if there is a completely bounded one from B(H) onto S.Finally, we compare the space En with the linear span Sn of a free system of randomvariables {x1, ..., xn} in a C∗-probability space (A, ϕ) in the sense of Voiculescu [V1,2].

Inparticular, in the case of a semicircular (or circular) system in Voiculescu’s sense, we showthat there is an isomorphism u from En onto the operator space Sn such that∥u∥cb∥u−1∥cb ≤2.6

§1. Operators between C∗-algebras.We will use repeatedly the following fact which has been known to the first authorfor some time.

The main point ((1.2) below) is a refinement of one of the inequalities of[H2]. (We remind the reader that we denote simply by C∗λ(Fn) ⊗A the minimal or spatialtensor product which is often denoted by C∗λ(Fn) ⊗min A.

)Proposition 1.1. Let Fn denote the free group on n generators g1, .

. ., gn, and let C∗λ(Fn)be the reduced C∗-algebra of Fn, i.e.

the C∗-algebra generated by the left regular repre-sentation λ: Fn →B(ℓ2(Fn)). Then(1) For any C∗-algebra A and for any set (ag)g∈S of elements of A indexed by a finitesubset S of Fn:(1.1)Xg∈Sλ(g) ⊗agC∗λ(Fn)⊗A≥maxXg∈Sa∗gag1/2,Xg∈Saga∗g1/2.

(2) For any C∗-algebra A and for any set (ag)g∈G of elements of A indexed by a subset Sof {g1, . .

., gn, g−11 , . .

., g−1n }:(1.2)Xg∈Sλ(g) ⊗agC∗λ(Fn)⊗A≤2 maxXg∈Sa∗gag1/2,Xg∈Saga∗g1/2.Proof. (1) let (δg)g∈G be the standard basis of ℓ2(Fn).

We may assume that A ⊂B(K)for some Hilbert space K. Since the min-tensor product coincide with the spatial tensorproduct, we have for all unit vectors ξ ∈K:Xλ(g) ⊗agC∗λ(Fn)⊗A ≥Xg∈S(λ(g) ⊗ag)(δe ⊗ξ)=Xg∈Sδg ⊗agξ=Xg∈G∥agξ∥21/2=Xg∈Ga∗gagξ, ξ1/27

Taking supremum over all unit vectors ξ ∈K we getXg∈Sλ(g) ⊗agC∗λ(Fn)⊗A≥Xg∈Sa∗gag1/2.The same argument applied to the norm of (λ(g) ⊗ag)∗= λ(g−1) ⊗a∗g givesXg∈Gλ(g) ⊗agC∗λ(Fn)⊗A≥Xg∈Saga∗g1/2.This proves (1). Note that the statement (1) actually holds in C∗λ(Γ) ⊗A for any discretegroup Γ.

(2) Consider first the case S = {g1, . .

. , gn, g−11 , .

. ., g−1n }.

We can write Fn as a disjointunion:Fn = {e} ∪( n[i=1Γ+i)∪( n[i=1Γ−i)whereΓ+i = set of reduced words starting with a positive power of gi,Γ−i = set of reduced words starting with a negative power of gi.Let e0, e+iand e−idenote the orthogonal projection of ℓ2(Fn) onto the subspaces Cδe,ℓ2(Γ+i ) and ℓ2(Γ−i ) respectively. Then these projections are pairwise orthogonal ande0 +nXi=1e+i +nXi=1e−i = Iℓ2(Fn).For any g ∈G and for any generator gi, the length of the reduced word for gig is either|gig| = |g| + 1or|gig| = |g| −1.The first case exactly occurs when gig starts with an element of Γ+i and the second casewhen g starts with an element of Γ−i .

Hence for all g ∈G:λ(gi)δg =e+i λ(gi)δgif|gig| = |g| + 1λ(gi)e−i δgif|gig| = |g| −1= e+i λ(gi)δg + λ(gi)e−i δg(all cases).8

Thereforeλ(gi) = e+i λ(gi) + λ(gi)e−iand by taking adjoints:λ(g−1i) = e−i λ(g−1i) + λ(g−1i)e+i .Setui = e+i λ(gi),un+i = e−i λ(g−1i)vi = λ(g−1i)e−i ,vn+i = λ(g−1i)e+i)i = 1, . .

., nand for simplicity of notation, set also gn+i = g−1i, i = 1, . .

., n. Thenλ(gi) = ui + vi,i = 1, . .

., 2n.SincenPi=1(e+i + e−i ) = 1 −e0 we have2nXi=1uiu∗i =2nXi=1v∗i vi = 1 −e0 ≤1.So2nXi=1uiu∗i ≤1and2nXi=1v∗i vi ≤1.For elements c1, . .

., cm, d1, . .

., dm of a C∗-algebra B one has easily thatmXi=1cidi ≤mXi=1cic∗i1/2 mXi=1d∗i di1/2.Hence, with u1, . .

. , u2n, v1, .

. ., v2n as above and a1, .

. ., a2n ∈A,2nXi=1ui ⊗aiC∗r (Fn)⊗A=nXi=1(ui ⊗1)(1 ⊗ai)C∗r (Fn)⊗A≤Xuiu∗i1/22nXi=1a∗i ai1/2≤Xa∗i ai1/29

and similarlyXvi ⊗aiC∗r (Fn)⊗A ≤nXi=1(1 ⊗ai)(vi ⊗1)C∗r (Fn)⊗A≤Xaia∗i1/2nXi=1v∗i vi1/2≤Xaia∗i1/2so altogether2nXi=1λ(gi) ⊗ai =2nXi=1ui ⊗ai +2nXi=1vi ⊗ai≤2nXi=1a∗i ai1/2+2nXi=1aia∗i1/2≤2 max2nXi=1a∗i ai1/2,2nXi=1aia∗i1/2.This proves (2) in the case S = {g1, . .

., gn, g−11 , . .

., g−1n }, and the remaining cases followsfrom this by setting some of the ag’s equal to 0.Remark. The preceding statement remains true (with the obvious modifications) for thefree group on infinitely many generators.

See also Proposition 4.9 below for a generalizationof (1.1) and (1.2).Remark 1.2. The proof of (2) is an illustration of the following general principle.

LetT1, . .

., Tn be operators on a Hilbert space H and let c be a constant. The followingproperties are essentially equivalent:(i)c For any C∗-algebra A and any set (ai)i≤n in A we haveXTi ⊗ai ≤c maxXa∗i ai1/2 ,Xaia∗i1/2.

(ii)c There are operators ui, vi in B(H) such that Ti = ui + vi andXu∗i ui1/2 +Xviv∗i1/2 ≤c.10

More precisely, we have (ii)c ⇒(i)c and (i)c ⇒(ii)2c The implication (ii)c ⇒(i)c followsas above from the triangle inequality. To prove the converse, note that (i)c equivalentlymeans that the operator u: En →B(H) which maps δi to Ti satisfies ∥u∥cb ≤1.

By theextension property of c.b. maps (cf.

[Pa, p.100]) there is an extension ˜u: Mn ⊕Mn →B(H)such that ˜u(δi) = Ti and ∥˜u∥cb ≤1. Letting ui = ˜u(ei1 ⊕0) and vi = ˜u(0 ⊕e1i) we obtaina decomposition satisfying (ii)2c.

This shows that (i)c implies (ii)2c.Proposition 1.3. Let En ⊂Mn ⊕Mn be the operator spaceEn =c1...⃝cn⊕c1.

. .cn⃝ c1, .

