{"title":"$W^*$-representations of subfactors and restrictions on the Jones index","authors":"S. Popa","doi":"10.4171/lem/1055","DOIUrl":"https://doi.org/10.4171/lem/1055","url":null,"abstract":"A {it W$^*$-representation} of a II$_1$ subfactor $Nsubset M$ with finite Jones index, $[M:N]<infty$, is a non-degenerate commuting square embedding of $Nsubset M$ into an inclusion of atomic von Neumann algebras $oplus_{iin I} Cal B(Cal K_i)=Cal N subset^{Cal E} Cal M=oplus_{jin J} Cal B(Cal H_j)$. We undertake here a systematic study of this notion, first introduced in [P92], giving examples and considering invariants such as the (bipartite) {it inclusion graph} $Lambda_{Cal N subset Cal M}$, the {it coupling vector} $(text{rm dim}(_MCal H_j))_j$ and the {it RC-algebra} (relative commutant) $M'cap Cal N$, for which we establish some basic properties. We then prove that if $Nsubset M$ admits a W$^*$-representation $Cal Nsubset^{Cal E}Cal M$, with the expectation $Cal E$ preserving a semifinite trace on $Cal M$, such that there exists a norm one projection of $Cal M$ onto $M$ commuting with $Cal E$, a property of $Nsubset M$ that we call {it weak injectivity/amenability}, then $[M:N]$ equals the square norm of the inclusion graph $Lambda_{Cal N subset Cal M}$.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130025545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Transformées de Fourier des fonctions d’invariance hyperbolique sur $mathbb{R}^2$","authors":"L. Clozel","doi":"10.4171/lem/1014","DOIUrl":"https://doi.org/10.4171/lem/1014","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114705543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Commission Internationale de l’Enseignement Mathématique. Once upon a time . . . Historical vignettes from the Archives of ICMI: The origins of the ICMEs","authors":"Bernard R. Hodgson","doi":"10.4171/lem/1016","DOIUrl":"https://doi.org/10.4171/lem/1016","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129139065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sur le developpement des courbes [Édité par Philippe Henry]","authors":"J. Lagrange","doi":"10.4171/lem/1005","DOIUrl":"https://doi.org/10.4171/lem/1005","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"222 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133754395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Un mémoire inédit de Lagrange sur le développement successif des courbes","authors":"P. Henry","doi":"10.4171/lem/1004","DOIUrl":"https://doi.org/10.4171/lem/1004","url":null,"abstract":"","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"181 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115566194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the Wasserstein (or earthmover) metric on the space $P(X)$ of probability measures on a metric space $X$. We show that, if a finite metric space $X$ embeds stochastically with distortion $D$ in a family of finite metric trees, then $P(X)$ embeds bi-Lipschitz into $ell^1$ with distortion $D$. Next, we re-visit the closed formula for the Wasserstein metric on finite metric trees due to Evans-Matsen cite{EvMat}. We advocate that the right framework for this formula is real trees, and we give two proofs of extensions of this formula: one making the link with Lipschitz-free spaces from Banach space theory, the other one algorithmic (after reduction to finite metric trees).
{"title":"Wasserstein distance and metric trees","authors":"Maxime Mathey-Prevot, A. Valette","doi":"10.4171/lem/1052","DOIUrl":"https://doi.org/10.4171/lem/1052","url":null,"abstract":"We study the Wasserstein (or earthmover) metric on the space $P(X)$ of probability measures on a metric space $X$. We show that, if a finite metric space $X$ embeds stochastically with distortion $D$ in a family of finite metric trees, then $P(X)$ embeds bi-Lipschitz into $ell^1$ with distortion $D$. Next, we re-visit the closed formula for the Wasserstein metric on finite metric trees due to Evans-Matsen cite{EvMat}. We advocate that the right framework for this formula is real trees, and we give two proofs of extensions of this formula: one making the link with Lipschitz-free spaces from Banach space theory, the other one algorithmic (after reduction to finite metric trees).","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"21 107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127138231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version of Helson's conjecture. We develop and simplify the ideas of Harper in this context, with the hope that the simplified proof would be of use to readers seeking a gentle entry-point to this fascinating area.
