arXiv subjects

Gilles Pisier

Publications and source records attributed to Gilles Pisier.

At least 73 records · Page 4Linked to original sources

arXiv1997

The similarity degree of an operator algebra

Let $A$ be a unital operator algebra. Let us assume that every {\it bounded\/} unital homomorphism $u\colon \ A\to B(H)$ is similar to a {\it contractive\/} one. Let $\text{\rm Sim}(u) = \inf\{\|S\|\, \|S^{-1}\|\}$ where the infimum runs over all invertible operators $S\colon \ H\to H$ such that the ``conjugate'' homomorphism $a\to S^{-1}u(a)S$ is contractive. Now for all $c>1$, let $Φ(c) = \sup\text{\rm Sim}(u)$ where the supremum runs over all unital homomorphism $u\colon\ A\to B(H)$ with $\|u\|\le c$. Then, there is $α\ge 0$ such that for some constant $K$ we have: $$Φ(c) \le Kc^α.\leqno (*)\qquad \forall c>1$$ Moreover, the smallest $α$ for which this holds is an integer, denoted by $d(A)$ (called the similarity degree of $A$) and $(*)$ still holds for some $K$ when $α=d(A)$. Among the applications of these results, we give new characterizations of proper uniform algebras on one hand, and of nuclear $C^*$-algebras on the other. Moreover, we obtain a characterization of amenable groups which answers (at least partially) a question on group representations going back to a 1950 paper of Dixmier.

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arXiv1997

The ``maximal" tensor product of operator spaces

In analogy with the maximal tensor product of $C^*$-algebras, we define the ``maximal" tensor product $E_1\otimes_μE_2$ of two operator spaces $E_1$ and $E_2$ and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: \ $E_1\otimes_h E_2 + E_2\otimes_h E_1$. Let $E$ be an $n$-dimensional operator space. As an application, we show that the equality $E^* \otimes_μE=E^* \otimes_{\rm min} E$ holds isometrically iff $E = R_n$ or $E=C_n$ (the row or column $n$-dimensional Hilbert spaces). Moreover, we show that if an operator space $E$ is such that, for any operator space $F$, we have $F\otimes_{\min} E=F\otimes_μ E$ isomorphically, then $E$ is completely isomorphic to either a row or a column Hilbert space.

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arXiv1997

Non-commutative martingale inequalities

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 an $L^p$-martingale via its integrand, and then extend the Ito-Clifford integral theory in $L^2$, developed by Barnett, Streater and Wilde, to $L^p$ for all $1<p<\infty$. We include an appendix on the non-commutative analogue of the classical Fefferman duality between $H^1$ and $BMO$.

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arXiv1996

A polynomially bounded operator on Hilbert space which is not similar to a contraction

Let $\eps >0$. We prove that there exists an operator $T_\eps:\ell_2\to\ell_2$, such that for any polynomial $P$ we have $\|{P(T)}\| \leq(1+\eps)\|{P}\|_\infty$, but which is not similar to a contraction, {\it i.e.} there does not exist an invertible operator $S:\ \ell_2\to\ell_2$ such that $\|{S^{-1}T_\eps S}\|\leq 1$. This answers negatively a question attributed to Halmos after his well known 1970 paper (``Ten problems in Hilbert space").

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arXiv1995

Quadratic forms in unitary operators

Let $u_1,\ldots,u_n$ be unitary operators on a Hilbert space $H$. We study the norm $$\left\|\sum^{i=n}_{i=1} u_i \otimes \bar u_i\right\|\leqno (1)$$ of the operator $\sum u_i \otimes \bar u_i$ acting on the Hilbertian tensor product $H\otimes_2 \overline H$. The main result of this note is Theorem 1. For any $n$-tuple $u_1,\ldots, u_n$ of unitary operators in $B(H)$, we have $$2\sqrt{n-1} \le \left\|\sum^n_1 u_i \otimes \bar u_i\right\|.\leqno (6)$$ In other words, the right side of (6) is minimal exactly when $u_i = λ(g_i)$.

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arXiv1995

A simple proof of a theorem of Kirchberg and related results on $C^*$-norms

Recently, E.\ Kirchberg [K1--2] revived the study of pairs of $C^*$-algebras $A,B$ such that there is only one $C^*$-norm on the algebraic tensor product $A\otimes B$, or equivalently such that $A \otimes_{\rm min}B = A\otimes_{\rm max}B$. Recall that a $C^*$-algebra is called nuclear cf.\ [L, EL] if this happens for any $C^*$-algebra $B$. Kirchberg [K1] constructed the first example of a non-nuclear $C^*$-algebra such that $A\otimes_{\rm min} A^{op} = A \otimes_{\rm max} A^{op}$. He also proved the following striking result [K2] for which we give a very simple proof and which we extend. \proclaim Theorem 0.1. {\bf (Kirchberg [K2]).} Let $F$ be any free group and let $C^*(F)$ be the (full) $C^*$-algebra of $F$, then $$C^*(F) \otimes_{\rm min} B(H) = C^*(F) \otimes_{\rm max} B(H).$$