. ., cn ∈C.Then there are linear mappingsw: En →C∗λ(Fn)andv: C∗λ(Fn) →Ensuch thatvw = IEnand∥v∥cb∥w∥cb ≤2.Similarly, for the von Neumann algebra V N(Fn) generated λ, there are linear mappingsw1: En →V N(Fn)andv1: V N(Fn) →Ensuch thatv1w1 = IEnand∥v1∥cb∥w1∥cb ≤2.In particular En is cb-isomorphic to a cb-complemented subspace of C∗λ(Fn) (resp.

ofV N(Fn)).Proof. Let (δ1, .

. ., δn) be the basis of En determined bynXi=1ciδi =c1...⃝cn⊕c1.

. .cn⃝for c1, .

. ., cn ∈C.

Define w: En →C∗λ(Fn) byw nXi=1ciδi!=nXi=1ciλ(gi)11

and v: C∗λ(Fn) →En byv(x) =nXi=1τ(λ(gi)∗x)δiwhere τ is the trace on C∗λ(Fn) given byτ(y) = (yδe, δe),y ∈C∗λ(Fn).(cf. [KR, p. 433]).For any set a1, .

. ., an of n elements in a C∗-algebra A(w ⊗IA) nXi=1δi ⊗ai!=nXi=1λ(gi) ⊗ai.SincenXi=1ei ⊗ai =a1...⃝an⊕a1.

. .an⃝= maxnXi=1a∗i ai1/2,nXi=1aia∗i1/2it follows from Theorem 1.1 (2), that ∥w ⊗IA∥≤2.

Hence ∥w∥cb ≤2. Sinceτ(λ(g)∗λ(h)) =1g = h0g ̸= hwe get for any finite subset S ⊂Fn and scalars (cg)g∈SvXg∈Scgλ(g)=nXi=1(gi∈S)cgiδiand hence(v ⊗IA)Xg∈Sλ(g) ⊗a(g)=nXi=1(gi∈S)δi ⊗a(gi).Let S1 = S ∩{g1, .

. .

, gn}. ThennXi=1gi∈Sδi ⊗a(gi)= maxXg∈S1a(g)∗a(g)1/2,Xg∈S1a(g)a(g)∗1/2≤maxXg∈Sa(g)∗a(g)1/2,Xg∈Sa(g)a(g)∗1/2,12

which by Theorem 1.1(1) is smaller than or equal toXg∈Sλ(g) ⊗a(g)C∗r (Fn)⊗A.Hence ∥v ⊗IA∥≤1 and thus ∥v∥cb ≤1. Therefore∥v∥cb∥w∥cb ≤2and by construction vw = IEn.

This implies that w is a cb-isomorphism of En onto itsrangew(En) = span{λ(gi) | i = 1, . .

., n}and∥w∥cb∥w−1∥cb ≤2.Moreover P = wv is a completely bounded projection of C∗λ(Fn) onto w(En) and ∥P∥cb ≤2. The proof with V N(Fn) in the place of C∗λ(Fn) is easy since v admits an extensionv1 : V N(Fn) →En with ∥v1∥cb ≤1.

We leave the details to the reader.Lemma 1.4. ([P1, H1, LPP]).

Let u: A →B be a bounded linear operator between twoC∗-algebras A and B. Then for every n ∈N∥IEn ⊗u∥En⊗A→En⊗B ≤√2 ∥u∥.Proof.

The statement of the lemma is equivalent to: For all a1, . .

., an ∈A(1.3)maxnXu(ai)∗u(ai) ,Xu(ai)u(ai)∗o≤2∥u∥2 maxnXa∗i ai ,Xaia∗io.This is essentially [P1], (see also [H1,LPP]). However to get the constant 2 in (1.3) one hasto modify the proof of [H1, Cor.

3.4] slightly:Let T: A →H be a bounded linear operator from the C∗-algebra A with values in aHilbert space. By [H1, Thm.

3.2],(1.4)X∥T(ak)∥2 ≤∥T∥2 Xa∗kak +Xaka∗k.13

We can assume, that B ⊆B(K) for some Hilbert space K. By the above inequality (1.4)we get for any ξ ∈K, thatX∥u(ak)ξ∥2 ≤∥ξ∥2∥u∥2 Xa∗kak +Xaka∗k.Clearly (1.4) also holds for conjugate linear maps, soX∥u(ak)∗ξ∥2 ≤∥ξ∥2∥u∥2 Xa∗kak +Xaka∗k.ThusmaxnXu(ak)∗u(ak) ,Xu(ak)u(ak)∗o≤∥u∥2 Xa∗kak +Xaka∗kwhich implies (1.3).Theorem 1.5. Let u: A →B be a bounded linear operator between two C∗-algebras Aand B.

Then for every k, n ∈N∥IEkn ⊗u∥Ekn⊗A→Ekn⊗B ≤232 k−1∥u∥.Proof. The theorem is proved by induction on k. By Lemma 1.4 the theorem holds fork = 1.

Assume next that the theorem is true for a particular k ∈N. Letw: En →C∗λ(Fn)andv: C∗λ(Fn) →Enbe as in Proposition 1.2, and let u: A →B be a linear map between two C∗-algebras Aand B.

Clearly(1.5)IEn ⊗u = (v ⊗u)(w ⊗IA)where∥v ⊗u∥= ∥(v ⊗IB)(IEn ⊗u)∥≤∥v∥cb∥IEn ⊗u∥≤√2 ∥u∥∥v∥cb14

by Lemma 1.4. Moreover v ⊗u maps the C∗-algebra C∗λ(Fn) ⊗A into the C∗-algebraMn(B) ⊕Mn(B), so by the induction hypothesis∥IEkn ⊗v ⊗u∥≤232 k−1∥v ⊗u∥≤232 k−12 ∥u∥∥v∥cb.On the other hand by (0.5)∥IEkn ⊗w ⊗IA∥= ∥IEkn⊗A ⊗w∥≤∥w∥cbNow by (1.5)IEk+1n⊗u = (IEkn ⊗v ⊗u)(IEkn ⊗w ⊗IA).Thus, by Proposition 1.3∥IEk+1n⊗u∥≤232 k−12 ∥u∥∥v∥cb∥w∥cb≤232 k+ 12 ∥u∥= 232 (k+1)−1∥u∥.Hence Theorem 1.5 follows by induction on k.15

§2. Description of Ekn.In this section, we will identify the norm in the space Ekn⊗A with the norm previouslyintroduced in (0.3) and (0.4) as [](k).Proposition 2.1.

Let A be any C∗-algebra. Let n ≥1, k ≥1 and let {aJ|J ∈[n]k} beelements of A.

Then(2.1)[(aJ)](k) =XJ∈[n]kδJ ⊗aJEkn⊗Awhere we denote if J = (j1, ..., jk)δJ = δj1 ⊗... ⊗δjkThe proof below is easy but the notation is a bit painful. Using Proposition 2.1 we cancomplete the proof of the results announced in the introduction.Proof of Theorem 0.k : Consider an operator u : A →B between C∗-algebras.

ByTheorem 1.5 we have for all (aJ) in AXδJ ⊗u(aJ)Ekn⊗B ≤2(3k/2)−1 ∥u∥XδJ ⊗aJEkn⊗A .Taking (2.1) into account this immediately implies (0.1)k and completes the proof of The-orem 0.k.We now check (2.1). We will need the following elementary factLemma 2.2.