{"title":"A model problem for multiplicative chaos in number theory","authors":"K. Soundararajan, Asif Zaman","doi":"10.4171/LEM/1031","DOIUrl":"https://doi.org/10.4171/LEM/1031","url":null,"abstract":"Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version of Helson's conjecture. We develop and simplify the ideas of Harper in this context, with the hope that the simplified proof would be of use to readers seeking a gentle entry-point to this fascinating area.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129261336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wave functions associated to the scale S. Then we use these functions to condition the canonical spectral triple of the circle of length L=2 Log(S) in such a way that they belong to the kernel of the perturbed Dirac operator. We give numerical evidence that, when one varies L, the low lying spectrum of the perturbed spectral triple resembles the low lying zeros of the Riemann zeta function. We justify conceptually this result and show that, for each eigenvalue, the coincidence is perfect for the special values of the length L of the circle for which the two natural ways of realizing the perturbation give the same eigenvalue. This fact is tested numerically by reproducing the first thirty one zeros of the Riemann zeta function from our spectral side, and estimate the probability of having obtained this agreement at random, as a very small number whose first fifty decimal places are all zero. The theoretical concept which emerges is that of zeta cycle and our main result establishes its relation with the critical zeros of the Riemann zeta function and with the spectral realization of these zeros obtained by the first author.
{"title":"Spectral triples and $zeta$-cycles","authors":"A. Connes, C. Consani","doi":"10.4171/lem/1049","DOIUrl":"https://doi.org/10.4171/lem/1049","url":null,"abstract":"We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wave functions associated to the scale S. Then we use these functions to condition the canonical spectral triple of the circle of length L=2 Log(S) in such a way that they belong to the kernel of the perturbed Dirac operator. We give numerical evidence that, when one varies L, the low lying spectrum of the perturbed spectral triple resembles the low lying zeros of the Riemann zeta function. We justify conceptually this result and show that, for each eigenvalue, the coincidence is perfect for the special values of the length L of the circle for which the two natural ways of realizing the perturbation give the same eigenvalue. This fact is tested numerically by reproducing the first thirty one zeros of the Riemann zeta function from our spectral side, and estimate the probability of having obtained this agreement at random, as a very small number whose first fifty decimal places are all zero. The theoretical concept which emerges is that of zeta cycle and our main result establishes its relation with the critical zeros of the Riemann zeta function and with the spectral realization of these zeros obtained by the first author.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130360813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The literature contains interesting examples of inclusions of simple C$^*$-algebras with the property that all intermediate C$^*$-algebras likewise are simple. In this article we take up a systematic study of such inclusions, which we refer to as being C$^*$-irreducible by the analogy that all intermediate von Neumann algebras of an inclusion of factors are again factors precisely when the given inclusion is irreducible. We give an intrinsic characterization of when an inclusion of C$^*$-algebras is C$^*$-irreducible, and use this to revisit known and exhibit new C$^*$-irreducible inclusions arising from groups and dynamical systems. Using a theorem of Popa one can show that an inclusion of II$_1$-factors is C$^*$-irreducible if and only if it is irreducible with finite Jones index. We further show how one can construct C$^*$-irreducible inclusions from inductive limits, and we discuss how the notion of C$^*$-irreducibility behaves under tensor products.
{"title":"Irreducible inclusions of simple $C^*$-algebras","authors":"M. Rørdam","doi":"10.4171/lem/1051","DOIUrl":"https://doi.org/10.4171/lem/1051","url":null,"abstract":"The literature contains interesting examples of inclusions of simple C$^*$-algebras with the property that all intermediate C$^*$-algebras likewise are simple. In this article we take up a systematic study of such inclusions, which we refer to as being C$^*$-irreducible by the analogy that all intermediate von Neumann algebras of an inclusion of factors are again factors precisely when the given inclusion is irreducible. We give an intrinsic characterization of when an inclusion of C$^*$-algebras is C$^*$-irreducible, and use this to revisit known and exhibit new C$^*$-irreducible inclusions arising from groups and dynamical systems. Using a theorem of Popa one can show that an inclusion of II$_1$-factors is C$^*$-irreducible if and only if it is irreducible with finite Jones index. We further show how one can construct C$^*$-irreducible inclusions from inductive limits, and we discuss how the notion of C$^*$-irreducibility behaves under tensor products.","PeriodicalId":344085,"journal":{"name":"L’Enseignement Mathématique","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125697122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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