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arXiv1993

Regular operators between non-commutative $L_p$-spaces

We introduce the notion of a regular mapping on a non-commutative $L_p$-space associated to a hyperfinite von Neumann algebra for $1\le p\le \infty$. This is a non-commutative generalization of the notion of regular or order bounded map on a Banach lattice. This extension is based on our recent paper [P3], where we introduce and study a non-commutative version of vector valued $L_p$-spaces. In the extreme cases $p=1$ and $p=\infty$, our regular operators reduce to the completely bounded ones and the regular norm coincides with the $cb$-norm. We prove that a mapping is regular iff it is a linear combination of bounded, completely positive mappings. We prove an extension theorem for regular mappings defined on a subspace of a non-commutative $L_p$-space. Finally, let $R_p$ be the space of all regular mappings on a given non-commutative $L_p$-space equipped with the regular norm. We prove the isometric identity $R_p=(R_\infty,R_1)^θ$ where $θ=1/p$ and where $(\ .\ ,\ .\ )^θ$ is the dual variant of Calderón's complex interpolation method.

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arXiv1993

Bilinear forms on exact operator spaces and B(H)\otimes B(H)

Let $E,F$ be exact operators (For example subspaces of the $C^*$-algebra $K(H)$ of all the compact operators on an infinite dimensional Hilbert space $H$). We study a class of bounded linear maps $u\colon E\to F^*$ which we call tracially bounded. In particular, we prove that every completely bounded (in short $c.b.$) map $u\colon E\to F^*$ factors boundedly through a Hilbert space. This is used to show that the set $OS_n$ of all $n$-dimensional operator spaces equipped with the $c.b.$ version of the Banach Mazur distance is not separable if $n>2$. As an application we show that there is more than one $C^*$-norm on $B(H)\otimes B(H)$, or equivalently that $$B(H)\otimes_{\min}B(H)\not=B(H)\otimes_{\max}B(H),$$ which answers a long standing open question. Finally we show that every ``maximal" operator space (in the sense of Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the ``exactness constant".

math.FA↗

arXiv1993

Exact operator spaces

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 subspace $E\subset B(H)$, with $H$ Hilbert. We will be mainly concerned here with the ``geometry'' of {\it finite dimensional\/} operator spaces. In the Banach space category, it is well known that every separable space embeds isometrically into $\ell_\infty$. Moreover, if $E$ is a finite dimensional normed space then for each $\vp>0$, there is an integer $n$ and a subspace $F\subset \ell^n_\infty$ which is $(1+\vp)$-isomorphic to $E$, i.e. there is an isomorphism $u\colon \ E\to F$ such that $\|u\|\ \|u^{-1}\|\le 1+\vp$. Here of course, $n$ depends on $\vp$, say $n=n(\vp)$ and usually (for instance if $E = \ell^k_2$) we have $n(\vp)\to \infty$ when $\vp\to 0$. Quite interestingly, it turns out that this fact is not valid in the category of operator spaces:\ although every operator space embeds completely isometrically into $B(H)$ (the non-commutative analogue of $\ell_\infty$) it is not true that a finite dimensional operator space must be close to a subspace of $M_n$ (the non-commutative analogue of $\ell^n_\infty$) for some $n$. The main object of this paper is to study this phenomenon.

math.FA↗

arXiv1993

The k_t--functional for the interpolation couple L^\infty(dμ;L^1(dν)), L^\infty(dν;L^1(dμ))

Let $(M,μ)$ and $(N,ν)$ be measure spaces. In this paper, we study the $K_t$--\,functional for the couple $$A_0=L^\infty(dμ\,; L^1(dν))\,,~~A_1=L^\infty(dν\,; L^1(dμ))\,. $$ Here, and in what follows the vector valued $L^p$--\,spaces $L^p(dμ\,; L^q(dν))$ are meant in Bochner's sense. One of our main results is the following, which can be viewed as a refinement of a lemma due to Varopoulos [V]. \proclaim Theorem 0.1. Let $(A_0,A_1)$ be as above. Then for all $f$ in $A_0+A_1$ we have $${1\over 2}\,K_t(f;\,A_0\,,A_1)\leq \sup\,\bigg\{ \Big(μ(E)\vee t^{-1}ν(F)\Big)^{-1} \int_{E\times F} \vert f\vert\,dμ\,dν\,\bigg\} \leq K_t(f;\,A_0\,,A_1)\,,$$ where the supremum runs over all measurable subsets $E\subset M\,,~ F\subset N$ with positive and finite measure and $u\!\vee\!v$ denotes the maximum of the reals $u$ and $v$.

math.FA↗

arXiv1993

Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps

Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator space satisfies the natural properties to be expected with respect to e.g. duality and interpolation. This notion leads to the definition of a ``completely p-summing" map which is the operator space analogue of the $p$-absolutely summing maps in the sense of Pietsch-Kwapień. These notions extend the particular case $p=1$ which was previously studied by Effros-Ruan.