Let H, H1, H2, H3, H4 be Hilbert spaces. Let e ∈H1, f ∈H4 be norm onevectors.

Let (ϕj)j∈J and (ψi)i∈I be orthonormal finite sequences in H2 and H3 respectively.Let aij be elements of a C∗-algebra A embedded into B(H). Then we have(2.2)Xi∈Ij∈J(e ⊗ϕj) ⊗(ψi ⊗f) ⊗aij = supyi∈Hxj∈HnXi,j< yi, aij−xj > ,X∥xj∥2 ≤1,X∥yi∥2 ≤1o.Here the norm on the left hand side means the norm in the space of all bounded operatorsfrom H1 ⊗2 H2 ⊗2 H into H3 ⊗2 H4 ⊗2 H.Proof.

We may clearly assume without loss of generality that H1 =|Ce, H4 =|Cf andthat (ϕj) (resp. (ψi)) is a basis of H2 (resp.

H3). Then the norm we want to compute is16

clearly equal to the norm of the operator˜T =Xijϕj ⊗ψi ⊗aijas an operator from H2 ⊗2 H to H3 ⊗2 H. But then the general form of an element in theunit ball of H2 ⊗2 H (resp. H3 ⊗2 H) is given by P ϕj ⊗xj (resp.

P ψi ⊗yi) with xj ∈H2(resp. yi ∈H3) such that P ∥xj∥2 ≤1 (resp.

P ∥yi∥2 ≤1). Hence the norm of ˜T (or of T)is equal to the right hand side of (2.2).We need to introduce more notation.Recall that En ⊂Mn⊕Mn and δi = ei1 −⊕e1i.

We consider of course Mn⊕Mn as a subsetof the set of all operators on ℓn2 ⊕ℓn2. It will be convenient to denote e0ij = eij ⊕0 ande1ij = 0⊕eij in Mn⊕Mn.

Also e0i = ei⊕0 and e1i = 0⊕ei in ℓn2 ⊕ℓn2. As is usual, for e, f in H,we will identify the tensor e⊗f −with the operator x →< e, x > f ( defined on H).

Hencein tensor product notation we have (with the usual matricial conventions) eij = ej ⊗eiand δi = e01 ⊗e0i + e1i ⊗e11. Let us denote by H0 the span of {ei1|i = 1, ..., n} in Mn and byH1 the span of {e1i|i = 1, ..., n} in Mn, so that En ⊂H0 ⊕H1.

Let P0 : H0 ⊕H1 →H0(resp. P1 : H0 ⊕H1 →H1) denote the canonical projection.

We have E⊗kn⊂(H0 ⊕H1)⊗k.For α ∈{0, 1}k we denotePα : (H0 ⊕H1)k →(H0 ⊕H1)kthe projection defined byPα = Pα(1) ⊗Pα(2) ⊗... ⊗Pα(k).Let us denote by IX the identity on X. Then we have(2.3)I(H0⊕H1)⊗k = (IH0⊕H1)⊗k= (P0 + P1)⊗k=Xα∈{0,1}kPα(0) ⊗... ⊗Pα(k)=XαPα17

Proof of Proposition 2.1 : Let T = PJ∈[n]k δJ ⊗aJ. By (2.3) we haveT =XαTαwhereTα =XJPα(δJ) ⊗aJ.We now claim that(2.4)∥Tα∥= ∥(aJ)∥α.To check this, we can assume for simplicity (up to a permutation of the factors in the tensorproduct) that α is the indicator function of the set {1, 2, ..., p} for some p with 1 ≤p ≤k.Then if J = (j1, ..., jk) we have(2.5)Pα(δJ) = e1j11 ⊗... ⊗e1jp1 ⊗e01jp+1 ⊗...

⊗e01jk. (Recall the convention that the tensor e ⊗f represents the operator x →< e, x > f).

Lete1(α) = e11 ⊗... ⊗e11-(p times) and f 0(α) = e01 ⊗... ⊗e01(k −p times).Then (2.5) yieldsPα(δJ) = (e1(α) ⊗e0jp+1 ⊗... ⊗e0jk) ⊗(e1j1 ⊗... ⊗e1jp ⊗f 0(α)).If we now write eε{j1,...,jp} instead of eεj1 ⊗... ⊗eεjp for ε = 0 or 1, we can rewrite the lastidentity as(2.6)Pα(δJ) = (e1(α) ⊗e0πα(J)) ⊗(e1παc (J) ⊗f 0(α)),where we recall that πα : [n]k →[n]α denotes the canonical projection. Then the abovelemma 2.2 gives in the present particular case∥Tα∥=XJ(e1(α) ⊗e0πα(J)) ⊗(e1παc(J) ⊗f 0(α)) ⊗aJ = ∥(aJ)∥α.This proves our claim (2.4).18

Now, we can conclude. Let us denote h0 = ℓn2 ⊕0 and h1 = 0 ⊕ℓn2 in ℓn2 ⊕ℓn2.

Let Kα bethe support of Tα (i.e. the orthogonal of its kernel) and let Rα be the range of Tα.

Thenthe preceding formula (2.6) shows that Kα is equal to the tensor product F1 ⊗F2 ⊗...⊗FkwhereFj =|Ce11 −if −j ∈αandFj = h0 −if −j ̸∈α.It follows that the subspaces (Kα) are mutually orthogonal. Similarly, the family (Rα) ismutually orthogonal.

By a well known estimate it follows thatXTα = maxα∥Tα∥.This completes the proof.19

§3. Random series in non-commutative L1-spaces.Let A be a von Neumann algebra with a predual denoted by A∗.Let ξ1, .

. ., ξn ∈A∗and let (recall (0.2))[(ξi)]∗(1) = supX⟨ξi, ai⟩ ai ∈A [(ai)](1) ≤1.For instance, if A = B(H), A∗= C1(H) (the space of trace class operators on H) and wehave clearly[(ξi)]∗(1) = inftrXx∗i xi1/2+ trXyiy∗i1/2where the infimum runs over all decompositions ξi = xi + yi in C1(H).Let TN be the infinite dimensional torus equipped with its normalized Haar measureµ.

The following result is proved in [LPP].For all ξ1, . .

. , ξn in A∗(3.1)12[(ξi)]∗(1) ≤Z nXj=1eitjξjA∗dµ(t) ≤[(ξi)]∗(1).

(See Theorem 3.3 below and its proof. )It is easy to deduce from (3.1) a necessary and sufficient condition for a series of theformS(t) =∞Xj=1eitjξj ,t = (tj)j∈N ∈TNto converge in L2(TN, µ; A∗).

The aim of this section is to prove a natural extension of(3.1) to double series of the formS(t′, t′′) =∞Xj,k=1eit′jeit′′k ξjkwith ξjk ∈A∗, t′, t′′ ∈TN. More generally, we will consider for any k ≥1, elements ξj1j2...jkin A∗and will find an equivalent for the expressionZ Xj1≤n,...,jk≤neit1j1 .

. .eitkjk ξj1j2...jkA∗dµ(t1) .

. .

dµ(tk).20

See Theorem 3.6 below for an explicit statement.Let A be a C∗-algebra throughout this section. We will denote simplyCn = C∗λ(Fn)andCkn = Cn ⊗· · · ⊗Cn(k times).We always equip the tensor products such as En ⊗A, Cn ⊗A, Ckn ⊗A with the spatial (orminimal) tensor product.

More precisely, whenever S ⊂B(K) is an operator space andA ⊂B(H) is a C∗-algebra, we will denote by S ⊗A the linear tensor product equippedwith the norm induced by B(K ⊗2 H).Let G be a discrete group. For t ∈G, let λ∗(t) denote the element of C∗λ(G)∗given by∀a ∈C∗λ(G)⟨λ∗(t), a⟩= ⟨aδe, δt⟩.Clearly⟨λ∗(s), λ(t)⟩=1if s = t0otherwise.Note that if C∗λ(G)∗is identified with Bλ(G) in the usual way (see for instance [E]) thenλ∗(t) simply corresponds to the function δt.For any J = (j1, .

. ., jk) ∈[n]k we denote by gJ the element of (Fn)k defined bygJ = (gj1, .

. ., gjk).Then with the obvious identificationC∗λ((Fn)k) = Cknwe have λ(gJ) = λ(gj1) ⊗· · · ⊗λ(gjk).

We will also consider the dual E∗n of the space Enconsidered in section 1 and will denote by {δ∗j } the basis of E∗n which is biorthogonal to{δj}. We will also consider Ekn = En ⊗· · · ⊗En (k times) and its dual (Ekn)∗.

We willdenote for any J = (j1, . .

., jk) in [n]kδ∗J = δ∗j1 ⊗· · · ⊗δ∗jk ∈(Ekn)∗21

andλ∗(gJ) = λ∗(gj1) ⊗· · · ⊗λ∗(gjk) ∈(Ckn)∗.We will denote by Ωthe infinite dimensional torus i.e. we setΩ= TNand we equip Ωwith the normalized Haar measure µ.

(In most of what follows, it wouldbe more appropriate to replace Ωby Ωn = Tn, but we try to simplify the notation.) Wewill denote byεj: Ω→Tthe sequence of the coordinate functions on Ω.

Moreover, we will consider the product Ωkequipped with the product measure µk. For any J = (j1, .

. ., jk) ∈[n]k, let εJ: Ωk →Tbe the function defined by∀(t1, .

. ., tk) ∈ΩkεJ(t1, .

. ., tk) = εj1(t1) .

. .

εjk(tk).Equivalently εJ = εj1 ⊗· · · ⊗εjk. We first record a simple consequence of Proposition 1.3.Lemma 3.1.

For any {εj | j ≤n} in A∗we have12Xδ∗j ⊗ξj(En⊗A)∗≤Xλ∗(gj) ⊗ξj(Cn⊗A)∗≤Xδ∗j ⊗ξj(En⊗A)∗.Proof. Let v, w be as in Proposition 1.3.

Since wδj = λ(gj) and v(λ(gj)) = δj we have(w ⊗IA)∗(λ∗(gj)⊗ξ) = δ∗j ⊗ξj and (v ⊗IA)∗(δ∗j ⊗ξj) = λ∗(gj)⊗ξj. Hence, recalling (0.5),Lemma 3.1 follows from ∥w∥cb ≤2 and ∥v∥cb ≤1.The next lemma is rather elementary.Lemma 3.2.

(i) Consider {ξij | i, j = 1, ..., n} in A∗. For any orthonormal systems ϕ1, ..., ϕn andψ1, ..., ψn in L2(µ) (where µ is a probability as above) we have(3.2)Z∥Xϕi(t)ψj(s)ξij∥A∗dµ(t)dµ(s) ≤∥(ξij)∥Mn(A)∗.22

(ii) For any k ≥1 and any (ξJ) in A∗we have(3.3)XJ∈[n]kεJξJL1(µ;A∗)≤XJ∈[n]kδ∗J ⊗ξJ(Ekn⊗A)∗.Proof: (i) To prove this, it clearly suffices to assume that A is a von Neumann algebraand that ξij ∈A∗. Since Mn(A) is a subspace of Mn(B(H)) for some Hilbert space H,by duality its predual Mn(A)∗is a quotient of Mn(B(H))∗.

This shows that it sufficesto prove (i) for A = B(H) and ξij ∈B(H)∗. Then we can identify Mn(B(H))∗with theprojective tensor product ℓn2(H)∗ˆ⊗ℓn2(H).

Consider an element x (resp. y) in the unit ballof ℓn2(H) (resp.

ℓn2(H)∗). Let ξ be the element of Mn(B(H))∗defined by ξ = y ⊗x orequivalently, ξ = (ξij) with ξij = yj ⊗xi.

For such a ξ we have(Z∥Xϕi(t)ψj(s)ξij∥2A∗dµ(t)dµ(s))1/2 = (Z∥Xϕi(t)xi∥2dµ(t)Z∥Xψj(s)yj∥2dµ(s))1/2= ∥x∥∥y∥≤1.Since the unit ball of Mn(B(H))∗is the closed convex hull of elements of this form, weobtain (3.2). (ii) Consider a subset α ⊂{1, ..., k}.

We denote by αc its complement. Recall that forelements (aJ)J∈[n]k in A the norm ∥(aJ)∥α defined in (0.3) can be viewed as the norm of amatrix acting from ℓ2([n]α, H) into ℓ2([n]αc, H).

Therefore we deduce from (3.2) that forany (ξJ)J∈[n]k in A∗we have(3.4)Z XJ∈[n]kεJξJA∗dµk ≤∥(ξJ)∥∗α.Observe that by duality (2.1) has the following consequence.IfPJ∈[n]k δ∗J ⊗ξJ(Ekn⊗A)∗≤1 then there is a decompositionξJ =Xα⊂{1,...,k}ξαJwithXα∥(ξαJ )∥∗α ≤1.Therefore (3.3) follows from (3.4) and the triangle inequality.We now reformulate the main result of [LPP] in our framework.23

Theorem 3.3. For any {ξj | j ≤n} in A∗we have(3.5)XεjξjL1(µ;A∗) ≤Xδ∗j ⊗ξj(En⊗A)∗≤2XεjξjL1(µ;A∗) .Proof.

The left side is (3.3) above for k = 1. By our earlier analysis of En ⊗A, the rightside is clearly equivalent to the following fact.Assume P εjξjL1(µ;A∗) < 1.

Then there is a decomposition ξj = xj +yj in A∗such that∀(aj) ∈AX⟨xj, aj⟩ ≤Xa∗jaj1/2and −X⟨yj, aj⟩ ≤Xaja∗j1/2 .This is precisely what is proved in section II of [LPP], except that the sequence (εj) on Ωis replaced by the sequence (ei3jt) on the one dimensional torus. By a routine averagingargument, one can then obtain the preceding fact as stated above with (εj).

(Note actuallythat the approach of [LPP] can be developed directly for the functions (εj), this is explicitedin [P2]. )We now relate certain series on Zn (formed by iterating the expressions appearing inTheorem 3.3) with the corresponding series on the free group Fn = Z ∗· · · ∗Z.

In otherwords, our aim is to compare for these series the free group Fn with n generators with itscommutative counterpart Zn.Lemma 3.4. For any {ξJ | J ∈[n]k} in A∗we have (the summation being over all J in[n]k)2−k XεJξJL1(µk;A∗) ≤Xλ∗(gJ) ⊗ξJ(Ckn⊗A)∗≤2k XεJξJL1(µk;A∗) .Proof.

By the preceding three statements, we know that this holds for k = 1. We nowargue by induction.

Assume Lemma 3.4 proved for an integer k ≥1, and let us prove itfor k + 1.Consider elements {ξJj | J ∈[n]k, j ∈[n]} in A∗. We haveXJ′∈[n]k+1λ∗(gJ′) ⊗ξJ′ =XJ∈[n]kλ∗(gJ) ⊗Xj≤nλ∗(gj) ⊗ξJj.24

By the induction hypothesis, we have(3.6)XJ′λ∗(gJ′) ⊗ξJ′(Ck+1N⊗A)∗≤2kZΩkXεJ(t)ηJ(C⊗A)∗dµk(t)where ηJ = Pjλ∗(gj) ⊗ξJj.Now for each fixed t in Ωk, we have by (3.5) and Lemma 3.1XεJ(t)ηJ(C⊗A)∗≤2Z XεJ(t)Xεj(s)ξJjA∗dµ(s).Integrating over t ∈Ωk this yields(3.7)Z XεJ(t)ηJ(C⊗A)∗dµk(t) ≤2Z XJ′∈[n]k+1εJ′ξJ′A∗dµ(k+1),hence (3.6) and (3.7) yield the induction step for k + 1. This concludes the proof for theright side inequality in Lemma 3.4.

The proof of the other inequality is entirely similar.We now come to the main result of this section.Theorem 3.5. For any {ξJ | J ∈[n]k} in A∗, we haveXεJξJL1(µk,A∗) ≤Xδ∗J ⊗ξJ(Ekn⊗A)∗≤22k XεJξJL1(µk,A∗) .Proof.

With v and w as in Proposition 1.1, we have ∥w⊗k∥cb ≤2k, hence by (0.5)∥w⊗k ⊗IA∥Ekn⊗A→Ckn⊗A ≤2k.Moreover, we have w⊗k(δJ) = λ(gJ) hence (w⊗k ⊗IA)∗(λ∗(gJ)⊗ξJ) = δ∗J ⊗ξJ. This yieldsXδ∗J ⊗ξJ(Ekn⊗A)∗≤2k Xλ∗(gJ) ⊗ξJ(Ckn⊗A)∗.Combined with Lemma 3.4 this gives the right side in Theorem 3.5.

The left side hasalready been proved in Lemma 3.2.Remark. A slight modification of our proof yields Theorem 3.5 with the constant 22k−1instead of 22k.25

Remark. Let k be a fixed integer.

Consider the mappingQk: C(Ωk) →Ekndefined by∀f ∈C(Ωk)Qk(f) =XJ∈[n]kˆf(J)δJ,where ˆf is the Fourier transform of f, i. e. ˆf(J) =Rf(t)¯ǫJ(t)dµk(t). Let Nk = Ker(Qk).Dualizing (3.3) we find that ∥Qk∥cb ≤1.

Hence, considering Qk modulo its kernel andequipping C(Ωk)/Nk with its quotient operator space structure (in the sense of [BP,ER]),we find a mapUk : C(Ωk)/Nk →Eknwith∥Uk∥cb ≤1.Then Theorem 3.5 admits the following dual reformulation: Uk : C(Ωk)/Nk →Ekn is acomplete isomorphism and ∥U −1k ∥cb ≤22k. In other words, the space C(Ωk)/Nk is, foreach k, completely isomorphic (uniformly with respect to n) to Ekn.Assume now that A is a von Neumann algebra and let A∗be its predual.

We definefor any family (xJ)J∈[n]k in A∗the norm which is dual to the norm ∥∥α defined in (0.3).We set(3.8)∥(xJ)∥∗α = supXJ∈[n]k⟨aJ, xJ⟩aJ ∈A,∥(aJ)∥α ≤1.Then we define(3.9)[(xJ)]∗(k) = infXα∈{0,1}k∥xαJ∥∗αwhere the infimum runs over all xαJ in A∗such that xJ =Pα∈{0,1}k xαJ.Assume that A = (A∗)∗is a von Neumann subalgebra of B(H) and let q: N(H) →A∗bethe quotient mapping which is the preadjoint of the embedding A ֒→B(H). We can alsowrite(3.10)∥(xJ)∥∗α = infnX|λm|o26

where the infimum runs over all the possibilities to write (xJ) as a seriesxJ =Xmλmhmπα(J) ⊗kmπαc(J)where (hmi )i∈[n]α and (kmj )j∈[n]αc are elements of H such that Pi∥hmi ∥2 ≤1 and Pj∥kmj ∥2 ≤1 for each m.The identity of (3.8) and (3.10) is clear since the dual norms are the same by (0.3). Similarlyit is clear that the dual space to (A∗)nk equipped with the norm []∗(k) can be identifiedwith (A)nk equipped with the norm [](k).

By Proposition 2.1, this means that (A∗)nkequipped with the norm []∗(k) can be viewed as a predual (isometrically) of Ekn ⊗A.Hence, we can now rewrite Theorem 3.5 a bit more explicitly. For all (xJ) in A∗, we have(as announced in the beginning of this section)(3.11)(22k)−1[(xJ)]∗(k) ≤XεJxJL1(Ωk,A∗) ≤[(xJ)]∗(k).In particular, we can state for emphasis.Theorem 3.6.

Let A ⊂B(H) be a von Neumann subalgebra with predual A∗and letq: H ˆ⊗H →A∗be the corresponding quotient mapping. Consider {xJ | J ∈[n]k} in A∗such thatXεJxJL1(Ωk,A∗) < 1.Then (xJ) admits a decomposition asxJ =Xα∈{0,1}kxαJwithxαJ = q Xnλαmhmπα(J) ⊗kmπαc(J)!where for each α,{hmi| i ∈[n]α} and {kmj| j ∈[n]αc} are elements of H such thatPi∥hmi ∥2 ≤1 and Pj∥kmj ∥2 ≤1 and where λαm are scalars such thatXαXm|λαm| < 22k.27

Conversely, if (xJ) admits such a decomposition, we must have P εJxJL1(Ωk,A∗) < 22k.Proof. The proof is nothing but (3.9), (3.10) and (3.11) spelt out explicitly.Remark.

The preceding theorem proves one of the conjectures formulated in [P2] in thecase A = B(H), A∗= H ˆ⊗H.28

§4. Complements.The following result shows that in Proposition 1.3, the algebra (C∗λ(Fn))∞n=1 cannotbe substituted by any sequence of nuclear algebras.Theorem 4.1.

Let A be either a nuclear C∗-algebra or an injective von Neumann algebra,and let IEn = vw be a factorization of IEn through A. Then∥v∥cb∥w∥cb ≥12(1 + √n).For the proof we need the following.Lemma 4.2. Consider the subspace Sn of Mn ⊕Mn given bySn =x1x2...⃝xn⊕y1.

. .yn⃝ x1, .

. ., xn, y1, .

. .

, yn ∈Cand define R: Sn →Sn byR(x ⊕y) = yt ⊕xt,x ⊕y ∈Sn.Then(a)12(ISn + R) is a projection of Sn onto En and12(ISn + R)cb= 12(1 + √n). (b) For any projection Q of Sn onto En (resp.

Mn ⊕Mn onto En) one has∥Q∥cb ≥12(1 + √n).Proof. a) Obviously R2 = ISn and En = {a ∈Sn | Ra = a}.

Hence 12(ISn + R) is aprojection of Sn onto En. Let A be a C∗-algebra.

ThenSn ⊗A =a1...⃝an⊕b1. .

.bn⃝ a1, . .

., an, b1, . .

., bn ∈A29

and(R ⊗IA)a1...⃝an⊕b1. .

.bn⃝=b1...⃝bn⊕a1. .

.an⃝.SincemaxXb∗i bi1/2,Xaia∗i1/2≤√n max{∥a1∥, . .

., ∥an∥, ∥b1∥, . .

., ∥bn∥}≤√n maxXa∗i ai1/2,Xbib∗i1/2it follows that ∥R ⊗1A∥≤√n. Hence ∥R∥cb ≤√n and thus12(ISn + R)cb≤12(1 + √n).To prove the converse inequality it suffices to consider n ≥2.

Let A be the Cuntz algebraOn (cf. [C]), which is generated by n isometries s1, .

. ., sn ∈B(H) satisfyings∗i sj = δijI(4.1)nXi=1sis∗i = 1.

(4.2)By (4.2) the elementz =s∗1...⃝s∗n⊕s1. .

.sn⃝in Sn ⊗A has norm ∥z∥= 1, while12(ISn + R) ⊗IA(z) = 12s1 + s∗1...⃝sn + s∗n⊕s1 + s∗1. .

.sn + s∗n⃝has norm12nXi=1(si + s∗i )21/2= 12 sup( nXi=1∥(si + s∗i )ξ∥2 | ξ ∈H, ∥ξ∥= 1)1/2= 12 sup( nXi=1s∗i si + sis∗i + s2i + (s∗i )2!ξ, ξ! ξ ∈H, ∥ξ∥= 1)1/2.30

By (4.1) and (4.2),nPi=1s∗i si = nI andnPi=1sis∗i = I. Setv =1√nnXi=1s2i .By (4.1), v∗v = I so v is an isometry. By (4.1) the range of v is orthogonal to the range ofthe isometry s1s2: Indeed for ξ, η ∈H(vξ, s1s2η) =1√n Xi(s∗2s∗1s2i ξ, η)!=1√n(s∗2s1ξ, η)= 0.Hence v is a non-unitary isometry.

Therefore the point spectrum of v∗contains the openunit disk D (cf. e.g.

[KP], p.253). Hence also the “numerical range” of v{(vξ, ξ) | ∥ξ∥= 1} = {(ξ, v∗ξ) | ∥ξ∥= 1}contains the open unit disk.

In particular the number 1 is in the closure of this set. Thereforesup∥ξ∥=1 nXi=1s∗i si + sis∗i + s2i + (s∗i )2!ξ, ξ!= n + 1 + 2√n sup∥ξ∥=1(Re(vξ, ξ))≥n + 1 + 2√n= (1 + √n)2.Hence 12(ISn + R) ⊗IA(z) ≥12(1 + √n)∥z∥, which proves (a).

(b) Let Q be a projection from Sn onto En. Set bQ = QR = RQR.

Then bQ is also aprojection from Sn to En. Let im denote the identity on Mm and tm the transposition ofMm.

ThenbQ ⊗im = (R ⊗tm)(Q ⊗im)(R ⊗tm).Since tn ⊗tm can be identified with transposition on Mnm, ∥tn ⊗tm∥= 1. Hence by thedefinition of R,∥R ⊗tm∥≤1.31

Therefore∥bQ ⊗im∥≤∥Q ⊗im∥,m ∈Nand so ∥bQ∥cb ≤∥Q∥cb, and therefore also12(Q + bQ)cb≤∥Q∥cb.But12(Q + bQ) = Q12(ISn + R)and since Q is the identity on En, which is the range of 12(ISn + R), we have 12(Q + bQ) =12(ISn + R). Thus∥Q∥cb ≥12(ISn + R) = 12(1 + √n).If ψ: Mn ⊕Mnonto−→En is a projection of norm 1, then from the above∥ψ∥cb ≥∥ψ|Sn∥cb ≥12(1 + √n),proving (b).Proof of Theorem 4.1.

Let IEn = vw be a factorization of IEn through an injectivevon Neumann algebra A. By the injectivity of A, w can be extended to a linear mapew: Mn ⊕Mn →A such that ∥ew∥cb ≤∥w∥cb (cf.

[Pa] Theorem 7.2). Clearly Q = v ew is aprojection of Mn ⊕Mn onto En.

Hence by (b) in the preceding lemma∥v∥cb∥w∥cb ≥∥v∥cb∥ew∥cb ≥∥Q∥cb ≥12(1 + √n).This proves the announced result when A is an injective von Neumann algebra. If A is anuclear C∗-algebra, and IEn = vw as above, we can extend v to a σ(A∗∗, A∗)-continuouslinear map ˜v: A∗∗→En such that ∥˜v∥cb = ∥v∥cb.

Since A∗∗is an injective von Neumannalgebra (cf. e.g.

[CE]), we are now reduced to the preceding case.Remark 4.3. The constant12(1 + √n) is best possible in Theorem 4.1 : Namely letA = Mn ⊕Mn, let w: En →Mn ⊕Mn be the inclusion map and define a projectionv: Mn ⊕Mn →En byv(x ⊕y) = 12(ISn + R)(xp ⊕py),x ⊕y ∈Mn ⊕Mn32

where p =10. .

.00...⃝0. Then clearly vw = IEn and∥v∥cb = ∥v∥cb∥w∥cb ≤12(ISn + S)cb= 12(1 + √n),which indeed shows that Theorem 4.1 is sharp.In connection with Lemma 4.2 (b), note that there is obviously a projection P: Mn ⊕Mn →En with (ordinary) norm ∥P∥≤1 (simply take P = v with v as in Remark 4.3).However, we will show below that the projection constant of En ⊗Mn in (Mn ⊕Mn)⊗Mngoes to infinity when n →∞.

To see this it is clearer to place the discussion in a broadercontext.Let S ⊂B(H) be a closed subspace. We define λ(S) (resp.

λcb(S), λn(S)) to bethe infimum of the constants λ such that there is a projection P: B(H) →S satisfying∥P∥≤λ (resp. ∥P∥cb ≤λ,resp.

∥IMn ⊗P∥Mn(B(H))→Mn(S) ≤λ). Then by the extensiontheorem of c.b.

maps (cf. [W,Pa]), these constants are invariants of the “operator space”structure of S. By this we mean that if S1 ⊂B(K) is another operator space which iscompletely isometric to S (resp.

such that for some constant λ there is an isomorphismu: S →S1 with ∥u∥cb∥u−1∥cb ≤λ) then λ(S1) = λ(S), λcb(S1) = λcb(S), λn(S1) = λn(S)(resp. 1λλ(S) ≤λ(S1) ≤λλ(S) and similarly for the other constants).By a simple averaging argument we can proveProposition 4.4.

Let S ⊂B(H) be a closed subspace. Consider Mn(S) = Mn ⊗S ⊂B(ℓn2(H)).

Then(i) λn(S) = λ(Mn(S)). (ii) If S is σ(B(H), B(H)∗)-closed in B(H) then(4.3)λcb(S) = supn≥1λn(S).For any infinite dimensional Hilbert space K we have(4.4)λcb(S) ≤λ(B(K) ⊗S).33

Moreover let B(K)⊗S denote the weak-∗closure of B(K)⊗S in B(K ⊗H). Then,λcb(S) = λ(B(K)⊗S).Proof.

(i) The inequality λ(Mn ⊗S) ≤λn(S) is obvious, so we turn to the converse.Assume that there is a projectionP: Mn ⊗B(H) →Mn ⊗Swith ∥P∥≤λ.Let Un be the group of all n×n unitary matrices. Consider then the group G = Un×Unequipped with its normalized Haar measure m. We will use the representationπ: G →B(Mn, Mn)defined byπ(u, v)x = uxv∗.We can define an operator eP: Mn ⊗B(H) →Mn ⊗B(H) by the following formula(4.5)eP =Z(π(u, v) ⊗IB(H))P(π(u, v) ⊗IB(H))−1dm(u, v).Note that π(u, v) leaves Mn ⊗S invariant so that the range of eP is included in Mn ⊗Sand eP restricted to Mn ⊗S is the identity, hence eP is a projection from Mn ⊗B(H) ontoMn ⊗S.

