arXiv subjects
Publications and source records attributed to Gilles Pisier.
A linear map $u\colon \ E\to F$ between operator spaces is called completely co-bounded if it is completely bounded as a map from $E$ to the opposite of $F$. We give several simple results about completely co-bounded Schur multipliers on $B(\ell_2)$ and the Schatten class $S_p$. We also consider Herz-Schur multipliers on groups.
Motivated by a question of Vincent Lafforgue, we study the Banach spaces $X$ satisfying the following property: there is a function $\vp\to Δ_X(\vp)$ tending to zero with $\vp>0$ such that every operator $T\colon L_2\to L_2$ with $\|T\|\le \vp$ that is simultaneously contractive (i.e. of norm $\le 1$) on $L_1$ and on $L_\infty$ must be of norm $\le Δ_X(\vp)$ on $L_2(X)$. We show that $Δ_X(\vp)\in O(\vp^α)$ for some $α>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $θ$-Hilbertian spaces for some $ θ>0$ (see Corollary \ref{comcor4.3}), where $θ$-Hilbertian is meant in a slightly more general sense than in our previous paper \cite{P1}. Let $B_{r}(L_2(μ))$ be the space of all regular operators on $L_2(μ)$. We are able to describe the complex interpolation space \[ (B_{r}(L_2(μ), B(L_2(μ))^θ. \] We show that $T\colon L_2(μ)\to L_2(μ)$ belongs to this space iff $T\otimes id_X$ is bounded on $L_2(X)$ for any $θ$-Hilbertian space $X$. More generally, we are able to describe the spaces $$ (B(\ell_{p_0}), B(\ell_{p_1}))^θ{\rm or} (B(L_{p_0}), B(L_{p_1}))^θ$$ for any pair $1\le p_0,p_1\le \infty$ and $0<θ<1$. In the same vein, given a locally compact Abelian group $G$, let $M(G)$ (resp. $PM(G)$) be the space of complex measures (resp. pseudo-measures) on $G$ equipped with the usual norm $\|μ\|_{M(G)} = |μ|(G)$ (resp. \[ \|μ\|_{PM(G)} = \sup\{|\hatμ(γ)| \big| γ\in\hat G\}). \] We describe similarly the interpolation space $(M(G), PM(G))^θ$. Various extensions and variants of this result will be given, e.g. to Schur multipliers on $B(\ell_2)$ and to operator spaces.
We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general "lacunary" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the "Z(2)-sequences" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to $R_k+C_k$ with $k\sim n^{\frac12}$. This naturally raises several interesting questions concerning the best possible such $k$. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for $0<p<1$ but we can prove several would be corollaries. For instance, given an infinite scalar matrix $[x_{ij}]$, we give a necessary and sufficient condition for $[\pm x_{ij}]$ to be in the Schatten class $S_p$ for almost all (independent) choices of signs $\pm 1$. We also characterize the bounded Schur multipliers from $S_2$ to $S_p$. The latter two characterizations extend to $0<p<1$ results already known for $1\le p\le2$. In addition, we observe that the hypercontractive inequalities, proved by Carlen and Lieb for the Fermionic case, remain valid for operator space valued functions, and hence the Kahane inequalities are valid in this setting.
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 theorem. Let $1 0}$ be a holomorphic semigroup on $L_p$ (relative to a probability space). Assume the following mild form of hypercontractivity: for some large enough number $s>0$, $T(s)$ is bounded from $L_p$ to $L_2$. Then for any $t>0$, $T(t)$ is in the norm closure in $B(L_p)$ (denoted by $\bar{Γ_2}$) of the subset (denoted by ${Γ_2}$) formed by the operators mapping $L_p$ to $L_2$ (a fortiori these operators factor through a Hilbert space).
We prove that a discrete group $G$ is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation $π$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $π$. Analogously a $C^*$-algebra $A$ is nuclear iff any bounded homomorphism $u: A\to B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $ξ$ in the von Neumann algebra generated by the range of $u$ such that $a\to ξu(a) ξ^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations. We apply this to answer several questions left open in our previous work concerning the length $\ell(A,B)$ of the maximal tensor product $A\otimes_{\max} B$ of two unital $C^*$-algebras, when we consider its generation by the subalgebras $A\otimes 1$ and $1\otimes B$. We show that if $\ell(A,B)<\infty$ either for $B=B(\ell_2)$ or when $B$ is the $C^*$-algebra (either full or reduced) of a non Abelian free group, then $A$ must be nuclear. We also show that $\ell(A,B)\le d$ iff the canonical quotient map from the unital free product $A\ast B$ onto $A\otimes_{\max} B$ remains a complete quotient map when restricted to the closed span of the words of length $\le d$.
We review the existing proofs that the min and max norms are different on $B(H)\otimes B(H)$ and give a shortcut avoiding the consideration of non-separable families of operator spaces.
Let $\Free_n$ denote the free group with $n$ generators $g_1, g_2, ..., g_n$. Let $λ$ stand for the left regular representation of $\Free_n$ and let $τ$ be the standard trace associated to $λ$. Given any positive integer $d$, we study the operator space structure of the subspace $\Word_p(n,d)$ of $L_p(τ)$ generated by the family of operators $λ(g_{i_1}g_{i_2} ... g_{i_d})$ with $1 \le i_k \le n$. Moreover, our description of this operator space holds up to a constant which does not depend on $n$ or $p$, so that our result remains valid for infinitely many generators. We also consider the subspace of $L_p(τ)$ generated by the image under $λ$ of the set of reduced words of length $d$. Our result extends to any exponent $1 \le p \le \infty$ a previous result of Buchholz for the space $\Word_{\infty}(n,d)$. The main application is a certain interpolation theorem, valid for any degree $d$ (extending a result of the second author restricted to $d=1$). In the simplest case $d=2$, our theorem can be stated as follows: consider the space $\mathcal{K}_p$ formed of all block matrices $a=(a_{ij})$ with entries in the Schatten class $S_p$, such that $a$ is in $S_p$ relative to $\ell_2 \otimes \ell_2$ and moreover such that $(\sum_{ij} a_{ij}^* a_{ij} )^{1/2}$ and $(\sum_{ij} a_{ij} a_{ij}^*)^{1/2}$ both belong to $S_p$. We equip $\mathcal{K}_p$ with the maximum of the three corresponding norms. Then, for $2 \le p \le \infty$ we have $\mathcal{K}_p \simeq (\mathcal{K}_2, \mathcal{K}_\infty)_θ$ with $1/p = (1-θ)/2$.
We show that a $C^*$-algebra $A$ is nuclear iff there is a constant $K$ and $α<3$ such that, for any bounded homomorphism $u\colon A \to B(H)$, there is an isomorphism $ξ\colon H\to H$ satisfying $\|ξ^{-1}\|\|ξ\| \le K\|u\|^α$ and such that $ ξ^{-1} u(.) ξ$ is a $*$-homomorphism. In other words, an infinite dimensional $A$ is nuclear iff its length (in ths sense of our previous work on the Kadison similarity problem) is equal to 2.
We give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group $G$ are similar to unitary ones, is the group amenable? In §5, we give a new proof of Haagerup's theorem that, on non-commutative free groups, there are Herz-Schur multipliers that are not coefficients of uniformly bounded representations. We actually prove a refinement of this result involving a generalization of the class of Herz-Schur multipliers, namely the class $M_d(G)$ which is formed of all the functions $f\colon G\to {\bb C}$ such that there are bounded functions $ξ_i\colon G\to B(H_i, H_{i-1})$ ($H_i$ Hilbert) with $H_0 = {\bb C}$, $H_d ={\bb C}$ such that $$f(t_1t_2... t_d) = ξ_1(t_1) ξ_2(t_2)... ξ_d(t_d).\qquad \forall t_i\in G$$ We prove that if $G$ is a non-commutative free group, for any $d\ge 1$, we have $$M_d(G)\not= M_{d+1}(G),$$ and hence there are elements of $M_d(G)$ which are not coefficients of uniformly bounded representations. In the case $d=2$, Haagerup's theorem implies that $M_2(G)\not= M_{4}(G).$
We prove a factorization of completely bounded maps from a $C^*$-algebra $A$ (or an exact operator space $E\subset A$) to $\ell_2$ equipped with the operator space structure of $(C,R)_θ$ ($0<θ<1$) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if $F$ denotes $\ell_2$ equipped with the operator space structure of $(C,R)_θ$, then $u: A \to F$ is completely bounded iff there are states $f,g$ on $A$ and $C>0$ such that \[ \forall a\in A\quad \|ua\|^2\le C f(a^*a)^{1-θ}g(aa^*)^θ.\] This extends the case $θ=1/2$ treated in a recent paper with Shlyakhtenko. The constants we obtain tend to 1 when $θ\to 0$ or $θ\to 1$. We use analogues of "free Gaussian" families in non semifinite von Neumann algebras. As an application, we obtain that, if $0<θ<1$, $(C,R)_θ$ does not embed completely isomorphically into the predual of a semifinite von Neumann algebra. Moreover, we characterize the subspaces $S\subset R\oplus C$ such that the dual operator space $S^*$ embeds (completely isomorphically) into $M_*$ for some semifinite von neumann algebra $M$: the only possibilities are $S=R$, $S=C$, $S=R\cap C$ and direct sums built out of these three spaces. We also discuss when $S\subset R\oplus C$ is injective, and give a simpler proof of a result due to Oikhberg on this question. In the appendix, we present a proof of Junge's theorem that $OH$ embeds completely isomorphically into a non-commutative $L_1$-space. The main idea is similar to Junge's, but we base the argument on complex interpolation and Shlyakhtenko's generalized circular systems (or ``generalized free Gaussian"), that somewhat unifies Junge's ideas with those of our work with Shlyakhtenko.
We prove that the operator Hilbert space OH does not embed completely isomorphically into the predual of a semi-finite von Neumann algebra. This complements Junge's recent result that it admits such an embedding in the non semi-finite case.
We prove several versions of Grothendieck's Theorem for completely bounded linear maps $T\colon E \to F^*$, when E and F are operator spaces. We prove that if E,F are $C^*$-algebras, of which at least one is exact, then every completely bounded $T\colon E \to F^*$ can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as $T=T_r+T_c$ where $T_r$ (resp. $T_c$) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on $C^*$-algebras. Moreover, our result holds more generally for any pair E,F of "exact" operator spaces. This yields a characterization of the completely bounded maps from a $C^*$-algebra (or from an exact operator space) to the operator Hilbert space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert operator spaces and their direct sums are the only operator spaces E such that both E and its dual $E^*$ are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class.
We study the space of functions $ϕ\colon \NN\to \CC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $ξ,η$ in $H$ such that $$ϕ(n) = < T^nξ,η>.$$ This implies that the matrix $(ϕ(i+j))_{i,j\ge 0}$ is a Schur multiplier of $B(\ell_2)$ or equivalently is in the space $(\ell_1 \buildrel {\vee}\over {\otimes} \ell_1)^*$. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of $H^1$ which we call ``shift-bounded''. We show that there is a $ϕ$ which is a ``completely bounded'' multiplier of $H^1$, or equivalently for which $(ϕ(i+j))_{i,j\ge 0}$ is a bounded Schur multiplier of $B(\ell_2)$, but which is not ``shift-bounded'' on $H^1$. We also give a characterization of ``completely shift-bounded'' multipliers on $H^1$.
The "similarity" degree of a unital operator algebra $A$ was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type $II_1$ factor with property $Γ$ is at most 5, improving on a previous bound ($\le 44$) due to E. Christensen.
This is a survey of the author's recent results on the Kadison and Halmos similarity problems and the closely connected notion of ``length'' of an operator algebra.
We give an alternate proof of one of the inequalities proved recently for martingales (=sums of martingale differences) in a non-commutative $L_p$-space, with $1<p<\infty$, by Q. Xu and the author. This new approach is restricted to $p$ an even integer, but it yields a constant which is $O(p)$ when $p\to \infty$ and it applies to a much more general kind of sums which we call $p$-orthogonal.
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 $a\otimes b\to ab$ does not define a surjective map from the projective tensor product $F(X) \hat\otimes F(X)$ onto F(X).
For every integer $d\ge 1$, there is a unital closed subalgebra $A_d\subset B(H)$ with similarity degree equal precisely to $d$, in the sense of our previous paper. This means that for any unital homomorphism $u\colon A_d\to B(H)$ we have $\|u\|_{cb} \le K\|u\|^d$ with $K>0$ independent of $u$, and the exponent $d$ in this estimate cannot be improved. The proof that the degree is larger than $d-1$ crucially uses an upper bound for the norms of certain Gaussian random matrices due to Haagerup and Thorbjørnsen. We also include several complements to our previous publications on the same subject.