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Uspekhi Mat. Nauk, 2017, Volume 72, Issue 2(434), Pages 67–146 (Mi umn9763)  

This article is cited in 8 scientific papers (total in 8 papers)

The theory of filtrations of subalgebras, standardness, and independence

A. M. Vershikabc

a St. Petersburg Department of the Steklov Mathematical Institute
b St. Petersburg State University
c Institute for Information Transmission Problems

Abstract: This survey is devoted to the combinatorial and metric theory of filtrations: decreasing sequences of $\sigma$-algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of the key notions, that of standardness, plays the role of a generalization of the notion of the independence of a sequence of random variables. Questions are discussed on the possibility of classifying filtrations, on their invariants, and on various connections with problems in algebra, dynamics, and combinatorics.
Bibliography: 101 titles.

Keywords: filtrations, $\sigma$-algebras, independence, standardness, graded graphs, central measures.

Funding Agency Grant Number
Russian Science Foundation 14-11-00581
Partially supported by the Russian Science Foundation (grant no. 14-11-00581).


DOI: https://doi.org/10.4213/rm9763

Full text: PDF file (1196 kB)
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English version:
Russian Mathematical Surveys, 2017, 72:2, 257–333

Bibliographic databases:

UDC: 517.518
MSC: 05A05, 37A60, 37M99, 28A06, 60A10
Received: 24.01.2017
Revised: 15.02.2017

Citation: A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence”, Uspekhi Mat. Nauk, 72:2(434) (2017), 67–146; Russian Math. Surveys, 72:2 (2017), 257–333

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russian Math. Surveys, 72:2 (2017), 199–256  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. A. M. Vershik, “Dvoistvennost i svobodnye mery v vektornykh prostranstvakh, spektralnaya teoriya deistvii ne lokalno kompaktnykh grupp”, Veroyatnost i statistika. 25, Posvyaschaetsya pamyati Vladimira Nikolaevicha SUDAKOVA, Zap. nauchn. sem. POMI, 457, POMI, SPb., 2017, 74–100  mathnet
    3. A. M. Vershik, A. V. Malyutin, “The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts”, Funct. Anal. Appl., 52:3 (2018), 163–177  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. M. Vershik, P. B. Zatitskii, “Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph”, Funct. Anal. Appl., 52:4 (2018), 258–269  mathnet  crossref  crossref  isi  elib
    5. A. M. Vershik, A. V. Malyutin, “The absolute of finitely generated groups: I. Commutative (semi)groups”, Eur. J. Math., 4:4 (2018), 1476–1490  crossref  mathscinet  isi  scopus
    6. P. E. Naryshkin, “Zamechanie ob izomorfizme skhemy Bernulli i mery Plansherelya”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXIX, Zap. nauchn. sem. POMI, 468, POMI, SPb., 2018, 98–104  mathnet
    7. A. M. Vershik, P. B. Zatitskii, “Ob universalnom borelevskom adicheskom prostranstve”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXIX, Zap. nauchn. sem. POMI, 468, POMI, SPb., 2018, 24–38  mathnet
    8. A. M. Vershik, “Asimptotika razbieniya kuba na simpleksy Veilya i kodirovanie skhemy Bernulli”, Funkts. analiz i ego pril., 53:2 (2019), 11–31  mathnet  crossref  elib
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