By Gilles Pisier

Influenced by way of a question of Vincent Lafforgue, the writer experiences the Banach areas X pleasant the next estate: there's a functionality \varepsilon\to \Delta_X(\varepsilon) tending to 0 with \varepsilon>0 such that each operator T\colon \ L_2\to L_2 with \|T\|\le \varepsilon that's concurrently contractive (i.e., of norm \le 1) on L_1 and on L_\infty has to be of norm \le \Delta_X(\varepsilon) on L_2(X). the writer indicates that \Delta_X(\varepsilon) \in O(\varepsilon^\alpha) for a few \alpha>0 if X is isomorphic to a quotient of a subspace of an ultraproduct of \theta-Hilbertian areas for a few \theta>0 (see Corollary 6.7), the place \theta-Hilbertian is intended in a touch extra common feel than within the author's previous paper (1979)

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By [15, Cor. 1) β(0) = (B0 , B1 )α(0) where α(0) = α(z) dm(z) = m(Γ1 ) ≥ θ. Consider now T in the open unit ball of B(θ, n). 1), we have T β(0) < 1. ∞ Therefore, there is an analytic function T (·) in the space H# relative to the family {β(z) | z ∈ ∂D} satisfying ess sup∂D T (z) β(z) < 1 and T (0) = T . Let Y (z) = n n 2 (X(z)). Clearly (see [14, 31]) we have Y (0) = 2 (X(0)) isometrically. Recall ∂D = Γ0 ∪ Γ1 . Note that for any z ∈ Γj (j = 0, 1) we have β(z) = Bj and hence T (z) Bj ≤ 1. Therefore ess sup T (z) ⊗ idX(z) : z∈∂D n 2 (X(z)) → n 2 (X(z)) ≤ 1.

It is easy to check that for any (non-void) B ⊂ Bn and any map u : E → F between n-dimensional Banach spaces we have γB (u) = inf γ{Ym } (u)d(Ym , B). m≥1 This shows that for any such u, the function z → γB(z) (u) is measurable on ∂D. 11, shows that {ΓB(z) (E, F ) | z ∈ ∂D} forms a compatible family if E, F are ﬁnite dimensional. Let us say that B ⊂ Bn is an SQ(p)-class if B contains all the n-dimensional spaces that are subquotients of ultraproducts of spaces of the form p ({Xi | i ∈ I}) with Xi ∈ B for all i in I, I being an arbitrary ﬁnite set.

Let (Xα ) be a directed family of ﬁnite dimensional subspaces of X with ∪Xα = X. Let εα > 0 be chosen so that εα → 0 when α → ∞ in the directed family. For each α, we can choose a ﬁnite εα -net (xi )i∈Iα in the unit ball of Xα . Let I be the disjoint union of the sets Iα and let Q : 1 (I) → X be the map deﬁned by Qei = xi . Clearly Q is a metric surjection onto X. Enlarging the sets Iα if necessary we may also ﬁnd a family (x∗i )i∈Iα in the unit ball of X ∗ such that x∗i|Xα is an εα -net in the unit ball of Xα∗ .