Julien Lavalle

GEV-TEV GALACTIC COSMIC RAYS

This short review aims at presenting the way we currently understand, model, and constrain the transport of cosmic rays in the GeV-TeV energy domain. This is a research field per se, but is also an important tool e.g. to improve our understanding of the cosmic-ray sources, of the diffuse non-thermal Galactic emissions (from radio wavelengths to gamma-rays), or in searches for dark matter annihilation signals. This review is mostly dedicated to particle physicists or more generally to non-experts.

Introduction:

The theoretical grounds of cosmic-ray (CR) physics have been developed for almost 5 decades [START_REF] Ginzburg | The Origin of Cosmic Rays[END_REF], but it is only since very recently that observations with sufficient precision have become available to test the main features of the models. The sources of GeV-TeV CRs are mostly byproducts of supernova explosions. Supernova remnants (SNRs) and pulsar wind nebulae (PWNe) are the main Galactic CR (GCR) accelerators b . The former accelerate hadrons and electrons at non-relativistic shocks, whereas the latter mostly accelerate electron-positron pairs produced from the annihilation of curvature photons with the strong magnetic fields around pulsars. Though the details of the acceleration mechanisms are not completely established, direct observations of these sources in radio, X-rays or gamma-rays prove that they do accelerate CRs up to 10-100 TeV.

The transport of Galactic cosmic rays: For reviews and/or books, see [START_REF] Ginzburg | The Origin of Cosmic Rays[END_REF][START_REF] Berezinskii | High energy astrophysics. Vol.1: Particles, photons and their detection[END_REF][START_REF] Strong | [END_REF] and references therein. In this section, I will introduce basic ingredients to help the non-experts understand CR transport and some degeneracies that currently plague even the most evolved models.

GCRs are charged particles that diffuse off magnetic inhomogeneities δB in the Galaxy, with typical amplitudes δB/B ∼ 1 (with B ≈ O(1 µG). Even though the Galactic magnetic field exhibits clear patterns in the spiral arms, these magnetic fluctuations still "isotropize" GCRs over scales of the order of a few times the Larmor radius r l ≃ 10 -3 pc × {1/|Z| (pc/1 TeV) (B/1 µG) -1 }, much shorter than the coherence length of the regular magnetic component. Let's first forget about the specific geometry of the Milky Way (MW) and the details of CR transport. Let's focus on stable CR nuclei, and consider only two different timescales: the confinement (or escape) time τ esc (magnetic confinement in the MW), and the spallation time τ s (inelastic interactions with the ISM gas) -for nuclei, energy losses mostly play a role at sub-GeV energies. The former must be a decreasing function of rigidity (R ≡ p/|Z|) because when a E-mail: jlavalle@um2.fr b A PWN must be paired with an SNR (not necessarily observable), though the contrary is not true, as for type 1A SNe r l reaches the size of the MW, CRs are no longer confined. The latter is a much weaker function of momentum as far as nuclear cross sections are considered. We can write a simple evolution equation for the differential CR density N = dn/dp:

dN dt = Q -N τesc -N τs ,
where Q is a source term. We can assume that the latter is constant in time (related to the SN explosion rate) and take the steady-state limit of the above equation to link the source features to the local CR density from τ esc and τ s (much smaller than the age of the Galaxy). This is the so-called leaky-box (LB) model, which allows to estimate the confinement time. Nevertheless, this requires to know the source term accurately, which we don't. This can be circumvented by considering two different kinds of CRs together: the primaries produced at sources (denoted A), and the secondaries produced only from inelastic scatterings of primaries with the ISM gas (denoted B, and assumed unique for simplicity). The source term for B is then merely N A /τ s,A , and using the steady state regime of the previous equation, we get:

NB NA = τescτs,B (τesc+τs,B )τs,A Rրր ≈ τesc(R)
τs,A , where the last approximation is obtained from the different rigidity dependences of τ esc and τ s,B , for large R (this can be checked a posteriori, taking 1/τ s,B = σ B v nism , where σ B is the B + ISM → X cross section and nism < n ism is the average ISM gas density encountered by a CR while it is confined). Therefore the secondary-to-primary (II/I) ratio, as derived in the LB model, allows to determine the energy dependence of τ esc . For its amplitude, one needs to refine the description of the spallation timescales by considering more elaborated distribution functions for the grammage (the column-density equivalent for CRs propagating in the ISM). The most widely used II/I ratio is the boron-to-carbon (B/C) because it is best measured. One typically finds a good fit to the data by taking τ esc = τ 0 esc (R/1 GV) -δ , with τ 0 esc ≈ 100 Myr and δ ≈ 0.3-0.5 (for kinetic energies above 1 GeV/nucleon).

We can now improve the physical picture by considering diffusion. It is a much involved theoretical field to try to infer the diffusion properties from those of magnetic turbulences, but we can first assume that the magnetic scatterers are distributed homogeneously in an extended slab that encompasses the Galactic disk, and that the diffusion coefficient is a scalar function of the rigidity K(R). The former is motivated by e.g. radio observations of the MW and of external disk galaxies. This slab has a half-height L, unknown a priori, but expected to be much larger than that of the disk, h ≃ 100 pc. Assuming that the ISM gas is confined to an infinitely thin disk (h ≪ L), we may write for a stable CR species:

∂ t N -∇(K(R) ∇N ) + 2 h n ism δ(z) σ v N = Q ,
where σ is the spallation cross section. For simplicity, we assume the slab radially extends to infinity. This reduces the above equation to one spatial coordinate z, perpendicular to the disk. We take the steady-state limit, and, to account for the escape, demand that N (|z| = L) = 0. For primary as well as secondary astrophysical CRs, the source term is confined to the disk, such that we may assume that Q = 2 h δ(z) Q 0 (R). For z = 0, a solution satisfying the bound-ary conditions is given by N (z = 0) = N (0)(1 -|z|/L). By integrating the previous equation in the range [-ǫ, +ǫ] in the limit ǫ → 0 (ǫ > 0), we get:

K h L N (0) + n ism σ v N (0) = Q 0 .
By analogy with the LB equation, one can readily conclude that the II/I ratios allow us to constrain K/L ∝ 1/τ esc ∝ R δ . This suggests that K(R) = K 0 (R/1 GV) δ provides a good description of the data -typical values are K 0 ∼ 0.01 kpc 2 /Myr and δ ∼ 0.3-0.5. Such a powerlaw shape for the diffusion coefficient is actually supported independently by theoretical results, in case diffusion originates from magnetic turbulence with a power-law spectrum (which is widely encountered in hydrodynamic systems) -an idealized example is the Kolmogorov turbulence spectrum, for which δ is close to 1/3. Note that a full theory connecting any turbulence properties to diffusion is yet to be derived, but models exist and can be tested [START_REF] Shalchi | Nonlinear Cosmic Ray Diffusion Theories[END_REF].

An important consequence of the above result is that the normalization of the diffusion coefficient K 0 and the slab size L, both unknown a priori, are found degenerate in the II/I ratio (K/L ∝ 1/τ esc ). Since we have seen from the LB model that the primary source term cancels out in the ratio, it is not surprising that any prediction made for other astrophysical secondary species from a primary spectrum should not be affected by uncertainties in K 0 and L, provided the K 0 /L is fixed. This remains roughly valid in any diffusion model, because the source terms for primary as well as secondary CRs originate in the Galactic disk. This is no longer the case for CRs that would be produced in the whole magnetic halo; a constant source term would then give N (0) ∼ ∝ L 2 /K: any prediction made for such a case is therefore very sensitive to L. This is typical of what happens for antimatter flux predictions made for dark matter signals, while the so-called secondary background prediction is under control (except for positrons, see below). Indeed, from the II/I ratios only, L can typically range from 1 to 15 kpc [START_REF] Maurin | [END_REF]. This K 0 -L degeneracy is rather endemic of all CR diffusion models, but, fortunately, there are ways to break it, based on two different approaches: (i) probe physical observables which are not sensitive to the spatial boundary L; (ii) probe physical observables that exhibit non-linear dependences on K 0 and L. For case (i), on may use unstable (radioactive) secondary CRs with decay timescales τ 0 such that the associated diffusion scalelength is λ d ∼ √ K γ τ 0 ≪ L, beyond which contributions are suppressed. One can therefore only observe those unstable CRs which have been produced within a distance ∼ λ d . The most used CR "clocks" range from τ 0 = 0.301 Myr ( 36 Cl) to 1.36 Myr ( 10 Be), for which, assuming a kinetic energy of 1 GeV/nucleon one finds λ d ranging from 100 to 240 pc. Using such constraints, the available data strongly favor large-halo models, with L > 5 kpc [START_REF] Strong | [END_REF]. Nevertheless, there are two important drawbacks. First, measuring fluxes of radioactive species is very challenging, and current data have large error bars (AMS02 should improve). Second, the fact that λ d ∼ 200 pc implies that this method is very sensitive to the details of the local ISM, and there are hints that the local ISM is underdense over a scale of ∼ 100 pc (known as the local bubble). Therefore, this procedure could be affected by large systematic errors [6,7]. As for case (ii), different observables can be used. Studies based on predictions of the diffuse Galactic gamma-ray emission [8] (see I. Moskalenko's contribution in these proceedings), and of the diffuse Galactic radio emission [9], strongly favor values of L > 5 kpc. In both cases, this comes from the contribution of CR electrons (gamma-rays from inverse Compton on CMB outside the disk, radio from synchrotron losses). Nevertheless, these are indirect constraints which also strongly depend on the descriptions of the radiation fields in the disk and of the magnetic field. More direct constraints come from the local secondary positron flux at low energy, for which small values of K 0 lead to predictions in excess with respect to the data (the correlation with L induced by the II/I ratios implies strong constraints on small values of L). This is clear from the predictions made in [10,11], and was pointed out in the context of dark matter searches [12].