math.FA↗

arXiv1993

Complex Interpolation and Regular Operators Between Banach

We study certain interpolation and extension properties of the space of regular operators between two Banach lattices. Let $R_p$ be the space of all the regular (or equivalently order bounded) operators on $L_p$ equipped with the regular norm. We prove the isometric identity $R_p = (R_\infty,R_1)^θ$ if $θ= 1/p$, which shows that the spaces $(R_p)$ form an interpolation scale relative to Calderón's interpolation method. We also prove that if $S\subset L_p$ is a subspace, every regular operator $u : S \to L_p$ admits a regular extension $\tilde u : L_p \to L_p$ with the same regular norm. This extends a result due to Mireille Lévy in the case $p = 1$. Finally, we apply these ideas to the Hardy space $H^p$ viewed as a subspace of $L_p$ on the circle. We show that the space of regular operators from $H^p$ to $L_p$ possesses a similar interpolation property as the spaces $R_p$ defined above.

math.FA↗

arXiv1993

Bounded linear operators between C^*-algebras

Let $u:A\to B$ be a bounded linear operator between two $C^*$-algebras $A,B$. The following result was proved by the second author. Theorem 0.1. There is a numerical constant $K_1$ such that for all finite sequences $x_1,\ldots, x_n$ in $A$ we have $$\leqalignno{&\max\left\{\left\|\left(\sum u(x_i)^* u(x_i)\right)^{1/2}\right\|_B, \left\|\left(\sum u(x_i) u(x_i)^*\right)^{1/2}\right\|_B\right\}&(0.1)_1\cr \le &K_1\|u\| \max\left\{\left\|\left(\sum x^*_ix_i\right)^{1/2}\right\|_A, \left\|\left(\sum x_ix^*_i\right)^{1/2}\right\|_A\right\}.}$$ A simpler proof was given in [H1]. More recently an other alternate proof appeared in [LPP]. In this paper we give a sequence of generalizations of this inequality.

math.FA↗

arXiv1993

Projections from a von~Neumann algebra onto a subalgebra

This paper is mainly devoted to the following question:\ Let $M,N$ be von~Neumann algebras with $M\subset N$, if there is a completely bounded projection $P\colon \ N\to M$, is there automatically a contractive projection $\widetilde P\colon \ N\to M$? We give an affirmative answer with the only restriction that $M$ is assumed semi-finite.

math.FA↗

arXiv1992

Multipliers and lacunary sets in non-amenable groups

Let $G$ be a discrete group. Let $λ: G \to B(\ell_2(G),\ell_2(G))$ be the left regular representation. A function $\ph : G \to \comp$ is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined on the linear span $K(G)$ of $\{λ(x),x \in G\}$ by $$\sum_{x \in G} f(x) λ(x) \to \sum_{x \in G} f(x) \ph(x) λ(x)$$ is completely bounded (in short c.b.) on the $C^*$-algebra $C_λ^*(G)$ which is generated by $λ$ ($C_λ^*(G)$ is the closure of $K(G)$ in $B(\ell_2(G),\ell_2(G))$.) One of our main results gives a simple characterization of the functions $\ph$ such that $\eps \ph$ is a c.b. multiplier on $C_λ^*(G)$ for any bounded function $\eps$, or equivalently for any choice of signs $\eps(x) = \pm 1$. We also consider the case when this holds for ``almost all" choices of signs.

math.FA↗

arXiv1992

Espace de Hilbert d'opérateurs et Interpolation complexe

Let $H$ be an infinite dimensional Hilbert space. We show that there exists a subspace of $B(H)$ which is isometric to $\ell_2$ and completely isometric to its antidual in the sense of the theory of operator spaces recently developed by Blecher-Paulsen and Effros-Ruan. Moreover this space is unique up to a complete isometry. We denote it by $OH$. This space has several remarkable properties in particular with respect to the complex interpolation method.

math.FA↗

arXiv1992

Interpolation Between $H^p$ Spaces and Non-Commutative Generalizations II

We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the $\d$-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution of the $\d$-equation, which satisfies simultaneously a good $L_ı$ estimate and a good $L_1$ estimate. This appears as a special case of our main result which can be stated as follows: Let $(Ω,\cal{A},μ)$ be any measure space. Consider a bounded operator $u:H^1\ra L_1(μ)$. Assume that on one hand $u$ admits an extension $u_1:L^1\ra L_1(μ)$ bounded with norm $C_1$, and on the other hand that $u$ admits an extension $u_ı:L^ı\ra L_ı(μ)$ bounded with norm $C_ı$. Then $u$ admits an extension $\w{u}$ which is bounded simultaneously from $L^1$ into $ L_1(μ)$ and from $L^ı$ into $ L_ı(μ)$ and satisfies $$\eqalign{&\|\tilde u\colon \ L_\infty \to L_\infty(μ)\|\le CC_\infty\cr &\|\tilde u\colon \ L_1\to L_1(μ)\|\le CC_1}$$ where $C$ is a numerical constant.

math.FA↗

arXiv1992

On the ``local theory'' of operator spaces

In Banach space theory, the ``local theory'' refers to the collection of finite dimensional methods 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 of operator 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 future publication.

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