286 citations · 878 across the 61 of their papers we have counts for
35 papers · 1 filter
Ideals in
William B. Johnson, Gilles Pisier, Gideon Schechtman
The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra of bounded linear operators on . This answers a q…
Impossibility of dimension reduction in the nuclear norm
Assaf Naor, Gilles Pisier, Gideon Schechtman
Let (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators whose nuclear norm $\|T\|_{\mathsf{S}_1}=\su…
A note on Sidon sets in bounded orthonormal systems
Gilles Pisier
We give a simple example of an -tuple of orthonormal elements in (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant…
Spectral gap properties of the unitary groups: around Rider's results on non-commutative Sidon sets
Gilles Pisier
We present a proof of Rider's unpublished result that the union of two Sidon sets in the dual of a non-commutative compact group is Sidon, and that randomly Sidon sets are Sidon. M…
On uniformly bounded orthonormal Sidon systems
Gilles Pisier
In answer to a question raised recently by Bourgain and Lewko, we show, with their paper's terminology, that any uniformly bounded -orthonormal system ( is a vari…
On the metric entropy of the Banach-Mazur compactum
Gilles Pisier
We study the metric entropy of the metric space of all n-dimensional Banach spaces (the so-called Banach-Mazur compactum) equipped with the Banach-Mazur (multiplicative) "dis…
Real interpolation and transposition of certain function spaces
Gilles Pisier
Our starting point is a lemma due to Varopoulos. We give a different proof of a generalized form this lemma, that yields an equivalent description of the -functional for the int…
Completely co-bounded Schur multipliers
Gilles Pisier
A linear map between operator spaces is called completely co-bounded if it is completely bounded as a map from to the opposite of . We give several simple…
Random Series of Trace Class Operators
Gilles Pisier
In this lecture, we present some results on Gaussian (or Rademacher) random series of trace class operators, mainly due jointly with F. Lust-Piquard. We will emphasize the probabil…
Grothendieck's Theorem, past and present
Gilles Pisier
Probably the most famous of Grothendieck's contributions to Banach space theory is the result that he himself described as "the fundamental theorem in the metric theory of tensor p…
Complex Interpolation between Hilbert, Banach and Operator spaces
Gilles Pisier
Motivated by a question of Vincent Lafforgue, we study the Banach spaces satisfying the following property: there is a function $\vp\to Δ_X(\vp)$ tending to zero with $\vp>0$ s…
A Remark on Hypercontractive Semigroups and Operator Ideals
Gilles Pisier
In this note, we answer a question raised by Johnson and Schechtman \cite{JS}, about the hypercontractive semigroup on $\{-1,1\}^{\NN}$. More generally, we prove the folllowing the…
Remarks on the similarity degree of an operator algebra
Gilles Pisier
The "similarity" degree of a unital operator algebra was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of…
Multipliers of the Hardy space H^1 and power bounded operators
Gilles Pisier
We study the space of functions $ϕ\colon \NN\to \CC$ such that there is a Hilbert space , a power bounded operator in and vectors in such that $$ϕ(n) = < T^…
On a question of Niels Grønbaek
Gilles Pisier
Let F(X) denote the norm closure of the space of all finite rank operators on a Banach space X. We show that there are Banach spaces X for which the product map …
The similarity degree of an operator algebra
Gilles Pisier
Let be a unital operator algebra. Let us assume that every {\it bounded\/} unital homomorphism is similar to a {\it contractive\/} one. Let $\text{\rm Sim…
Non-commutative martingale inequalities
Gilles Pisier, Quanhua Xu
We prove the analogue of the classical Burkholder-Gundy inequalites for non-commutative martingales. As applications we give a characterization for an Ito-Clifford integral to be a…
A polynomially bounded operator on Hilbert space which is not similar to a contraction
Gilles Pisier
Let $\eps >0$. We prove that there exists an operator $T_\eps:\ell_2\to\ell_2$, such that for any polynomial we have $\|{P(T)}\| \leq(1+\eps)\|{P}\|_\infty$, but which is not s…
Quadratic forms in unitary operators
Gilles Pisier
Let be unitary operators on a Hilbert space . We study the norm $$\left\|\sum^{i=n}_{i=1} u_i \otimes \bar u_i\right\|\leqno (1)$$ of the operator $\sum u_i \ot…
A simple proof of a theorem of Kirchberg and related results on -norms
Gilles Pisier
Recently, E.\ Kirchberg [K1--2] revived the study of pairs of -algebras such that there is only one -norm on the algebraic tensor product , or equivalen…
Bilinear forms on exact operator spaces and B(H)\otimes B(H)
Marius Junge, Gilles Pisier
Let be exact operators (For example subspaces of the -algebra of all the compact operators on an infinite dimensional Hilbert space ). We study a class of boun…
Regular operators between non-commutative -spaces
Gilles Pisier
We introduce the notion of a regular mapping on a non-commutative -space associated to a hyperfinite von Neumann algebra for . This is a non-commutative gene…
Exact operator spaces
Gilles Pisier
In this paper, we study {\it operator spaces\/} in the sense of the theory developed recently by Blecher-Paulsen [BP] and Effros-Ruan [ER1]. By an operator space, we mean a closed…
The k_t--functional for the interpolation couple L^\infty(dμ;L^1(dν)), L^\infty(dν;L^1(dμ))
Albrecht Hess, Gilles Pisier
Let and be measure spaces. In this paper, we study the --\,functional for the couple Here,…