State-of-the-Art: Evolved CR transport models include more ingredients than those discussed above. They generally rely on the following general CR transport equation (and then may differ in the assumptions used to solve it):

∂ t N -∇• K x ∇ -V c N +∂ p ṗ -p 3 ∇ • V c -p 2 K p ∂ p 1 p 2 N + 1 τs( x,p) + 1 τ dec (p) N = Q .
New ingredients have appeared: the convection velocity V c that encodes the effect of stellar winds in the disk; the energy loss term ṗ (a spatial dependence is implicit), which is very important for the transport of electrons and positrons (inverse Compton and synchrotron losses dominate above a few GeV); reacceleration, through a diffusion coefficient in momentum space K p , predicted to be related to spatial diffusion by the relation

K x K p ∝ p 2 V 2
a , where V a is the average velocity of the magnetic scatterers (i.e. the Alfvèn velocity in some idealized cases); the decay and spallation timescales τ dec , τ s , which only affect nuclei (radioactive nuclei for the former).

These parameters may be theoretically constrained, like K x (R) ∝ R δ , but hardly be fully inferred from first principles (except for the cross sections), as they often result from large-scale averages of complex microphysics that depends on the details of many Galactic components. In particle physics language, this is an effective theory. All parameters have to be calibrated from observational data (mostly II/I, but also diffuse emissions). Depending on the assumptions, the above transport equation can be solved semi-analytically or fully numerically. The former method (e.g. [START_REF] Maurin | [END_REF]7,13]) is well suited to study theoretical uncertainties, while the latter (e.g. the popular Galprop code [14] or derivatives), more CPU-time demanding, is often used to probe more complex configurations of the different input parameters or observables (e.g. implement more detailed source/gas distributions for diffuse gamma-ray predictions).

The present generation of CR experiments have reached sensitivities that push the most popular transport models to their limits, as the predictions get more and more sensitive to the theoretical assumptions or prejudice. Though critical for data interpretations, this is good news for CR theorists. Some issues related to cosmic-ray transport: In this section, as a conclusion to this contribution, I briefly review some issues related to cosmicray transport.

The rising positron fraction: In 2009, the PAMELA data [15] confirmed with clear statistical significance previous indications [16] that the positron fraction (φ e + /(φ e + + φ e -)) was increasing with energy, in contrast with what is expected when assuming CR positrons are pure secondaries [10,11,17]: there must be sources of primary positrons (AMS02 confirmed this rise [18] up to a flattening at energies around 350 GeV). It has actually been known for decades that pulsars are sources of electron-positron pairs that are further accelerated in PWNe [19]. Moreover, at energies beyond 10 GeV, the discreteness of the source distribution becomes manifest for CR electrons and positrons, and one has to account for the known local objects [20]. This is because of energy losses b(E) = -dE/dt ∝ E 2 , which imply that the spatial horizon λ of positrons decreases with energy (λ 2 ≈ K(E) E/b(E)), down to ∼ 1 kpc for 100 GeV. Therefore, PWNe are natural candidates to explain the increase of the positron fraction [21]. Nevertheless, a detailed and constrained dynamical modeling of these local sources is still lacking (chi-square analyses relying on simplistic descriptions are no longer sufficient). Another interesting astrophysical scenario relies on the possibility that this increase be due to secondary positrons produced in SNRs and accelerated in situ at non-relativistic shocks [22]; this scenario also predicts increases in other II/I ratios (p/p, B/C), which will be tested very soon with AMS02. Finally, a more speculative source of positrons is dark matter annihilation or decay [23,24], while this would require unconventionally large annihilation or decay rates and very fine-tuned scenarios. Though this can formally not be excluded from present astrophysical knowledge, the very existence of astrophysical sources precludes the positron data from being interpreted as a clean dark matter signal, and one can at best use the data to put constraints [25]. Solutions to the rising positron fraction illustrate that an accurate (local) description of the spatial source distribution is a critical ingredient for predictions related to high-energy CR electrons and positrons. This also holds for inverse Compton predictions.

Puzzles at the Galactic center (GC): Recently, many gamma-ray excesses have been reported toward the GC, where CR physics is fundamental for interpretation or criticism (contributions from hadronic or inverse Comptonleptonic -processes). Three cases: (i) the Fermi bubbles [26], (ii) the 130 GeV gamma-ray line [27], (iii) 10-50 GeV "excess" at the GC [28]. Case (i) is observationally established (large-scale gamma-ray emission around the GC), and interpretation mostly relies on CR acceleration and transport, with both hadronic and leptonic scenarios [29][30][31], or astrophysical jet physics. What is interesting with these features is that they challenge the idea that CR transport could be described everywhere assuming a simplistic slab geometry wherein the diffusion coefficient would be homogeneous (the bubbles extend up to 8 kpc away from the GC). Though not firm detections, cases (ii) and (iii) are interesting in the sense that their putative interpretations depend on diffuse gamma-ray background models (both the intensity and the spectral shape), and thereby on CRs (sources, transport, and ISM at the GC). I hope it is clear from above that uncertainties are large even at the stage of modeling. It is therefore not surprising at first sight to find observational departures from current background models, even though a gamma-ray line, if detected at high significance, would be difficult to associate with a CR origin. Anyway, strong theoretical and modeling improvements will be necessary to fully understand the high-energy phenomena at the GC in the near future. This is a very active research field.

Spatial diffusion and reacceleration: To conclude, I emphasize that most CR transport models rely on the assumption that the spatial diffusion coefficient is homogeneous and isotropic in a slab, and has a power-law dependence on rigidity. Such approximations are sufficient to get a consistent physical picture for what concerns CR nuclei, but some problems remain unsolved. For instance, II/I data seem to prefer values of the power-law slope of the diffusion coefficient which are in significant tension with the absence of anisotropy [32]. Understanding the intimate relations between the magnetic turbulence existing in astrophysical environments and the diffusion properties of CRs is a therefore promising line of research for the near future [START_REF] Shalchi | Nonlinear Cosmic Ray Diffusion Theories[END_REF]33]. To some extent, this will affect the interpretation of the rising positron fraction (diffusion a in very local environment), as well as that of any other fine-structure observation (e.g. the GC, or features in the diffuse gamma-ray emission). Finally, since reacceleration is linked to diffusion and strongly affects the sub-GeV CR budget (and fits to the II/I data), more systematic theoretical inspections are now necessary [34]. Current and new generations of multiwavelength observations and CR experiments will bring important matter to the field in the next decade.


Antonio M Peralta
email: aperalta@ugr.es

Hermann Pfitzner
email: pfitzner@labomath.univ-orleans.fr

PERTURBATION OF ℓ 1 -COPIES IN PREDUALS OF JBW * -TRIPLES

)). This is shown to be stable under small norm perturbations in the following sense: if the linear span of the two functionals is isometric up to δ > 0 to ℓ1(2), then the functionals are less far (in norm) than ε > 0 from two orthogonal functionals, where ε → 0 as δ → 0. Analogous statements for finitely and even infinitely many functionals hold as well. And so does a corresponding statement for non-normal functionals. Our results have been known for C * -algebras.

Introduction

The starting point of this note consists in two well-known facts. First, two elements in the predual of a JBW * -triple are orthogonal if and only if they span the two-dimensional ℓ 1 (2) isometrically and second, in preduals of von Neumann algebras this still makes sense after small norm perturbations, moreover not only for two but for finitely and, up to subsequences, even infinitely many elements.

For example, if a sequence (ϕ n ) in L 1 ([0, 1]) is such that

|α n | ≥ α n ϕ n ≥ r |α n |,
then there are pairwise orthogonal ϕ n such that ϕ n -ϕ n < ε, with ε → 0 as r → 1 (and, of course, with α n ϕ n = |α n |), see [START_REF] Dor | On projections in L1[END_REF]. Briefly, in L 1 a sequence near to an isometric copy of ℓ 1 is near to an orthogonal sequence. Up to subsequences the same follows from [START_REF] Pfitzner | Perturbations of ℓ 1 -copies and measure convergence in preduals of von Neumann algebras[END_REF]Th. 1.2] for arbitrary von Neumann preduals. Analogous non-normal versions hold, too: it can be deduced from [START_REF] Pfitzner | Perturbations of ℓ 1 -copies and measure convergence in preduals of von Neumann algebras[END_REF]Prop. 1.3] that if a sequence (ϕ n ) in the dual of a C * -algebra A is as above then for any ε > 0 there are pairwise orthogonal elements c k ∈ A such that ϕ n k (c k ) > (1 -ε)r for some subsequence (ϕ n k ); a similar formulation (reminiscent of Pe lczyński's property (V ) or Grothendieck's criterion of weak compactness in the dual of a C(K)-space) is that if one accepts to replace r by a worse constant (e.g. r 2 /2) in the last First author partially supported by the Spanish Ministry of Science and Innovation, D.G.I. project no. MTM2011-23843, Junta de Andalucía grant FQM375 and Deanship of inequality, then the c k 's can be considered to be selfadjoint elements of a commutative subalgebra of A [36, §6, Lem. 6.3].

In view of the numerous generalizations of geometric (=Banach space theoretic) properties from C * -algebras to JB * -triples it is natural to conjecture similar results for JB * -triples. The aim of this article is to state and prove them.

Let us describe these results. They divide into two parts, contained in Sections 3 and 4, respectively, depending on whether the ℓ 1 -copies are of finite or infinite dimension. Basic to all this, as already alluded to in the first paragraph, is the result of Y. Friedman, B. Russo [START_REF] Friedman | Conditional expectation and bicontractive projections on Jordan C * -algebras and their generalizations[END_REF]Lem. 2.3], according to which two functionals are algebraically orthogonal if and only if they are L-orthogonal where the latter means that the two functionals span an isometric copy of ℓ 1 (2) (see Section 2 for definitions).

The main result of Section 3, Theorem 3.6, yields a quantification of algebraic orthogonality for finitely many arbitrary elements in the dual of a JB * -triple E: if functionals ϕ 1 , . . . , ϕ n in E * span ℓ 1 (n) (1-δ)-isomorphically then they are near to pairwise orthogonal functionals as δ is near to 0 and moreover they attain their norm up to a given ε > 0 on some pairwise orthogonal elements in E. A quantification of orthogonality in E is not possible in general but it is for tripotents, see Proposition 3.10.

