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

This article is cited in 10 scientific papers (total in 10 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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    Citing articles on Google Scholar: Russian citations, English citations
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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, “Duality and free measures in vector spaces; spectral theory and the actions of non locally compact groups”, J. Math. Sci. (N. Y.), 238:4 (2019), 390–405  mathnet  crossref
    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, “A remark on the isomorphism between the Bernoulli scheme and the Plancherel measure”, J. Math. Sci. (N. Y.), 240:5 (2019), 567–571  mathnet  crossref
    7. A. M. Vershik, P. B. Zatitskii, “On a universal Borel adic space”, J. Math. Sci. (N. Y.), 240:5 (2019), 515–524  mathnet  crossref
    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
    9. A. M. Vershik, “The problem of combinatorial encoding of a continuous dynamics and the notion of transfer of paths in graphs”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye i algoritmicheskie metody. XXX, Zap. nauchn. sem. POMI, 481, POMI, SPb., 2019, 12–28  mathnet
    10. A. M. Vershik, “Kombinatornoe kodirovanie skhem Bernulli i asimptotika tablits Yunga”, