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On the “local theory” of operator spaces

arXiv:math/9212205v1 [math.FA] 4 Dec 1992On the “local theory” of operator spacesby Gilles Pisier*In Banach space theory, the “local theory” refers to the collection of finite dimensionalmethods and ideas which are used to study infinite dimensional spaces (see e.g. [P4,TJ]).It is natural to try to develop an analogous theory in the recently developed category ofoperator spaces [BP,B1-2,BS,ER1-7,Ru].

The object of this paper is to start such a theory.We plan to present a more thorough discussion of the associated tensor norms in a futurepublication.We refer to [BP,B1-2, ER1-7] for the definition and the main properties of operatorspaces. We merely recall that an operator space is a Banach space isometrically embeddedinto the space B(H) of all bounded operators on a Hilbert space H, and that in thecategory of operator spaces, the morphisms are the completely bounded maps (in shortcb) for which we refer the reader to [Pa1].

If E, F are operator spaces, we denote byE ⊗min F their minimal (or spatial) tensor product. We denote by E (or H) the space E(or H) equipped with the conjugate complex multiplication.

Note that E∗can be identifiedwith the antidual of E and the elements of (E ⊗E)∗can be viewed as sesquilinear formson E × E.Recently [P1-3] we introduced the analogue of Hilbert space in the category of operatorspaces. We proved that there is a Hilbert space H and a sequence of operators Tn ∈B(H)such that for all finitely supported sequence (an) in B(ℓ2) we have(1)XTn ⊗anB(H⊗ℓ2) =Xan ⊗¯an1/2B(ℓ2⊗ℓ2) .We denoted by OH the closed span of (Tn) and by OHn the span of T1, ...Tn.

We call OHthe operator Hilbert space. For any operator u: OHn →E we have (cf.

[P1-2])∥u∥cb = ∥nX1u(Ti) ⊗Ti∥E⊗minOHn.Our main tool will be a variation (one more!) on the notion of 2-summing operator.Let E be an operator space and let Y be a Banach space.

Following our previous work* Supported in part by N.S.F. grant DMS 90035501

[P3] we will say that an operator u: E →Y is (2, oh)-summing if there is a constant Csuch that for all finite sequences (xi) in E we have(2)X∥u(xi)∥2 ≤C2 Xxi ⊗¯xiE⊗minE .We will denote by π2,oh(u) the smallest constant C for which this holds. Moreover for anyinteger n we denote by πn2,oh(u) the smallest constant C such that (2) holds for all n-tuplesx1, ..., xn in E. Recall that the usual 2-summing norm π2(u) of an operator u: E →Fbetween Banach spaces (resp.

the 2-summing norm on n vectors πn2,oh(u)) is the smallestconstant C such that for all finite sequences (resp. all n-tuples) (xi) in E we haveX∥u(xi)∥2 ≤C2 sup{X|ξ(xi)|2| ξ ∈E∗∥ξ∥≤1}.Equivalently this means that (2) holds when E is embedded isometrically into a commuta-tive C∗-subalgebra of B(H).

An alternate definition of π2(u) (resp. πn2 (u)) is the smallestconstant C such that(3)π2(uv) ≤Cfor all finite rank operators v: ℓ2 →E (resp.

v: ℓn2 →E) with ∥v∥≤1. As observed in[P1], it is easy to see using (1) that for every bounded operator v: OHn →OHn we have∥v∥= ∥v∥cb.

It follows that for any operator u: OH →E we haveπ2,oh(u) = π2(u)andπn2,oh(u) = πn2 (u).Similarly when E is an operator space for any u: E →F the norm π2,oh(u) (resp. πn2,oh(u))is the smallest constant C such that (3) holds for all finite rank v: OH →E (resp.v: OHn →E) with ∥v∥cb ≤1.Since the cb-norm dominates the usual norm of an operator v: OH →E, it is easy to checkthat for if E is an operator space and F a Banach space then every 2-summing u: E →Fis necessarily (2, oh)-summing and we have(4)π2,oh(u) ≤π2(u).2

By an important inequality due to Tomczak-Jaegermann (see [TJ] p.143) we have for anyrank n operator u: E →F between Banach spaces π2(u) ≤√2πn2 (u). This fact and thepreceding equalities yield that for any rank n operator u: E →F between operator spaceswe have(5)π2,oh(u) ≤√2πn2,oh(u).In [P3] the following result (which is crucial for the present note) is mentioned.Anyoperator u: E →OH (with domain an arbitrary operator space but with range OH)which is (2, oh)-summing is necessarily completely bounded and we have(6)∀u: E →OH∥u∥cb ≤π2,oh(u).An element u in E ⊗E is called positive if u can be written asu =nX1xi ⊗¯xiwithxi ∈E.In that case we will write u ≥0.

Equivalently this means that ⟨u, ξ ⊗¯ξ⟩≥0 for all ξ inE∗, so that u ≥0 iff⟨u, v⟩≥0 for all v ≥0 in E∗⊗E∗. More generally, a linear formϕ ∈(E ⊗E)∗will be called positive if ϕ(x ⊗¯x) ≥0 for all x in E. Note that this impliesthat ϕ is symmetric, i.e.

ϕ satisfies ϕ(x ⊗¯y) = ϕ(y ⊗¯x) (or equivalently ϕ(u) ∈R for allsymmetric u in E ⊗E). We will denote by K(E) the set of all the positive linear forms ϕin (E ⊗E)∗such thatsup{ϕ(u) | u ∈E ⊗E, u ≥0, ∥u∥E⊗minE ≤1} ≤1.Then it is rather easy to check that for all u ≥0 in E ⊗E we have(7)∥u∥E⊗minE =supϕ∈K(E)ϕ(u).Indeed, consider u =nP1xi ⊗¯yi in E ⊗E.

Assume E ⊂B(H). Let C2 be the space of allHilbert-Schmidt operators on H, equipped with the Hilbert-Schmidt norm ∥∥2.

Observe3

that for any y in C2 there is a decomposition y = a+ −a−+ i(b+ −b−) with a+, a−, b+, b−hermitian positive and such that∥a+∥22 + ∥a−∥22 + ∥b+∥22 + ∥b−∥22 = ∥y∥22.By definition of E ⊗min E we have∥u∥= supntrXxiyy∗i z | y, z ∈C2 ∥y∥2 ≤1, ∥z∥2 ≤1o.Now assume u ≥0, say u =nP1xi ⊗¯xi. Let F(y, z) = tr (P xiyx∗i z).

Note that F(y, z)is positive when y, z are both positive. Then, by the decomposition recalled above thesupremum of F(y, z) when y, z run over the unit ball of C2 is unchanged if we restrict itto positive operators y, z in the unit ball of C2.

But if y, z are positive in the unit ball ofC2 then the form defined by ∀u =nP1xi ⊗¯yi ∈E ⊗¯Eϕ(u) = trXxiyy∗i z= trz1/2xiyy∗i z1/2=Xtr[(z1/2xiy1/2)(z1/2yiy1/2)∗]is clearly positive so that (7) follows.Proposition 1. Let E be an operator space, let F ⊂E be a closed subspace, and let Ybe a Banach space.

Let u: F →Y be an operator and let C be a constant. The followingare equivalent.

(i) u is (2, oh)-summing with π2,oh(u) ≤C. (ii) There is a ϕ in K(E) such that∀x ∈F∥u(x)∥2 ≤C2ϕ(x ⊗¯x).

(iii) There is an extension ˜u: E →Y such that ˜u|F = u andπ2,oh(˜u) ≤C.Proof: Assume (i). Note that for all w ∈F ⊗F we have∥w∥F ⊗minF = ∥w∥E⊗minE.4

Hence by (7) we haveX∥u(xi)∥2 ≤C2supϕ∈K(E)Xϕ(xi ⊗¯xi),for all finite sequences xi in F. By a classical application of the Hahn-Banach theorem itfollows that there is a ϕ in K(E) such that∀x ∈E∥u(x)∥2 ≤C2ϕ(x ⊗¯x). (Indeed, one can reproduce the argument included e.g.

in [P4] p. 11 for 2-summing op-erators and observe that K(E) is convex so that the barycenter of a probability measureon K(E) belongs to K(E).) This proves (i) ⇒(ii).

Now assume (ii). Consider the scalarproduct ⟨x, y⟩= ϕ(x ⊗¯y) on E. Let us denote by L2(ϕ) the resulting Hilbert space (afterpassing to the usual quotient and completing) and let J: E →L2(ϕ) be the natural inclu-sion.

Observe that we trivially have by (7) π2,oh(J) ≤1. We now introduce an operatorv: J(F) →Y .

For any element y in J(F) we can define if y = J(x) with x ∈Fv(y) = u(x).Note that (ii) ensures that this definition is unambiguous and ∥v∥≤C. Hence v extends toan operator v: J(F) →Y such that ∥v∥≤C.

Finally let P be the orthogonal projectionfrom L2(ϕ) onto J(F) and let ˜u = vPJ. Clearly π2,oh(˜u) ≤∥v∥π2,oh(J) ≤C and ˜u extendsu.

This proves (ii) ⇒(iii). Finally (iii) ⇒(i) is trivial.A fundamental inequality in Banach space theory (originally due to Garling and Gor-don, see [P4,p.15]) says that for any n-dimensional Banach space the identity operator IEsatisfies π2(IE) = n1/2.

By (4) it follows that for any n-dimensional operator space wehaveπ2,oh(IE) ≤n1/2.In that case the equality no longer holds, as shown by the examples below. However thefollowing consequence of the upper bound still holds in the category of operator spaces.5

Theorem 2. Let E be any n-dimensional operator space then there is an isomorphismu: E →OHnsuch thatπ2,oh(u) = n1/2and ∥u−1∥cb = 1.Corollary 3.

For any n-dimensional operator space E there are n elements x1, ..., xn inE such thatnX1xi ⊗¯xiE⊗minE≤1andnX1∥xi∥2 ≥π2,oh(IE)2/2.Corollary 4. For any n-dimensional E there is an isomorphism u: OHn →E such that∥u∥cb ∥u−1∥cb ≤√n.Corollary 5.

For any n-dimensional subspace E ⊂B(H) there is a projection P: B(H) →E such that∥P∥cb ≤√n.Proof of Theorem 2. We adapt an argument well known in the ”local theory” of Banachspaces.

By Lewis’ version of Fritz John’s theorem (cf. [P5] p. 28) there is an isomorphismu: E →OHn such that π2,oh(u) = √n and π∗2,oh(u−1) = √n.

It is rather easy to check(cf. [P3]) directly from the definition of the norm π2,oh that for all v: OHn →Eπ∗2,oh(v) = inf{∥B∥HS∥A∥cb}where B: OHn →OHn, A: OHn →E and v = AB.Hence u−1 = AB with ∥A∥cb = 1 and ∥B∥HS = √n.

Clearly ∥uA∥HS ≤√n by definitionof π2,oh, hence∥B∥HS ∥uA∥HS ≤n = tr(uu−1) = tr(uA · B)so by the equality case of the Cauchy Schwarz inequality we must have(uA) = B∗hence B−1 = B∗, so that B is unitary. It follows that∥u−1∥cb ≤∥A∥cb∥B∥cb ≤16

since for B: OHn →OHn we clearly have (cf. [P1-2]) ∥B∥cb ≤∥B∥.

Conversely we have√n = π∗2,oh(u−1) ≤√n ∥u−1∥cb, which proves that ∥u−1∥cb = 1.Proof of Corollary 3. By Theorem 2 and by (5) we haveπn2,oh(IE)2 ≥π2,oh(IE)2/2,from which the corollary follows.Proof of Corollary 4.

By (6) we have∥u∥cb ≤π2,oh(u)hence ∥u∥cb∥u−1∥cb ≤√n.Proof of Corollary 5. Let u: E →OHn be as in Theorem 2.

By Proposition 1 thereis an extension ˜u: B(H) →OHn such that π2,oh(˜u) ≤√n. By (6) we have ∥˜u∥cb ≤√n,hence letting P = u−1˜u we find ∥P∥cb ≤√n.Finally we have (by going through OHn).Corollary 6.

Let E, F be arbitrary n-dimensional operator spaces. There is an isomor-phism u: E →F such that ∥u∥cb ∥u−1∥cb ≤n.Note that this is optimal (asymptotically) already in the category of Banach spaces,as shown by the well known spaces constructed by E.Gluskin [Gl1,2].

We refer the readerto [Pa2] for a discussion of the problem considered in corollary 6 when E and F are thesame underlying Banach space. Even when the Banach space underlying E and F is then-dimensional Euclidean space, the asymptotic order of growth of corollary 6 cannot beimproved (see Theorem 2.15 in [Pa2]).Finally we turn to some examples.1) If En = OHn, then clearly π2,0h(IEn) = π2(IEn) = √n.

More generally since we havea completely contractive inclusion (cf [P1]) OHn →Rn+Cn, we have π2,oh(IRn+Cn) =√n.7

2) If En = Rn or Cn, we claim thatπ2,oh(IRn) = π2,oh(ICn) = n1/4.Indeed, let e1i, . .

. , e1n be the canonical basis of Rn.It is easy to check thatnP1e1i ⊗¯e1i1/2= n1/4.SinceP ∥e1i∥21/2 = n1/2 we find π2,oh(IRn) ≥n1/4.On the other hand, by the interpolation theorem for operator spaces (cf.

[P1] Remark2.11), for all u: OH →Rn we have ∥u∥cb = (tr|u|4)1/4 where |u| = (u∗u)1/2 isthe modulus of u as an operator between Hilbert spaces. Hence if (Ti) denotes anothonormal basis of OH, we have since Rn is n-dimensionalX∥u(Ti)∥21/2= (tr|u|2)1/2 ≤n1/4(tr|u|4)1/4≤n1/4∥u∥cbhence π2,oh(IRn) ≤n1/4.This proves the above claim for Rn.

The proof for Cn is similar.3) Let u1, . .

. , un be unitary operators in B(H) such that ui = u∗i , u2i = I anduiuj + ujui = 0ifi ̸= j.These are the canonical generators of a Clifford algebra.

It is known that such oper-ators can be constructed inside the space M2n. Let En = span(u1, .

. ., un).

We claimthat π2,oh(IEn) ≤√2. Let i: ℓn2 →M2n be the map defined by i(x) = P xiui.Clearly we have(8)∀x ∈ℓn2i(x)∗i(x) + i(x)i(x)∗= 2∥x∥2I.This implies(9)∀x ∈ℓn2∥x∥2 ≤∥i(x)∥2 ≤2∥x∥2.Moreover let us denote∀a ∈M2nτ(a) = 2−ntr(a).8

Then the identity (8) yieldsτ(i(x)∗i(x)) = ∥x∥2hence we have by (9)(10)∀x ∈ℓn212∥i(x)∥2 ≤∥x∥2 = τ(i(x)∗i(x)) ≤∥i(x)∥2.Let us denote ∥a∥2 = (tr a∗a)1/2 for all a in M2n. Now consider a finite sequence (aj) inEn.

Let aj = i(xj) with xj ∈ℓn2. We haveXaj ⊗¯aj =sup∥y∥2≤1Xajya∗j2≥2−n/2 Xaja∗j2 ≥2−ntrXaja∗j=Xτ(aja∗j)hence by (10)≥2−1 X∥aj∥2.Hence we have π2,oh(IEn) ≤21/2.In [ER7], Effros and Ruan proved an analogue of the Dvoretzky-Rogers theorem foroperator spaces.

We can deduce a similar (and somewhat more precise) result from theabove corollary 5. Indeed, let E ⊂B(H) be any n-dimensional operator space.

Let usdenote by iE: E →B(H) the embedding. Then the “operator space nuclear norm” of iE,denoted by ν(iε), as introduced in [ER6] satisfiesν(iE) ≥n1/2.Indeed, by corollary 5 there is a projection P: B(H) →E with ∥P∥cb ≤n1/2, hencen = tr(IE) = tr(PiE) ≤∥P∥cbν(iE)≤n1/2ν(iE).This implies that the identity of an operator space X is “operator 1-summing” in the senseof [ER7] iffX is finite dimensional.Acknowledgement: I am grateful to Vern Paulsen for stimulating conversations.9

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