Section 4 contains what has been described in the second paragraph. More specifically, if a bounded sequence (ϕ n ) in a JBW * -predual W * spans ℓ 1 almost isometrically, then according to Theorem 4.1 there are pairwise orthogonal ϕ k such that ϕ n k -ϕ k → 0 for some subsequence ϕ n k . The non-normal case, treated in Theorem 4.2, can be resumed by saying that if the ϕ n 's span ℓ 1 r-isomorphically in the dual of a JB * -triple E, then E contains an abelian subtriple such that the restrictions of an appropriate subsequence of the ϕ n to this subtriple still span ℓ 1 (1 -ε)-isomorphically for any given ε > 0. A quantitative version of Theorem 4.2 is already contained in [START_REF] Francisco | Weak compactness in the dual space of a JB * -triple is commutatively determined[END_REF]Th. 2.3] and, what is more, the arguments in [START_REF] Francisco | Weak compactness in the dual space of a JB * -triple is commutatively determined[END_REF] and [START_REF] Chu | JB * -triples have Pe lczynski's property V. Manuscr[END_REF] seem to lend themselves to a quantification that gives our Theorem 4.2. We refrained from pursuing this approach for it seems more natural to deduce the infinite dimensional case from the finite dimensional one, all the more because the latter has an interest in its own.

Preliminaries

We shall follow the standard notation employed, for example in [START_REF] Francisco | Closed tripotents and weak compactness in the dual space of a JB * -triple[END_REF], [START_REF] Francisco | Weak compactness in the dual space of a JB * -triple is commutatively determined[END_REF] or [START_REF] Bunce | Dunford-Pettis properties, Hilbert spaces and projective tensor products[END_REF]. For Banach space theory we refer, e.g., to [START_REF] Diestel | Sequences and series in a Banach space[END_REF][START_REF] Fabian | Banach space theory[END_REF][START_REF] Johnson | Handbook of the geometry of Banach spaces[END_REF].

We recall that a JB * -triple [START_REF] Kaup | A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces[END_REF] is a complex Banach space E equipped with a continuous ternary product {., ., .} symmetric and bilinear in the outer variables and conjugate linear in the middle one satisfying (2.1) L(x, y) {a, b, c} = {L(x, y)a, b, c} -{a, L(y, x)b, c} + {a, b, L(x, y)c} , such that L(a, a) = a 2 and L(a, a) is an hermitian operator on E with non-negative spectrum, where L(a, b) is given by L(a, b)y = {a, b, y}.

Every C * -algebra is a JB * -triple with respect to the triple product given by {x, y, z} = 1 2 (xy * z + zy * x). The same triple product equipes the space B(H, K), of all bounded linear operators between complex Hilbert spaces H and K, with a structure of JB * -triples. Among the examples involving Jordan algebras, we can say that every JB * -algebra is a JB * -triple under the triple product {x, y, z}

= (x • y * ) • z + (z • y * ) • x -(x • z) • y * .
An element u in a JB * -triple E is said to be a tripotent when it is a fixed point of the triple product, that is, when u = {u, u, u}. Given a tripotent u ∈ E, the mappings P i (u) : E → E i (u), (i = 0, 1, 2), defined by

P 2 (u) = L(u, u)(2L(u, u) -id E ), P 1 (u) = 4L(u, u)(id E -L(u, u)),
and P 0 (u) = (id E -L(u, u))(id E -2L(u, u)), are contractive linear projections, called the Peirce projections associated with u. The range of P i (u) is the eigenspace E i (u) of L(u, u) corresponding to the eigenvalue i 2 , and

E = E 2 (u) ⊕ E 1 (u) ⊕ E 0 (u)
is the Peirce decomposition of E relative to u. Furthermore, the following Peirce rules are satisfied,

(2.2) {E 2 (u), E 0 (u), E} = {E 0 (u), E 2 (u), E} = {0}, (2.3) {E i (u), E j (u), E k (u)} ⊆ E i-j+k (u),
where E i-j+k (u) = {0} whenever i -j + k / ∈ {0, 1, 2} ( [START_REF] Friedman | Structure of the predual of a JBW * -triple[END_REF] or [12, Th. 1.2.44]). For x, y, z in a JB * -triple E we have [START_REF] Friedman | The Gelfand-Naimark theorem for JB * -triples[END_REF]Cor. 3] {x, y, z} ≤ x y z .

(2.4)

A tripotent u is called complete if E 0 (u) reduces to {0}.
The Peirce-2 subspace E 2 (u) is a unital JB * -algebra with unit u, product a • u b = {a, u, b} and involution a ♯u = {u, a, u} (c.f. [7, Theorem 2.2] and [START_REF] Kaup | Jordan algebras and symmetric Siegel domains in Banach spaces[END_REF]Theorem 3.7]; [12, p. 185]).

A JBW * -triple is a JB * -triple which is also a dual Banach space. Every JBW * -triple admits a unique isometric predual and its triple product is separately weak * -continuous ( [START_REF] Barton | Weak * -continuity of Jordan triple products and its applications[END_REF], [START_REF] Horn | Characterization of the predual and ideal structure of a JBW * -triple[END_REF], [START_REF] Chu | Jordan structures in geometry and analysis[END_REF]Th. 3.3.9]). Consequently, the Peirce projections associated with a tripotent in a JBW * -triple are weak *continuous. The second dual of a JB * -triple is a JBW * -triple such that its triple product reduces to the original one (cf. [START_REF] Dineen | Complete holomorphic vector fields on the second dual of a Banach space[END_REF], [START_REF] Chu | Jordan structures in geometry and analysis[END_REF]Cor. 3.3.5]). The class of JBW * -triples includes all von Neumann algebras. Functionals on a JBW * -triple W are called normal if they belong to the predual W * .

JBW * -triples play, in the category of JB * -triples, a similar role to that played by von Neumann algebras in the setting of C * -algebras. A JB *triple need not have any non-zero tripotent. However, since the complete tripotents of a JB * -triple E coincide with the complex and the real extreme points of its closed unit ball (cf. [START_REF] Braun | A holomorphic characterization of Jordan C * -algebras[END_REF]Lem. 4.1], [START_REF] Kaup | Jordan algebras and symmetric Siegel domains in Banach spaces[END_REF]Prop. 3.5], [START_REF] Chu | Jordan structures in geometry and analysis[END_REF]Th. 3.2.3]), the Krein-Milman theorem implies that every JBW * -triple contains an abundant set of tripotents.

Given elements a, b in a JB * -triple E, the symbol Q(a, b) will denote the conjugate linear operator on E defined by Q(a, b)(x) := {a, x, b}. We write [START_REF] Loos | Bounded symmetric domains and Jordan pairs[END_REF] or [39, page 305]). In the particular case of u being a tripotent, we have P 0 (u) = B(u, u).

Q(a) instead of Q(a, a). The Bergmann operator B(a, b) : E → E is the mapping given by B(a, b)(z) = z -2L(a, b)(z) + Q(a)Q(b)(z), for all z in E (compare
Throughout the paper, given a Banach space X, we consider X as a closed subspace of X * * , via its natural isometric embedding, and for each closed subspace Y of X we shall identify Y σ(X * * ,X * ) , the weak * -closure of Y in

X * * , with Y * * . A normalized sequence (x n ) in a Banach space X is said to span ℓ 1 r- isomorphically if n α n x n ≥ r n |α n | for all scalars α n .
If there is a sequence (δ m ) such that 0 ≤ δ m → 0 and (x n ) n≥m spans ℓ 1 δ m -isomorphically for all m then (x n ) is said to span ℓ 1 almost isometrically.

The strong- * -topology. Given a norm-one element ϕ in the predual W * of a JBW * -triple W , and a norm-one element z in W with ϕ(z) = 1, it follows from [4, Proposition 1.2] that the assignment (x, y) → ϕ {x, y, z} defines a positive sesquilinear form on W. Moreover, for every norm-one element w in W satisfying ϕ(w) = 1, we have ϕ {x, y, z} = ϕ {x, y, w} , for all x, y ∈ W . The mapping x → x ϕ := (ϕ {x, x, z}) 1 2 , defines a prehilbertian seminorm on W . The strong * -topology of W , introduced by T.J. Barton and Y. Friedman in [START_REF] Barton | Bounded derivations of JB * -triples[END_REF], is the topology on W generated by the family { • ϕ : ϕ ∈ W * , ϕ = 1}, and will be denoted by s * (W, W * ). From [4, page 258] we get |ϕ(x)| ≤ x ϕ for any x ∈ W, and it is clear from this that s * (W, W * ) is stronger than the weak * -topology of W .

It is known that the triple product of a JBW * -triple is jointly strong *continuous on bounded sets ([34, Th. 9], [37, Th., page 103]). Another interesting property tells us that the strong*-topology of a JBW * -triple W is compatible with the duality (W, W * ) (i.e. a linear functional on W is weak * -continuous if, and only if, it is strong * -continuous, see [START_REF] Peralta | Grothendieck's inequalities for real and complex JBW * -triples[END_REF]Th. 9]). The bipolar theorem implies that for convex sets of W , weak * -closure and strong * -closure coincide. It follows that the closed unit ball of a weak * -dense JB * -subtriple E of a JBW * -triple W is strong * -dense in the closed unit ball of W . This result, known as Kaplansky Density theorem for JBW * -triples, was established by J.T. Barton and Y. Friedman in [4,Cor. 3.3].

In 2001, L.J. Bunce culminated the description of the fundamental properties of the strong * -topology showing that for every JBW * -subtriple F of a JBW * -triple W , the strong * -topology of F coincides with the restriction to F of the strong * -topology of W , that is, s * (F, F * ) = s * (W, W * )| F [START_REF] Bunce | Norm preserving extensions in JBW*-triple[END_REF]. It is also known that a linear map between JBW * -triples is strong * -continuous if, and only if, it is weak * -continuous (compare [34, page 621]).

Functional calculus, open and closed tripotents. Let x be an element in a JB * -triple E. Throughout the paper, the symbol E x will stand for the norm-closed subtriple of E generated by x. It is known that E x is JB * -triple isomorphic to the abelian C * -algebra C 0 (L) of all complex-valued continuous functions on L vanishing at 0, where L is a locally compact subset of (0, x ] satisfying that L ∪ {0} is compact. Further, there exists a JB * -triple isomorphism Ψ : E x → C 0 (L) satisfying Ψ(x)(t) = t, for all t in L (compare [30, 1.15]). Given a continuous complex-valued function f : L ∪ {0} → C vanishing at 0, the continuous triple functional calculus f (x) will have its usual meaning f (x) = Ψ -1 (f ).