Moreover, by Jensen’s inequality (notice that π(u, v) ⊗IB(H) is an isometry onMn ⊗B(H)) we have∥eP∥≤∥P∥≤λ.Furthermore, using the translation invariance of m in (4.5) we find(4.6)∀(u0, v0) ∈GeP(π(u0, v0) ⊗IB(H)) = (π(u0, v0) ⊗IB(H)) eP,so that eP commutes with π(u0, v0) ⊗IB(H). By well known facts this implies that eP is ofthe formeP = IMn ⊗Q34

for some operator Q which has to be a projection onto S. Indeed, since Mn is spanned byUn, the above formula (4.6) is equivalent to: For all a, b in Mn and for all x in Mn ⊗B(H),(4.7)e((a ⊗1)x(b ⊗1)) = (a ⊗1)e(x)(b ⊗1).Let (eij)i,j=1,...,n denote the matrix units in Mn. Set x = eij ⊗y, where y is in B(H) andi, j are in {1, ..., n}.

Applying (4.7) to a = 1 −eii and b = 1 −ejj, one gets (1 −eii)e(eij ⊗y)(1−ejj) = 0 i.e. e(eij ⊗y) = eij ⊗z for some z in B(H) depending on y, i and j. Howeverapplying (4.7) again, this time with a = eki and b = ejl it follows that z is independentof i and j.

Hence e=IMn ⊗Q, for some operator Q (which has to be a projection onto S).Finally, we conclude∥IMn ⊗Q∥= ∥eP∥≤λhence λn(S) ≤λ(Mn ⊗S). This proves (i).We now check (ii).

Consider an arbitrary closed subspace S ⊂B(H) and let S be theσ(B(H), B(H)∗)-closure of S. We claim that there is an operator Q: B(H) →S such thatQ|S = IS and ∥Q∥cb ≤supn λn(S).Let εn > 0 be such that εn →0. For each n there is a projection Pn: B(H) →S such that(4.8)∥IMn ⊗Pn∥Mn(B(H))→Mn(S) ≤(1 + εn)λn(S).Let U be a non-trivial ultrafilter on N. For any bounded sequence (αn) of real numbers(or for any relatively compact sequence in a topological space) we will denote simply bylimU αn the limit of αn when n →∞along U.

For any x in B(H) letQ(x) = limU Pn(x)where the limit is in the σ(B(H), B(H)∗)-sense. Observe that ∥Q∥≤limU ∥Pn∥≤supnλn(S).More generally for any integer m ≥1 we clearly have∀y ∈Mm ⊗B(H)(IMm ⊗Q)(y) = limU (IMm ⊗Pn)(y)hence ∥IMm ⊗Q∥≤limU ∥IMm ⊗Pn∥but when n ≥m we have obviously∥IMm ⊗Pn∥≤∥IMn ⊗Pn∥35

hence by (4.8) we obtain∥IMm ⊗Q∥≤limU (1 + εn)λn(S) ≤supn λn(S),so that ∥Q∥cb ≤supn λn(S). Clearly Q(B(H)) ⊂S and Q|S = IS.

This proves our claimand in the case S = S we obtain (4.3). (Note that λcb(S) ≥supn λn(S) is trivial.) We nowturn to (4.4).

We may clearly assume K = ℓ2. Recall that there is obviously a completelycontractive projection πn: B(ℓ2) →Mn (here Mn is considered as a subspace of B(ℓ2) inthe usual way) henceλn(S) = λ(Mn ⊗S) ≤∥πn∥λ(B(ℓ2) ⊗S) = λ(B(ℓ2) ⊗S)which implies by (4.3)λcb(S) ≤λ(B(ℓ2) ⊗S).This concludes the proof of (4.4).To prove the last assertion, note that Mn(S) is clearly contractively complemented inB(ℓ2)⊗S hence we haveλcb(S) ≤supn≥1λn(S) = supn≥1λ(Mn(S)) ≤λ(B(ℓ2)⊗S).To prove the converse inequality, note that B(ℓ2)⊗B(H) can be identified with the space ofmatrices a = (aij)i,j∈IN which are bounded on ℓ2(H), and B(ℓ2)⊗S can be identified withthe subspace formed by all matrices with entries in S. Then if P is a bounded projectionfrom B(H) onto S, defininge(a) = (P(aij))i,j∈INwe obtain a projection from B(ℓ2)⊗B(H) to B(ℓ2)⊗S with ∥e∥≤∥P∥cb.

To check thislast estimate, observe that the norm of an element a = (aij)i,j∈IN in B(ℓ2)⊗B(H) is thesupremum over n of the norms in Mn(B(H)) of the matrices (aij)i,j≤n. This yields thelast assertion.Corollary 4.5.

Let H, K be Hilbert spaces. Consider a completely isometric embeddingEn →B(H).

Then, if dim K = ∞, for any projection P from B(K)⊗B(H) to B(K)⊗Enwe have∥P∥≥12(√n + 1).36

A fortiori the same holds for any projection P from B(K ⊗H) onto B(K) ⊗En.Proof. By the preceding statement, this follows from Theorem 4.1.Corollary 4.6.

Let M ⊂B(H) be a von Neumann subalgebra such that M is isomorphic(as a von Neumann algebra) to Mn(M) for some integer n ≥2. Then if there is a boundedlinear projection from B(H) onto M, there is also a completely bounded one.Proof: Note that if M is isomorphic to Mn(M), then obviously it is isomorphic toMn(Mn(M)) = Mn2(M), and similarly to Mn3(M), and so on.

Hence this follows clearlyfrom the first two parts of Proposition 4.4 and the observation preceding Proposition 4.4.In particular we have using [V1]Corollary 4.7. Let M ⊂B(H) be a von Neumann subalgebra.

If M is isomorphic to thevon Neumann algebra V N(Fn) (resp. V N(F∞)) associated to the free group with n > 1generators (resp.

countably many generators) then there is no bounded linear projectionfrom B(H) onto M.Proof: First note that V N(Fn) trivially embeds into V N(F∞) as a subalgebra which is therange of a completely contractive projection. Therefore by Proposition 1.3 and Theorem4.1 there is no completelybounded projection from B(H) onto M if M is isomorphicto V N(F∞).

By [V1] Mn(V N(F∞)) is isomorphic to V N(F∞) for all n. Hence Corollary4.7 for V N(F∞) follows from the preceding corollary. To obtain the case of finitely manygenerators, recall the well known fact that F∞can be embedded in Fn for all n ≥2.

(Ifa, b are two of the generators of Fn, then it is easy to check, that b, aba−1, ..., anba−n, ...are free generators of a subgroup isomorphic to F∞.) Therefore if M = V N(Fn) for n > 1,then V N(F∞) is isomorphic to a von Neumann subalgebra N ⊂M, and since M is afinite von Neumann algebra, N is the range of a conditional expectation, hence there isa bounded projection from M onto N. Since there is no bounded projection from B(H)onto N by the first part of the proof, a fortiori there cannot exist a bounded projectionfrom B(H) onto M.37

For two operator spaces E and F of the same finite dimension n, one can define thecomplete version of the Banach-Mazur distance between E and F bydcb(E, F) = inf{∥u∥cb, ∥u−1∥cb},where the infimum is taken over all invertible linear maps u from E to F. By Proposition 1.3it follows thatdcb(En, span{λ(gi) | i = 1, . .

., n}) ≤2for all n ∈N. The next proposition shows that the same inequality holds if the unitaryoperators λ(g1), .

. ., λ(gn) are replaced by a semicircular or circular system of operators inthe sense of Voiculescu [V1].Proposition 4.8.

Let n ∈N and let x1, . .

., xn be a semicircular or circular system ofoperators on a Hilbert space, then the map u: En →span{x1, . .

., xn} given byu:nXk=1ckδk −→nXk=1ckxk,c1 ∈Csatisfies ∥u∥cb∥u−1∥cb ≤2.Proof: Assume first that x1, . .

., xn is a semicircular system of selfadjoint operators in thesense of [V1]. By [V2], we can exchange x1, .

. ., xn with the operatorsxk = 12(sk + s∗k),k = 1, .

. ., nwhere s1, .

. .

, sn are the “creation operators” ξ →ei ⊗ξ on the full Fock spaceH = C ⊗ ∞Mn=1H⊗n!based on a Hilbert space H with orthonormal basis (e1, . .

., en). In particular s1, .

. ., snare n isometries with orthogonal ranges, and thereforenXk=1sks∗k ≤1.38

Hence, as in the proof of Proposition 1.1, we get that for any n-tuple a1, . .

., an of elementsin a C∗-algebra A,Xkxk ⊗ak ≤12 Xksk ⊗ak +Xks∗k ⊗ak!≤12Xksks∗k1/2 Xka∗kak1/2+Xks∗ksk1/2 Xkaka∗k1/2≤maxXka∗kak1/2,Xkaka∗k1/2.Hence ∥u∥cb ≤1. To prove that ∥u−1∥cb ≤2, notice that by [V1], [V2], the C∗-algebragenerated by x1, .

. ., xn and 1 has a traceτ: C∗(x1, .

. ., xn, 1) →C(namely the vector-state given by a unit vector in the C-part of the Fock space H), withthe properties:τ(1) = 1,τ(x2k) = 14andτ(xkxℓ) = 0k ̸= ℓ.Let a1, .

. ., an be n operators in a C∗-algebra A, and let S(A) denote the state space of A.ThenXkxk ⊗ak2≥supω∈S(A)(τ ⊗ω) Xkxk ⊗ak!∗ Xℓxℓ⊗aℓ!

!= 14supω∈S(A)ω Xka∗kak!= 14Xka∗kakand similarlyPkxk ⊗ak2≥14Pkaka∗k. HenceXkxk ⊗ak ≥12 maxXka∗kak1/2,Xkaka∗k1/239

proving ∥u−1∥cb ≤2.Assume finally that y1, . .

. , yn is a circular system.

Thenyk =1√2(x2k−1 + ix2k),k = 1, . .

., n,where (x1, . .

., x2n) is a semicircular system of selfadjoint operators. Therefore the state-ment about circular systems in Proposition 4.8 follows from the one on semicircular systemsby observing, that the mapnXk=1ckδk →1√2nXk=1ck(e2n−1 + e2k)defines a cb-isometry of En onto its range in E2n.To conclude this paper we give a generalization of Proposition 1.1 to free products ofdiscrete groups, or more generally free products of C∗-probability spaces in the sense of[V1] and [V2].

We refer to [V1] and [V2] for the terminology.Proposition 4.9. Let (A, ϕ) be a C∗-algebra equipped with a faithful state ϕ.

Let(Ai)i∈I be a free family of unital C∗-subalgebras of A in the sense of [V1] or [V2]. Considerelements xi ∈Ai such that for some δ > 0∀−i ∈I∥xi∥≤1, ϕ(xi) = 0 and −min{ϕ(x∗i xi), ϕ(xix∗i )} ≥δ2Then, for all finitely supported families (ai)i∈I in B(H) (H Hilbert) we have(4.9)δ max{Xa∗i ai1/2,Xaia∗i1/2} ≤Xxi ⊗ai ≤2 max{Xa∗i ai1/2,Xaia∗i1/2}.Proof.

We may assume that I is finite. The lower bound in (4.9) is proved exactly as inthe semicircular case.

To prove the upper bound we will prove that A can be faithfullyrepresented as a C∗-algebra of operators on a Hilbert space H, such that xi admits adecomposition xi = ui + vi with ui, vi in B(H) and(4.10)Xu∗i ui ≤1 −and −Xviv∗i ≤1.The upper bound in (4.9) then follows as in the semicircular case.40

Following the notation of [V 2, pp. 558-559], we let (Hi, ξi) be the space of the GNS-representation πi = πϕ|Ai.

In particular ξi is a unit-vector in Hi andϕ(x) = (πi(x)ξi, ξi) when x ∈Ai.Then A can be realized as the C∗-algebra of operators on the Hilbert space(H, ξ) = ∗i∈I −(Hi, ξi)generated by Si∈Iλi◦πi(Ai), where λi : B(Hi) →B(H) is the ∗-representation defined in[V2, sect. 1.2].

For simplicity of notation we will identify Ai with its range in B(H), i.e.we setλi◦πi(x) = x when x ∈Ai.Let x ∈Ai. Corresponding to the decompositionHi = H0i ⊕|Cξi.we can write πi(x) as a 2 × 2 matrixπi(x) =bηζ∗twhere b ∈B(H0i ), η, ζ ∈H0i and t ∈|C.

(Here we identify η, ζ with the correspondinglinear maps from|C to H0i , and we also identify|C with|Cξi.) The action of x = λi◦πi(x)on ∗i∈I(Hi, ξi) can now be explicitly computed from [V 2, sect.

1.2]. One finds :(4.11)xξ = η ⊗ξ + tξ,(4.12)x(h1 ⊗... ⊗hn) = bh1 ⊗... ⊗hn + (h1, ζ)h2 ⊗... ⊗hn when n ≥1,hk ∈H0ik, i = i1 ̸= i2 ̸= ... ̸= in,(4.13)x(h1 ⊗... ⊗hn) = η ⊗h1 ⊗... ⊗hn + th1 ⊗... ⊗hn, when n ≥1, hk ∈H0ik,i ̸= i1 ̸= i2 ̸= ... ̸= inwhere h2 ⊗... ⊗hn = ξ for n = 1.Let ei ∈B(H) be the orthogonal projection of H onto the subspaceHi = ⊕∞n=1(⊕(Hi1 ⊗... ⊗Hin))41

where the second direct sum contains all n-tuples (i1, ..., in) for which i = i1 ̸= i2 ̸= ... ̸= in.From (4.11), (4.12) and (4.13) one gets for all x in Ai(4.14)(1 −ei)x(1 −ei) = ϕ(x)(1 −ei)where we have used thatt = (πi(x)ξi, ξi) = ϕ(x).Let now xi ∈Ai, ∥xi∥−≤1, ϕ(xi) = 0. Then by (4.14)(1 −ei)xi(1 −ei) = 0Thus xi = ui + vi, whereui = xiei −and −vi = eixi(1 −ei).Since ∥xi∥−≤1, and since (ei)i∈I is a set of pairwise orthogonal projections,Xi∈Iu∗i ui ≤Xi∈Iei ≤1andXi∈Iviv∗i ≤Xi∈Iei ≤1.This completes the proof of proposition 4.9.42

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