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We prove that all isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher rank simple groups, this confirms a long-standing conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce that sequences of Cayley graphs of finite quotients of a higher rank lattice are super-expanders., 30 pages ; minor changes in v2

We present a simple tool to perform analysis with groups such as SL(n,R) and SL(n,Z), that has been introduced by Vincent Lafforgue in his study of non-unitary representations, in connection with the Baum-Connes conjecture and strong property (T). It has been later applied in various contexts: operator algebras, Fourier analysis, geometry of Banach spaces or dynamics. The idea is to first restrict to compact subgroups and then exploit how they sit inside the whole group., Prepared for the proceedings of the ICM2022 (20 pages)

Given a set $B$ of operators between subspaces of $L_p$ spaces, we characterize the operators between subspaces of $L_p$ spaces that remain bounded on the $X$-valued $L_p$ space for every Banach space on which elements of the original class $B$ are bounded. This is a form of the bipolar theorem for a duality between the class of Banach spaces and the class of operators between subspaces of $L_p$ spaces, essentially introduced by Pisier. The methods we introduce allow us to recover also the other direction --characterizing the bipolar of a set of Banach spaces--, which had been obtained by Hernandez in 1983., Comment: 34 pages. Old project, already announced at several occasions in 2016, and that took a long time to be completed. Comments welcome v2: 37 pages. Section 5 added on the duality between Banach spaces and operators on full Lp spaces. A few references added

Answering a question by Chatterji--Dru\c{t}u--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($02$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher. We use this to show that for every connected semisimple Lie group $G$ and for every lattice $\Gamma < G$, we have $p_\Gamma=p_G$., Comment: v1: 10 pages v2: 20 pages. Added : study of critical parameters for semisimple Lie groups and their lattices, and of their behaviour under quantitative measure equivalence; integrability properties of lattices v3: 24 pages. Added : discussion of harmonic cocycles and and actions on non-commutative Lp spaces coming from state-preserving actions

We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of $SL_n(\mathbf{R})$. Our main result is inspired by H��rmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing weight with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Lafforgue/de la Salle's rigidity theorem fits here. The proof includes a new relation between Fourier and Schur $L_p$-multipliers for nonamenable groups. By transference, matters are reduced to a rather nontrivial $RC_p$-inequality for $SL_n(\mathbf{R})$-twisted forms of Riesz transforms associated to fractional laplacians. Our second result gives a new and much stronger rigidity theorem for radial multipliers in $SL_n(\mathbf{R})$. More precisely, additional regularity and Mikhlin type conditions are proved to be necessary up to an order $\sim |\frac12 - \frac1p| (n-1)$ for large enough $n$ in terms of $p$. Locally, necessary and sufficient growth rates match up to that order. Asymptotically, extra decay for the symbol and its derivatives imposes more accurate and additional rigidity in a wider range of $L_p$-spaces. This rigidity increases with the rank, so we can construct radial generating functions satisfying our H��rmander-Mikhlin sufficient conditions in rank $n$ and failing the rigidity conditions for ranks $m >> n$. We also prove automatic regularity and rigidity estimates for first and higher order derivatives of $\mathrm{K}$-biinvariant multipliers in the rank 1 groups $SO(n,1)$., To appear in Duke Math. J

We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces $\mathcal{E}_{10}$ containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for $\mathrm{Sp}(2,\mathbb{R})$ and its universal covering group. As consequences of our main result, it follows that for $X \in \mathcal{E}_{10}$, connected higher rank simple Lie groups and their lattices have property (F$_X$) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into $X$., Comment: 33 pages, 1 figure

In this short note we give a short proof of a recent result by Potapov and Sukochev (arXiv:0904.4095v1), stating that a Lipschitz function on the real line remains Lipschitz on the (self-adjoint part of) non-commutative $L_p$ spaces with $1, Comment: 3 pages

��uk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1-\varepsilon$. This criterion applies to random groups in the triangular density model for densities $> \frac{1}{3}$. In this way, we are able to generalize recent results of Dru��u and Mackay to affine isometric actions of random groups on uniformly curved Banach spaces. Also, in the setting of actions on $L^p$-spaces, our results are quantitatively stronger, even in the case $p=2$. This naturally leads to new estimates on the conformal dimension of the boundary of random groups in the triangular model. Additionally, we obtain results on the eigenvalues of the $p$-Laplacian on graphs, and on the spectrum and degree distribution of Erd��s-R��nyi graphs., v2: Major revision; substantial improvements of the main results; 28 pages v3: Minor revision; 29 pages

We prove some noncommutative analogues of a theorem by Plotkin and Rudin about isometries between subspaces of Lp-spaces. Let 0, Comment: 30 pages; revised version of the paper with previous title "Equimeasurabily and isometries in noncommutative Lp-spaces". Changes in the title, presentation and content. Added results on unbounded operators, and applications

If $E$ is an operator space, the non-commutative vector valued $L^p$ spaces $S^p[E]$ have been defined by Pisier for any $1 \leq p \leq \infty$. In this paper a necessary and sufficient condition for a Hankel matrix of the form $(a_{i+j})_{0 \le i,j}$ with $a_k \in E$ to be bounded in $S^p[E]$ is established. This extends previous results of Peller where $E=\C$ or $E=S^p$. The main theorem states that if $1 \leq p < \infty$, $(a_{i+j})_{0 \le i,j}$ is bounded in $S^p[E]$ if and only if there is an analytic function $\phi$ in the vector valued Besov Space $B_p^{1/p}(E)$ such that $a_n = \hat \phi(n)$ for all $n \in \N$. In particular this condition only depends on the Banach space structure of $E$. We also show that the norm of the isomorphism $\phi \mapsto (\hat \phi(i+j))_{i,j}$ grows as $\sqrt p$ as $p \to \infty$, and compute the norm of the natural projection onto the space of Hankel matrices., Comment: 17 pages