We define x [1] := x and x [2n+1] = {x, x, x [2n-1] } for every n ∈ N. JB *triples are power associative, that is,

x [2k-1] , x [2l-1] , x [2m-1] = x [2(k+l+m)-3] ,
for every k, l, m ∈ N (cf. [32, §3.3] or [START_REF] Chu | Jordan structures in geometry and analysis[END_REF]Lem. 1.2.10] or simply apply the Jordan identity).

Suppose now that x = 1 and that E is a subtriple of a JBW * -triple W , for example W = E * * . It is known that (x [2n+1] ) converges in the strong *topology to the tripotent u(x) = χ {1} (x) ∈ E x w * ⊂ W , which is called the support tripotent of x ( [START_REF] Edwards | Compact tripotents in bi-dual JB *triples[END_REF]Lem. 3.3]). (By χ A we denote the characteristic function of a set A.) By functional calculus there exist, for each n ∈ N,

unique elements x [ 1 2n-1 ] in E x ∼ = C 0 (L) satisfying x [ 1 2n-1 ] [2n-1] = x. The latter are strong * -convergent to the tripotent r(x) = χ (0,1] (x) ∈ E x w *
⊂ W , which is called the range tripotent of x. The tripotent r(x) is the smallest tripotent e ∈ W satisfying that x is positive in the JBW * -algebra W 2 (e) (see, for example, [17, comments before Lemma 3.1] or [9, §2]). The inequalities

u(x) ≤ x [2n+1] ≤ x ≤ r(x) hold in W 2 (r(x)) for every norm-one element x ∈ E.
A tripotent u, in a JB * -triple E, is said to be bounded if there exists a norm-one element x ∈ E such that L(u, u)x = u. The element x is called a bound of u and we write u ≤ x. We shall write y ≤ u whenever y is a positive element in the JB * -algebra E 2 (u) (cf. [21, pages 79-80]). A JB *triple E need not have a cone of positive elements and the lacking of order implies that the symbol x ≤ y makes no sense for general elements x, y ∈ E. However, it should be remarked that, given x, y ∈ E and tripotents u, v ∈ E with x ≤ u and u ≤ y ≤ v, we have x ≤ u ≤ y ≤ v with respect to the natural order of the JB * -algebra E 2 (v). Note further, for Lemma 3.3 below, that we have u ≤ x in the just mentioned sense if we have u ≤ x in the JBW * -algebra E * * 2 (r(x)). Inspired by the notion of open projection in the bidual of a C * -algebra introduced and studied by C. Akemann, L. Brown, and G.K. Pedersen (cf. [START_REF] Akemann | The general Stone-Weierstrass problem[END_REF] or [START_REF] Akemann | Complications of semicontinuity in C *algebra theory[END_REF][START_REF] Akemann | Facial structure in operator algebra theory[END_REF] or [START_REF] Pedersen | C*-algebras and their automorphism groups[END_REF]Proposition 3.11.9]), C.M. Edwards and G.T. Rüttimann develop the notion of open tripotent in the bidual of a JB * -triple E: we say that a tripotent e in

E * * is open if E * * 2 (e) ∩ E is weak * -dense in E * * 2 (e) (see [18, page 167]). It is known that the range tripotent of a norm-one element of E is open (cf. [9, Proposition 2.1]).
A tripotent e in E * * is said to be compact-G δ (relative to E) if there exists a norm-one element x in E such that e coincides with u(x), the support tripotent of x. A tripotent e in E * * is said to be compact (relative to E) if there exists a decreasing net (e λ ) of tripotents in E * * which are compact-G δ with infimum e, or if e is zero (cf. [18, pages 163-164]). In the terminology of [START_REF] Francisco | Closed tripotents and weak compactness in the dual space of a JB * -triple[END_REF], we say that a tripotent u in

E * * is closed if E ∩ E * * 0 (u) is weak * -dense in E * * 0 (u).
The equivalence established in [START_REF] Francisco | Closed tripotents and weak compactness in the dual space of a JB * -triple[END_REF]Th. 2.6] shows that a tripotent e ∈ E * * is compact if and only if e is closed and bounded by an element of E.

Small perturbation of a normal functional. Let ϕ ∈ W * be a functional in the predual of a JBW * -triple W and let e be a tripotent in W . In [24, Proposition 1], Y. Friedman and B. Russo prove that ϕP 2 (e) = ϕ if and only if ϕ = ϕP 2 (e). Using the techniques of ultraproducts of Banach spaces, J. Becerra-Guerrero and A. Rodríguez Palacios obtained the following quantitative version of the above property which will be used throughout this article.

Lemma 2.1. [6, Lem. 2.2] Given ε > 0, there exists η = η(ε) > 0 such that, for every JB * -triple E, every non-zero tripotent e in E, and every ϕ in E * with ϕ ≤ 1 and ϕP 2 (e) ≥ 1 -η, we have ϕ -ϕP 2 (e) < ε.

E * * 2 (r(a)) ⊥ E * * 2 (r(b)); r(a) ∈ E * * 0 (r(b)); a ∈ E * * 0 (r(b)); b ∈ E * * 0 (r(a)); E a ⊥ E b .
It follows from Peirce rules (2.2) that, for each tripotent u in a JB * -triple

E, E 0 (u) ⊥ E 2 (u).
For each norm one functional ϕ in the predual of a JBW * -triple W , the square of the prehilbertian seminorm • ϕ is additive on orthogonal elements:

a + b 2 ϕ = a 2 ϕ + b 2 ϕ , ∀a ⊥ b.
We recall that a functional φ in the predual of a JBW * -algebra M is said to be faithful if for each a ≥ 0 in M , φ(a) = 0 implies a = 0.

Let ϕ be a norm-one functional in the predual of a JBW * -triple W . By [24, Prop. 2], there exists a unique tripotent e = e(ϕ) ∈ W satisfying ϕ = ϕP 2 (e) and ϕ| W 2 (e) is a faithful normal state of the JBW * -algebra W 2 (e). This unique tripotent e is called the support tripotent of ϕ, and will be denoted by e(ϕ). (Note that at the time of the writing of [START_REF] Friedman | Structure of the predual of a JBW * -triple[END_REF] condition [24, (1.13)] was not yet known to hold for all JBW * -triples.)

Now, according to [START_REF] Friedman | Conditional expectation and bicontractive projections on Jordan C * -algebras and their generalizations[END_REF] and [START_REF] Edwards | Orthogonal faces of the unit ball in a Banach space[END_REF], we define two functionals ϕ and ψ in the predual of a JBW * -triple W to be algebraically orthogonal or simply orthogonal, denoted by ϕ ⊥ ψ, if their support tripotents are orthogonal in W , that is e(ϕ) ⊥ e(ψ).

Elements x, y in a normed space X are said to be L-orthogonal (and we write x ⊥ L y) if x ± y = x + y , and are said to be M -orthogonal

(denoted by x ⊥ M y) if x ± y = max { x , y }.
Given a, b in E, it follows from [24, Lem. 1.3(a)] that a ⊥ M b whenever a ⊥ b. In general the reverse implication does not hold, for example (1/2, 1, 0) and (1/2, 0, 1) in the C * -algebra l ∞ (3) are M -orthogonal but not orthogonal. The following result is borrowed from [START_REF] Friedman | Conditional expectation and bicontractive projections on Jordan C * -algebras and their generalizations[END_REF] and [START_REF] Edwards | Orthogonal faces of the unit ball in a Banach space[END_REF]. Note that ϕ ⊥ L ψ in the lemma (with ψ , ϕ = 0) is equivalent to

α ϕ ϕ + β ψ ψ = |α| + |β|,
for any α, β ∈ C.

3.

Quantitative versions of M -and L-orthogonality in JBW * triples and their predual spaces

The main goal of this section is to establish quantitative versions of Lemma 2.2 (see Propositions 3.5, 3.10 and Theorem 3.6 below). The proof will follow from a series of technical results. The next two lemmas, which are included here for the sake of completeness, are borrowed from [START_REF] Francisco | Weak compactness in the dual space of a JB * -triple is commutatively determined[END_REF]. We shall also need an appropriate version of [22, Lem. 2.2], the argument is taken from the just quoted paper. Lemma 3.3. Let E be a JB * -triple, θ > 0, 1 > δ > 0, and let ϕ 1 , ϕ 2 be two norm-one functionals in E * . Suppose x is an element in the closed unit ball of E, satisfying |ϕ 1 (x)| ≥ 1 -δ and x ϕ 2 ≤ θ. Then, for every ε > 0 with 1 -δ ≥ 2ε there exist two elements a, y in the unit ball of E x , and two tripotents u, v in

Lemma
(E x ) * * such that a ≤ u ≤ y ≤ v = r(y), 1 ≥ |ϕ 1 ( a)| > 1 -δ -ε, and v ϕ 2 < 3θ ε . We can further find a ∈ E * * 2 (u) such that 1 ≥ ϕ 1 (a) > 1 -δ -ε. Proof. Let α > 0 and define f α , g α ∈ C 0 (L) by f α =    0, if 0 ≤ t ≤ α affine, if α ≤ t ≤ 2α t, if 2α ≤ t ≤ x , g α =    0, if 0 ≤ t ≤ α 2 affine, if α 2 ≤ t ≤ α 1, if α ≤ t ≤ x . Let 2ε/3 < ε ′ < ε and define a = f ε ′ (x) and y = g ε ′ (x) by the functional calculus recalled in Section 2. Then a ≤ 1 because 2ε ′ ≤ 1 -δ ≤ x . Since x -a ≤ ε ′ and |ϕ 1 (x)| ≥ 1 -δ it follows that |ϕ 1 ( a)| > 1 -δ -ε. We set u = χ [ε ′ , x ] , and v = r(y) = χ ( ε ′ 2 ,
x ] (also in (E x ) * * ) and get a ≤ u ≤ y ≤ v. Let us take α ∈ R such that ϕ 1 (e iα a) > 0 and define a = e iα a. Then a ∈ E * * 2 (u) because a ≤ u and we have

1 ≥ ϕ 1 (a) > 1 -δ -ε.
Since • ϕ is an order-preserving map on the set of positive elements in (E x ) * * (cf. [START_REF] Francisco | Closed tripotents and weak compactness in the dual space of a JB * -triple[END_REF]Lem. 3.3]), we deduce that

v ϕ 2 ≤ 2 ε ′ x ϕ 2 ≤ 2θ ε ′ < 3θ ε .
Proposition 3.4. Let E be a JB * -triple, and let ϕ 1 and ϕ 2 be two orthogonal norm-one functionals in E * . Then for every ε > 0 there exist norm-one

elements a, b in E satisfying a ⊥ b, ϕ 1 (a) > 1 -ε and ϕ 2 (b) > 1 -ε.
Proof. Let us fix an arbitrary ε > 0. Take η > 0 satisfying η < min{ 1 3 , ε 2 }. We can also find 0 < δ < εη 66 . We note that η and δ satisfy 2η < 1 -η, 1 -2η > 1 -ε, and 22 3 η δ < ε. Let e j in E * * be the support tripotent of ϕ j , j = 1, 2. Since e 1 ⊥ e 2 , e 1 +e 2 and e 1 -e 2 are the support tripotents of φ = ϕ 1 + ϕ 2 and ψ = ϕ 1 -ϕ 2 , respectively (see [START_REF] Edwards | Orthogonal faces of the unit ball in a Banach space[END_REF]Th. 5.4]). In particular, ϕ 1 (e 2 ) = 0 = ϕ 2 (e 1 ).

By the Kaplansky Density theorem for JBW * -triples [4, Cor. 3.3] (i.e. by the strong * -density of the closed unit ball of E in the one of E * * ), there are two nets (z λ ) and ( z µ ) in the closed unit ball of E converging in the strong *topology of E * * to e 1 and e 2 , respectively. Since s * (E * * , E * ) is stronger than the weak * -topology of E * * , we deduce that (z λ ) → e 1 and ( z µ ) → e 2 in the weak * -topology of E * * . In particular,

ϕ 1 (z λ ) → ϕ 1 (e 1 ) = 1, ϕ 1 ( z µ ) → ϕ 1 (e 2 ) = 0, ϕ 2 (z λ ) → ϕ 2 (e 1 ) = 0, ϕ 2 ( z µ ) → ϕ 2 (e 2 ) = 1, z λ ϕ 1 → e 1 ϕ 1 = 1, z µ ϕ 1 → e 2 ϕ 1 = 0, z λ ϕ 2 → e 1 ϕ 2 = 0, and z µ ϕ 2 → e 2 ϕ 2 = 1.
Find indices λ 0 and µ 0 such that

(3.2) |ϕ 1 (z λ 0 )| > 1 -η, z λ 0 ϕ 2 < δ, |ϕ 2 ( z µ 0 ) -1| < 3 η δ.
Applying Lemma 3.3 (with δ, η, z λ 0 for θ, δ = ε, x) we can find a 0 , a, y in the closed unit ball of E z λ 0 and two tripotents u, v in E * *

z λ 0 satisfying a ≤ u ≤ y ≤ v, 1 ≥ ϕ 1 (a 0 ) > 1 -2η > 1 -ε, v ϕ 2 < 3 η δ, and a 0 ∈ E * * 2 (u).
Define a = a 0 / a 0 ∈ E * * 0 (u). Then ϕ 1 (a) > 1-ε. By Lemma 3.2 we obtain

ϕ 2 z µ 0 -B(y, y)( z µ 0 ) < 21 z µ 0 v ϕ 2 < 21 3 η δ,
and by the third inequality in (3.2) we deduce that

ϕ 2 B(y, y)( z µ 0 ) -1 < 22 3 η δ < ε. Setting b = e iβ B(y, y)( z µ 0 ) for a suitable β ∈ IR we have ϕ 2 ( b) > 1 -ε and setting b = b/ b we still have ϕ 2 (b) > 1 -ε. By Lemma 3.1, b ∈ B(y, y)(E) ⊆ E * * 0 (u). Since, by construction, a lies in E * * 2 (u), it follows that a ⊥ b.
Remark. Proposition 3.4 remains valid (with practically the same proof) if the first sentence is replaced by "Let E be a weak * -dense subtriple of a JBW * -triple W and let ϕ 1 , ϕ 2 be two orthogonal norm-one functionals in W * ."

We shall require some results in the theory of ultraproducts of Banach spaces [START_REF] Heinrich | Ultraproducts in Banach space theory[END_REF]. To this end, we recall some basic facts and definitions. Let U be an ultrafilter on a non-empty set I, and let {X i } i∈I be a family of Banach spaces. Let ℓ ∞ (I, X i ) = ℓ ∞ (X i ) denote the Banach space obtained as the ℓ ∞ -sum of the family {X i } i∈I , and let

c 0 (X i ) := (x i ) ∈ ℓ ∞ (X i ) : lim U x i = 0 .
The ultraproduct of the family {X i } i∈I relative to the ultrafilter U , denoted by (X i ) U , is the quotient Banach space ℓ ∞ (X i )/c 0 (X i ) equipped with the quotient norm. Let [x i ] U be an equivalence class in (X i ) U represented by a family

(x i ) i ∈ ℓ ∞ (X i ). It is known that [x i ] U = lim U x i ,
independently of the representative of [x i ] U . In general, the ultraproduct of a family of dual Banach spaces is not a dual Banach space (not even in the case of von Neumann algebras). The ultraproduct (X * i ) U of the duals can be identified isometrically with a closed subspace of the dual ((X i ) U ) * via the canonical mapping

J : (X * i ) U → ((X i ) U ) * J [ϕ i ] U ([x i ] U ) = lim U ϕ i (x i ).
In [START_REF] Dineen | Complete holomorphic vector fields on the second dual of a Banach space[END_REF]Cor. 10] S. Dineen establishes that the class of JB * -triples (analogously to the class of C * -algebras [27, Prop. 3.1]), is stable under ultraproducts via the canonical triple product

{[u i ] U , [v i ] U , [w i ] U } = [{u i , v i , w i }] U .
Here is a simple argument to prove Dineen's theorem (cf. also [12, proof of Cor. 3.3.5]). Let {E i } i∈I be a family of JB * -triples. Then the Banach space ℓ ∞ (E i ) is a JB * -triple with pointwise operations ([30, page 523] or [START_REF] Chu | Jordan structures in geometry and analysis[END_REF]Ex. 3.1.4]). Let E be a JB * -triple. A subtriple I of a JB * -triple a JB * -triple E is said to be an ideal or a triple ideal of E if {E, E, I}+{E, I, E} ⊆ I. It is easy to see, under the above conditions, that {ℓ

∞ (E i ), ℓ ∞ (E i ), c 0 (E i )} ⊆ c 0 (E i ) and {ℓ ∞ (E i ), c 0 (E i ) , ℓ ∞ (E i )} ⊆ c 0 (E i ) , and hence c 0 (E i ) is a closed triple ideal of ℓ ∞ (E i ).
Since the quotient of a JB * -triple by a closed triple ideal is a JB * -triple ( [START_REF] Kaup | A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces[END_REF] or [START_REF] Chu | Jordan structures in geometry and analysis[END_REF]Cor. 3.1.18]), we deduce that

(E i ) U = ℓ ∞ (E i )/c 0 (E i ) is a JB * -triple.
Proposition 3.5. For each ε > 0 there exists δ > 0 such that for every JB *triple E and every pair of functionals ϕ 1 and ϕ 2 in the closed unit ball of

E * with 2 ≥ ϕ 1 ± ϕ 2 ≥ 2(1 -δ) there exist orthogonal norm-one elements a, b in E satisfying ϕ 1 (a) > 1 -ε and ϕ 2 (b) > 1 -ε.
Proof. Suppose, to the contrary, that there exists ε 0 > 0 such that for each natural n, we can find a JB * -triple E n and functionals ϕ 1,n and ϕ 2,n in the closed unit ball of

E * n with 2 ≥ ϕ 1,n ± ϕ 2,n ≥ 2(1 -1 n ) satisfying |ϕ 1,n (a)| ≤ 1 -ε 0 and |ϕ 2,n (b)| ≤ 1 -ε 0 , whenever a, b are elements of norm one in E n with a ⊥ b.
Take a non-trivial ultrafilter U in N, let 0 < ε 1 < ε and let J :

(E * n ) U → ((E n ) U ) * be the canonical isometric embedding defined by J [ϕ i ] U ([x n ] U ) = lim U ϕ n (x n ). Then J [ϕ 1,n ] U and J [ϕ 2,n ] U have norm one and are L-ortho- gonal in ((E n ) U ) * because so are [ϕ 1,n ] U and [ϕ 2,n ] U in (E * n ) U .
As explained above, ((E n ) U ) * is a JB * -triple and Proposition 3.4 applies: there exist normone elements

[a n ] U , [b n ] U in (E n ) U satisfying [a n ] U ⊥ [b n ] U , J [ϕ 1,n ] U ([a n ] U ) > 1 -ε 1 and J [ϕ 2,n ] U ([b n ] U ) > 1 -ε 1 . We note that the elements [a n ] U , [b n ] U are orthogonal in the quotient (E n ) U = ℓ ∞ (E n )/c 0 (E n ). Since the quotient mapping π : ℓ ∞ (E n ) → ℓ ∞ (E n )/c 0 (E n ) is a triple homomorphism between JB * -triples and π((a n ) n ) = [a n ] U ⊥ π((b n ) n ) = [b n ] U , by [10, Proposition 4.7] there exist ortho- gonal elements ( a n ) n and ( b n ) n in ℓ ∞ (E n ) satisfying π(( a n ) n ) = [a n ] U and π(( b n ) n ) = [b n ] U . We have lim U a n = lim U a n = 1 and likewise for (b n ) n , ( b n ) n . Now, 1-ε 1 < J [ϕ 1,n ] U ([ a n ] U ) = lim U ϕ 1,n ( a n ), 1-ε 1 < J [ϕ 2,n ] U ([ b n ] U ) = lim U ϕ 2,n ( b n ), and, for every n, a n ⊥ b n . Hence ϕ 1,n ( a n / a n ) > 1 -ε or ϕ 2,n ( b n / b n ) > 1-
ε can be achieved for infinitely many n which contradicts the assumption made in the beginning of the proof.

We shall establish now an analogous version of Proposition 3.5 for finite sets of functionals in the dual of a JB * -triple. Theorem 3.6. For each ε > 0 and each natural n, there exists δ = δ(n, ε) > 0 with the following property. Let E be a JB * -triple and let ϕ 1 , . . . , ϕ n be functionals in E * such that

n j=1 |α j | ≥ n j=1 α j ϕ j ≥ (1 -δ(n, ε)) n j=1 |α j | ∀α j ∈ C. (3.3)
Then there exist mutually orthogonal elements a 1 , . . . , a n of norm one in E and mutually orthogonal functionals ϕ 1 , . . . , ϕ n of norm one in E * satisfying ϕ j (a j ) > 1 -ε and ϕ j -ϕ j < ε ∀j = 1, . . . , n (3.4) where ϕ j = ϕ j P 2 (r(a j )) ϕ j P 2 (r(a j )) .

Proof. We shall proceed by induction over n ≥ 1. For n = 1 there is nothing to prove. Let us fix ε > 0 and n ∈ IN.

We claim that there is

ε ′ ∈ (0, ε) such that if an element b in a JB * -triple E and ξ ∈ E * satisfy b ≤ 1, ξ ≤ 1 and |ξ(b)| > 1 -ε ′ (3.5) then ξ -ξP 2 (r(b))
ξ - ψ ψ ≤ ξ -ψ + ψ - ψ ψ < ε 2 + 1 ψ -1 ψ = ε 2 + 1 -ψ < ε
which proves the claim. Choose δ(n, ε ′ ) > 0 according to the induction hypothesis, choose η 0 = η( δ(n,ε ′ )

2 ) > 0 according to Lemma 2.1 and choose

δ 0 = δ min ε ′ , 1 n η 0 > 0
according to Theorem 3.5. Furthermore, let δ(n + 1, ε ′ ) be such that

0 < δ(n + 1, ε ′ ) < min δ(n, ε ′ ) 2 , δ 0 .
Let E be a JB * -triple and let ϕ 1 , . . . , ϕ n+1 ∈ E * satisfy n+1 j=1

|α j | ≥ n+1 j=1 α j ϕ j ≥ 1 -δ(n + 1, ε ′ ) n+1 j=1 |α j | ∀α j ∈ C. Let us define φ = n j=1 1 n ϕ j . Clearly, 2 ≥ φ ± ϕ n+1 ≥ 1 -δ(n + 1, ε ′ ) (n 1 n + 1) > 2(1 -δ 0 ).
Thus, by the choice of δ 0 (see Theorem 3.5) there exist a ⊥ a n+1 of norm one in E satisfying

ϕ n+1 (a n+1 ) > 1 -min ε ′ , 1 n η 0 > 1 -ε ′ , (3.6) φ(a) > 1 -min ε ′ , 1 n η 0 > 1 - 1 n η 0 .
Since n j=1 ϕ j (a) = nφ(a) > n -η 0 and ϕ j ≤ 1 we have

|ϕ j (a)| > n -η 0 -(n -1) = 1 -η 0 ∀j = 1, . . . , n.
Thus, |ϕ j P 2 (r(a))(a)| = |ϕ j (a)| > 1 -η 0 and by the choice of η 0 we deduce

ϕ j -ϕ j P 2 (r(a)) < δ(n, ε ′ ) 2 ∀j = 1, . . . , n.
Therefore we have

n j=1 |α j | ≥ n j=1 α j ϕ j P 2 (r(a)) ≥ n j=1 α j ϕ j - n j=1 α j ϕ j -ϕ j P 2 (r(a)) ≥ 1 -δ(n + 1, ε ′ ) n j=1 |α j | - δ(n, ε ′ ) 2 n j=1 |α j | ≥ 1 -δ(n, ε ′ ) n j=1 |α j |
for all scalars α j . Recall that r(a) is an open tripotent which means that the subtriple

F := E ∩ E * * 2 (r(a)) is weak * -dense in E * * 2 (r(a)). Set ψ j = ϕ j P 2 (r(a)) |F for j ≤ n. Then n j=1 |α j | ≥ n j=1 α j ψ j = n j=1 α j ϕ j P 2 (r(a)) ≥ 1 -δ(n, ε ′ ) n j=1 |α j |,
for all α j ∈ C, and by the induction hypothesis, applied to F , there exist mutually orthogonal norm-one elements a 1 , . . . , a n ∈ F satisfying ψ j (a j ) = ϕ j (a j ) > 1 -ε ′ , for every j = 1, . . . , n. They are orthogonal to a n+1 because a is. Together with (3.6) this shows the first half of (3.4) (for n + 1) because 1 -ε ′ > 1 -ε. The second half follows from the claim. This ends the induction and the proof.

In passing we note an obvious reformulation of the conclusion of Theorem 3.6: There exists an abelian subtriple C of E such that if we set

ψ j = ϕ j |C then (ψ j ) n j=1 spans ℓ 1 (n) (1 -ε)-isomorphically in C * .
For the proofs of Theorems 4.1 and 4.2 we need the following technical strengthening of Theorem 3.6.

Lemma 3.7. In Theorem 3.6 the a j can be constructed such that additionally there are mutually orthogonal compact tripotents u 1 , . . . , u n in E * * such that a j ∈ E * * 2 (u j ) for j = 1, . . . , n. Proof. We claim that if a JB * -triple E, a ∈ E, ϕ ∈ E * , and ε ′ > 0 are given such that

ϕ ≤ 1, a ≤ 1, ϕ(a) > 1 -ε ′ (3.7) then there exist a compact tripotent u ∈ E * * a ⊂ E * * 2 (e) and b ∈ E a ∩ E * * 2 (u) such that b = 1 and ϕ(b) > 1 -ε ′ .
In order to show the claim suppose (3.7) holds. Define z m = f α (a) ∈ E a for α = b j /m where f α is as in the proof of Lemma 3.3. Since z m -a → 0 there is m 0 such that ϕ(e iθ z m 0 ) > 1 -ε ′ for an appropriate θ ∈ IR. Also z m 0 ≤ 1. It remains to set b = e iθ z m 0 / z m 0 and u = χ [ b j /2m 0 , b j ]∩L and the claim is proved. Now we apply the claim n times to pairwise orthogonal a j and note that

u j ∈ E * * a j ⊥ E * * a k ∋ u k if j = k.
The claim in the proof of Theorem 3.6 shows that it is enough to replace a j by b j in (3.4) in order to finish the proof.

Recalling that Peirce projections associated with a tripotent in a JBW *triple are weak * -continuous, and the fact that the range tripotent of an element in a JBW * -triple always lies in the JBW * -triple, the arguments given above show: Corollary 3.8. For each ε > 0 and each natural n, there exists a positive δ = δ(n, ε) such that for every JBW * -triple W, and every finite set of functionals ϕ 1 , . . . , ϕ n in W * satisfying (3.3) there exist orthogonal norm one elements a 1 , . . . , a n ∈ W and orthogonal functionals ϕ 1 , . . . , ϕ n ∈ W * of norm one such that (3.4) holds.

The following corollary will not be needed in the sequel but could perhaps be useful elsewhere. The argument leading to part (a) has already been used in the proof of Theorem 3.6. Corollary 3.9. For each ε > 0 and each natural n, there exists a positive δ = δ(n, ε) with the following properties. We can now establish the promised quantitative version of the last statement in Lemma 2.2. Proposition 3.10. Given ε > 0 there exits δ = δ(ε) satisfying that for every JB * -triple E and every couple of tripotents u, v in E with

1 -δ < u ± v < 1 + δ, we have u -P 0 (v)(u) < ε and v -P 0 (u)(v) < ε.
Proof. Let us note that the statement is true whenever the set of tripotents in a JB * -triple E reduces to the zero element. Suppose, contrary to our claim, that there exists ε 0 > 0 such that for each natural n, we can find a JB * -triple E n and tripotents u n and v n with 1 -

1 n < u n ± v n < 1 + 1 n satisfying u n -P 0 (v n )(u n ) ≥ ε 0 or v n -P 0 (u n )(v n ) ≥ ε 0 . Take a non-trivial ultrafilter U in N. The elements [u n ] U and [v n ] U are non- zero tripotents in (E n ) U with [u n ] U ± [v n ] U = 1, that is, [u n ] U ⊥ M [v n ] U in (E n ) U . The final statement in Lemma 2.2 implies that [u n ] U ⊥ [v n ] U in (E n ) U . In particular [P 0 (u n )(v n )] U = P 0 ([u n ] U )([v n ] U ) = [v n ] U (because P 0 ([u n ] U ) = [P 0 (u n )] U ) and [P 0 (v n )(u n )] U = P 0 ([v n ] U )([u n ] U ) = [u n ] U , which implies that lim U P 0 (u n )(v n )-v n = 0 and lim U P 0 (v n )(u n )-u n = 0, contradicting our assumptions.
4. Infinite dimensional copies of ℓ 1 in preduals of JBW * -triples Theorem 4.1. Let W be a JBW * -triple and let (ϕ m ) be a bounded sequence in its predual W * . If (ϕ m ) spans ℓ 1 almost isometrically then there are a subsequence (ϕ mn ) of (ϕ m ) and a sequence ( ϕ n ) of pairwise orthogonal functionals in W * such that ϕ mn -ϕ n → 0 when l → ∞.

Proof. We can assume that ϕ m = 1, for every m. Let (ν n ) be a sequence of strictly positive numbers such that ∞ n=1 ν n < ∞. We shall prove, by induction over n, the existence of m n ∈ N, and φ

(n) m 1 , . . . , φ (n) mn in W * satisfying m n-1 < m n , and for each natural n φ (n) m k ⊥ φ (n) m l ∀k = l ∈ {1, . . . , n}, φ (n) m k = 1, ∀k ≤ n, φ (n) m k -φ (n-1) m k < ν n , ∀k = 1, . . . , n -1 if n ≥ 2, and φ (n) mn -ϕ mn < ν n . When n = 1 we set m 1 = 1, φ (1) 1
= ϕ 1 and the statement is clear. Suppose that m 1 < m 2 < . . . < m n , {φ

m 1 }, {φ (1)
m 1 , φ (2)
m 2 }, ..., {φ (n) m 1 , . . . , φ (2)
mn } have been defined satisfying the above properties.

By Corollary 3.8, there exists δ 1 = min{δ(n, ν n+1 /2), ν n+1 /2} > 0. Choose a natural j satisfying 21 √ j < δ 1 . We use Corollary 3.8 again in order to choose

δ 0 = δ(nj, ν n+1 ) > 0. Since (ϕ m ) spans ℓ 1 almost isometrically there exists m 0 > m n satisfying (1 -δ 0 ) ∞ m=m 0 |α m | ≤ ∞ m=m 0 α m ϕ m ∀α m ∈ C. Set N = {m 0 + 1, . . . , m 0 + nj} ⊆ N. Since (1 -δ(nj, ν n+1 )) m 0 +nj m=m 0 +1 |α m | ≤ m 0 +nj m=m 0 +1 α m ϕ m ∀α m ∈ C,
Corollary 3.8 implies the existence of mutually orthogonal elements a 1 , . . . , a nj in the closed unit ball of W such that

(4.1) ϕ m - ϕ m P 2 (r(a m )) ϕ m P 2 (r(a m )) < ν n+1 ∀m ∈ N.
On the other hand, it is clear, by orthogonality, that 0

≤ m∈N n k=1 r(a m ) 2 φ (n) m k = n k=1 m∈N r(a m ) 2 φ (n) m k = n k=1 m∈N r(a m ) 2 φ (n) m k ≤ n.
Thus, there exists

m n+1 ∈ N satisfying r(a m n+1 ) 2 φ (n) m k ≤ 1 j ∀k = 1, . . . , n hence, by Lemma 3.2, (4.2) φ (n) m k -φ (n) m k P 0 (r(a m n+1 )) ≤ 21 1 √ j ∀k = 1, . . . , n.
We define φ

(n+1) m k = φ (n) m k P 0 (r(a m n+1 )), for k = 1, . . . , n and
φ (n+1) m n+1 = ϕ m n+1 P 2 (r(a m n+1 )) ϕ m n+1 P 2 (r(a m n+1 ))
.

By (4.1), ϕ m n+1 -φ (n+1)
m n+1 < ν n+1 , and by (4.2)

(4.3) φ (n) m k -φ (n+1) m k ≤ 21 1 √ j < δ 1 ≤ ν n+1 2 ∀k = 1, . . . , n.
Therefore we have

n k=1 |α k | ≥ n k=1 α k φ (n+1) m k ≥ n k=1 α k φ (n) m k - n k=1 α k ( φ (n+1) m k -φ (n) m k ) ≥ n k=1 α k φ (n) m k -δ 1 n k=1 |α k | = n k=1 |α k | -δ 1 n k=1 |α k | ≥ (1 -δ(n, ν n+1 /2)) n k=1 |α k | ∀α m ∈ C.
By Corollary 3.8, applied to the JBW * -triple W 0 (r(a m n+1 )) and the functionals { φ Fix a natural k and consider the sequence (φ

(n) m k ) n≥k . The inequalities φ (n) m k -φ (n-1) m k < ν n and φ (n) m k -φ (i) m k < n j=i+1 ν j → 0 if n > i → ∞
show that (φ

(n) m k ) n≥k is a Cauchy sequence which converges to some ϕ k ∈ W * . By construction φ (n) m k ⊥ φ (n)
m j for every k = j, and every n ≥ max{j, k}, therefore

ϕ k ± ϕ j = lim n→∞ φ (n) m k ± φ (n) m j = 2
for every k = j in N. This implies ϕ k ⊥ ϕ j for every j = k (cf. Lemma 2.2). Finally, the inequality

ϕ mn -ϕ n ≤ φ (n) mn -ϕ mn + φ (n) mn -ϕ n = φ (n) mn -ϕ mn + φ (n) mn -lim k→∞ φ (k) mn < ν n + ∞ k=n+1 ν k ,
gives the desired statement lim n→∞ ϕ mn -ϕ n = 0.

The study of isomorphic copies in the dual space of a JB * -triple requires an extra effort. It should be remarked here that the next proposition can be considered as a quantitative version of [START_REF] Pfitzner | Weak compactness in the dual of a C * -algebra is determined commutatively[END_REF]Theorem 1] and [START_REF] Francisco | Weak compactness in the dual space of a JB * -triple is commutatively determined[END_REF]Theorem 2.3]. Theorem 4.2. Let E be a JB * -triple and let (ϕ m ) be a normalized sequence in E * spanning ℓ 1 r-isomorphically (with 0 < r ≤ 1). Then for each ε > 0 there exist a subsequence (ϕ mn ) of (ϕ m ) and a sequence (c n ) of mutually orthogonal elements of norm one in E such that

ϕ mn (c n ) > r(1 -ε), ∀n ∈ N, (4.5)
and such that the restrictions ϕ mn |C span ℓ 1 (r(1 -ε))-isomorphically where C is the abelian subtriple of E generated by the c n 's and isometric to a commutative C * -algebra.

Proof. We may assume that 1 ≥ ε > 0, we consider a series n≥1 ε n with

ε n > 0 and ∞ n=1 ε n = ε 2 .
(a n )| > 1 -r n-1 i=1 ε i 3 r 1 - n i=1 ε i
(where 0 i=1 = 0) and, for

v n = u 1 + . . . + u n , (4.10)
m∈Nn α m ϕ m P 0 (v n ) ϕ m P 0 (v n ) ≥ r 1 - n i=1 ε i m∈Nn |α m | ∀α m ∈ C.
Let us note that since

1 -r n-1 i=1 ε i 3 r 1 - n i=1 ε i > r 1 -2 n i=1 ε i > r 1 -2 ∞ i=1 ε i = r (1 -ε) ,
the inequality in (4.9) proves |ϕ mn (a n )| > r (1 -ε), for every n ∈ N. The statement of the proposition will follow for c n = e iθn a n for a suitable choice of θ n ∈ R such that ϕ mn (c n ) = |ϕ mn (a n )| for every natural n.

We deal first with the case n = 1. Set N 0 = N. Let us take a natural number j 1 such that 3 21

√ j 1 < rε 1 . Let δ 1 = δ(j 1 , ε 1 /
2) > 0 be given by Lemma 3.7. By James' distortion theorem there exist mutually disjoint finite subsets G

(1) k ⊂ N 0 , finite sequences (λ (1) m ) m∈F (1) k ⊂ C such that (4.11) m∈G (1) k |λ (1) m | ≤ 1 r , for every k ∈ N,
and the functionals φ

(1) k = m∈G (1) k λ (1) m ϕ m satisfy (4.12) k∈N 0 |α k | ≥ k∈N 0 α k φ (1) k ≥ (1 -δ 1 ) k∈N 0 |α k | ∀α k ∈ C.
By Lemma 3.7 and the choice of δ 1 , we find mutually orthogonal elements a

j 1 in E * * satisfying a (1) k ∈ E * * 2 (u (1)
k ), φ

k -φ

k P 2 (r(a

(1) k )) φ (1) k P 2 (r(a (1) k )) < ε 1 2
, and φ

k (a

k ) > 1 -ε 1 /2, (1)
for every k = 1, . . . , j 1 . Keeping in mind that u

1 , . . . , u

j 1 are mutually orthogonal we deduce that 0 ≤

j 1 k=1 u (1) k 2 ϕm = j 1 k=1 u (1) k 2 ϕm ≤ 1 ∀m ∈ N 0 .
It follows that there exist k 1 ≤ j 1 in N 0 and an infinite subset

N 1 ⊂ N 0 such that k 1 / ∈ N 1 , max G (1)
k < min N 1 , for every k = 1, . . . , j 1 , and

(1) k 1 2 ϕm ≤ 1 j 1 ∀m ∈ N 1 . Lemma 3.2 implies that (4.14) ϕ m -ϕ m P 0 (u (4.13) u
k 1 ) ≤ 21 √ j 1 ∀m ∈ N 1 . (1)
Define a 1 = a

(1)

k 1 and u 1 = u (1)
k 1 . By (4.13) and the choice of j 1 we obtain (4.8) for n = 1.

We claim that there exists m 1 ∈ G (1)

k 1 satisfying |ϕ m 1 (a 1 )| > r(1 -ε 1 ). Otherwise, |ϕ m (a 1 )| ≤ r(1 -ε 1 ), for every m ∈ G (1) k 1 . Recalling that φ (1) k 1 (a 1 ) > 1 -ε 1 2 , we have m∈N 1 |α m | ∀α m ∈ C.
This proves (4.10) for n = 1, which concludes the first induction step.

Suppose now, by the induction hypothesis, that m k , N k , a k , and u k have been defined for k ≤ n according to (4.6) -(4.10). By [START_REF] Francisco | Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals[END_REF]Theorem 3.9] the element

v n = n k=1 u k is a compact tripotent in E * * , therefore F n := E ∩ E * * 0 (v n ) is a weak * -dense subtriple of E * * 0 (v n ) whose second dual, F * * n , identifies with E * * 0 (v n ).
To simplify notation, we write ψ

(n) m = ϕ m P 0 (v n ) ϕ m P 0 (v n ) (m ∈ N n ),
m∈Nn |α m | ≥ m∈Nn α m ψ (n) m ≥ r 1 - n i=1 ε i m∈Nn |α m | ∀α m ∈ C that is, (ψ (n) m ) is a normalized basis spanning ℓ 1 (r (1 -n i=1 ε i ))-isomorphically.
Let us take a natural number j n+1 such that 3 21 √ j n+1 < rε n+1 . Let δ n+1 = δ(j n+1 , ε n+1 /2) > 0 given by Lemma 3.7. By James' distortion theorem there exist mutually disjoint finite subsets G

(n) k ⊂ N n (k ∈ N), finite sequences (λ (n) m ) m∈G (n) k ⊂ C such that (4.15) m∈G (n) k |λ (n) m | ≤ 1 r 1 - n i=1 ε i for all k ∈ N,
and the functionals φ

(n) k = m∈G (n) k λ (n) m ψ (n) m satisfy (4.16) k∈Nn |α k | ≥ k∈Nn α k φ (n) k ≥ (1 -δ n+1 ) k∈Nn |α k | ∀α m ∈ C.
By Lemma 3.7 and the choice of δ n+1 , we find mutually orthogonal elements a

(n) 1 , . . . , a (n) j n+1 of norm one in F n = E ∩ E * * 0 (v n ) and mutually orthogonal compact tripotents u (n) 1 , . . . , u (n) j n+1 in F * * n ∼ = E * * 0 (v n ) satisfying a (n) k ∈ E * * 2 (u (n) k ), and
φ (n) k - φ (n) k P 2 (r(a (n) k )) φ (n) k P 2 (r(a (n) k )) < ε n+1 2 , and
φ (n) k (a (n) k ) > 1 - ε n+1 2 ,
for every k = 1, . . . , j n+1 . We remark that v n ⊥ u

(n) k , for every k = 1, . . . , j n+1 because u (n) k ∈ E * * 0 (v n ). Keeping in mind that u (n) 1 , . . . , u (n) j n+1 are mutually orthogonal we deduce that 0 ≤ j n+1 k=1 u (n) k 2 ϕm = k n+1 k=1 u (n) k 2 ϕm ≤ 1 ∀m ∈ N n .
It follows that there exist k n+1 ≤ j n+1 in N n , and an infinite subset

N n+1 ⊂ N n such that k n+1 / ∈ N n+1 , max G (n)
k < min N n+1 , for k = 1, . . . , j n+1 , and (4.17)

u (n) k n+1 2 ϕm ≤ 1 j n+1
, for all m ∈ N n+1 .

Lemma 3.2 implies that (4.18) ϕ m -ϕ m P 0 (u

(n) k n+1 ) ≤ 21 √ j n+1 ∀m ∈ N n+1 .
Define a n+1 = a ϕ m (a n+1 ), which gives

1 - ε n+1 2 < φ (n) k n+1 (a n+1 ) = m∈G (n) k n+1 λ (n) m ψ (n) m (a n+1 ) ≤ r 1 - n+1 i=1 ε i m∈G (n) k n+1 |λ (n) m | ≤ r 1 - n+1 i=1 ε i 1 r 1 - n i=1 ε i = 1 - ε n+1 1 - n i=1 ε i < 1 - ε n+1 2
which is impossible. This proves the claim in (4.19).

By (4.8) for i ≤ n, we have By an extraction lemma of Simons [START_REF] Simons | On the Dunford-Pettis property and Banach spaces that contain c0[END_REF] we may (after passing to appropriate subsequences of (ϕ mn ) and (c n ) which we still denote by (ϕ mn ) and (c n )) suppose that k =n |ϕ mn (c k )| < ε ′ for all n where ε ′ > 0 is such that r(1 -ε) -ε ′ > r(1 -2ε). That the subtriple C generated by the c n is isometric to a commutative C * -algebra can be seen as in [START_REF] Francisco | Weak compactness in the dual space of a JB * -triple is commutatively determined[END_REF]Th. Up to an adjustment of ε this ends the proof.

  Orthogonality and geometric M -and L-orthogonality. We recall that elements a, b in a JB * -triple E are said to be algebraically orthogonal or simply orthogonal (written a ⊥ b) if L(a, b) = 0. If we consider a C * -algebra as a JB * -triple then two elements are orthogonal in the C * -sense if and only if they are orthogonal in the triple sense. It is known (compare [11, Lem. 1]) that a ⊥ b if and only if one of the following statements holds: {a, a, b} = 0; a ⊥ r(b); r(a) ⊥ r(b);

Lemma 2 . 2 .

 22 [START_REF] Friedman | Conditional expectation and bicontractive projections on Jordan C * -algebras and their generalizations[END_REF] Lem. 2.3] and [19, Theorem 5.4 and Lemma 5.5] Let ϕ and ψ be two functionals in the predual of a JBW * -triple W . Then ϕ ⊥ ψ if, and only if, ϕ ⊥ L ψ. Furthermore, given two tripotents e and u in W , then e ⊥ u if, and only if, e ⊥ M u.

ξP 2

 2 (r(b)) < ε In fact, define ε ′ ∈ (0, ε/2) by Lemma 2.1 such that (3.5) entails ξ -ψ < ε/2 where ψ = ξP 2 (r(b)). Hence, if (3.5) holds then |ψ(b)| = |ξ((b)| > 1 -ε/2 and

  (a) Let E be a JB * -triple, e ∈ E * * an open tripotent and let ϕ 1 , . . . , ϕ n be functionals in the closed unit ball of E * . If n j=1 |α j | ≥ n j=1 α j ϕ j P 2 (e) ≥ (1 -δ(n, ε)) n j=1 |α j |, ∀α j ∈ C (3.8) then there exist orthogonal norm-one elements a 1 , . . . , a n in E ∩ E * * 2 (e) and mutually orthogonal norm-one functionals ϕ 1 , . . . , ϕ n in E * satisfying (3.4). (b) Let W be a JBW * -triple, e ∈ E * * an arbitrary tripotent and let ϕ 1 , . . . , ϕ n be functionals in the closed unit ball of W * satisfying (3.8). Then there exist mutually orthogonal elements a 1 , . . . , a n of norm one in W 2 (e) and mutually orthogonal norm-one functionals ϕ 1 , . . . , ϕ n in W * satisfying (3.4). Proof. (a) Set F = E ∩ E * * 2 (e) and ψ j = ϕ j P 2 (e) |F . Then F w * = E * * 2 (e) because e is open. The ψ j satisfy (3.3) and it is enough to apply Theorem 3.6 to F . Similarly, for part (b) identify ϕ j P 2 (e) with ϕ |W 2 (e) ∈ (W 2 (e)) * and apply Corollary 3.8 to W 2 (e).

P 2

 2 mn } ⊂ W 0 (r(a m n+1 )) * , we can find mutually orthogonal elements b 1 , . . . , b n in the closed unit ball of W 0 (r(a m n+1 )) such that (4.4)φ (n+1) m k -φ (n+1) m k P 2 (r(b j )) φ (n+1) m k P 2 (r(b j )) < ν n+1 2 ∀k = 1, . . . , n. (r(b j ))for k = 1, . . . , n. The inequalities (4n (k = 1, . . . , n), which finishes the induction argument.

j 1

 1 of norm one in E and mutually orthogonal compact tripotents u

k

  n+1 and u n+1 = u (n) k n+1 . By (4.17) and the choice of j n+1 we obtain (4.8) for n + 1. Sincea n+1 ∈ F n = E ∩ E * * 0 (v n ), and a 1 , . . . , a n ∈ E * * 2 (v n ), it follows that a n+1 ⊥ a k , for k = 1, . . . , n.Therefore, a 1 , . . . , a n+1 are mutually orthogonal elements in the closed unit ball of E.We claim that there existsm n+1 ∈ G n+1 P 0 (v n ) |ϕ m n+1 (a n+1 )| > r 1m P 0 (v n ) |ϕ m (a n+1 )| ≤ r 1 -n+1 i=1 ε i , for all m ∈ G (n) k n+1 . Recalling that φ (n) k n+1 (a n+1 ) > 1 -ε n+12 , anda n+1 ∈ F n = E ∩ E * * 0 (v n ), we have ψ (n) m (a n+1 ) = ϕ m P 0 (v n ) ϕ m P 0 (v n ) (a n+1 ) = 1 ϕ m P 0 (v n )

  n+1 . Now, since 1 = ϕ m n+1 ≥ ϕ m n+1 P 0 (v n ) we deduce via Lemma 3.2 that0 ≤ 1 -ϕ m n+1 P 0 (v n ) ≤ ϕ m n+1 -ϕ m n+1 P 0 (v n ) < 21r n+1 P 0 (v n ) hence |ϕ m n+1 (a n+1 )| > 1 -r 19). We have thus defined m n+1 /∈ N n+1 such that m n < m n+1 and such that (4.9) holds for n + 1.Finally we show (4.10) for n + 1. Since (4.8) holds for i ≤ n + 1 we get(4.20) v n+1 ϕm = ∀m ∈ N n+1 .To simplify notation, let us denote ϕ (n+1) m := ϕ m P 0 (v n+1 ), m ∈ N n+1 . Lemma 3.2 and (4.20) imply thatϕ m -ϕ (n+1) ϕ m P 0 (v n+1 ) ϕ m P 0 (v n+1 ) = ϕ m -ϕfor all m ∈ N n+1 . By hypothesis, (ϕ m ) is a normalized sequence in E * spanning ℓ 1 r-isomorphically, hencem∈N n+1 α m ϕ m P 0 (v n+1 ) ϕ m P 0 (v n+1 ) ≥ m∈N n+1 α m ϕ m -m∈N n+1 α m ϕ m -ϕ m P 0 (v n+1 ) ϕ m P 0 (v n+1 )for all α m ∈ C. This proves (4.10) for n + 1 and shows (4.5).

  2.3b) ⇒ b ′ )].

Fix (α n ) ∈ ℓ 1 ,≥

 1 choose θ n ∈ C such that θ n α n = |α n | and set c = k≥1 θ k c k . Then c ≤ 1 and n≥1 α n ϕ mn |C ≥ n≥1 α n ϕ mn (c) = n |α n |ϕ mn (c n ) + (r(1 -ε) -ε ′ ) n |α n | ≥ r(1 -2ε) n |α n |.

  Lemma 3.2. [22, Lem. 2.1] Let E be a JB * -triple, v be a tripotent in E, and ϕ an element in the closed unit ball of E * . Then for each y ∈ E 2 (v) with y ≤ 1 we have

	(3.1) for every x ∈ E.	|ϕ(x -B(y, y)(x))| ≤ 21 x v ϕ ,

3.1. [22, Lem. 1.2]

Let E be a JB * -triple, e a tripotent in E, and x a norm-one element in E with e ≤ x. Then B(x, x) is a contractive operator and B(x, x)(y) belongs to E 0 (e), for every y in E.

  By induction on n we shall define a strictly monotone sequence (m n ) in N, a strictly decreasing sequence (N n ) of infinite subsets of N (i.e. N n N n+1 ), a sequence (a n ) of mutually orthogonal normone elements in E, and a sequence (u n ) of mutually orthogonal compact tripotents in E * * satisfying, for all n ∈ IN,

	(4.6)	m n < min N n ,
	(4.7)	a n ∈ E * * 2 (u n ),
	(4.8)	u n ϕm < r	ε n 63	∀m ∈ N n ,
	(4.9)	|ϕ mn

Scientific Research at King Saud University (Saudi Arabia) research group no. RGP-361.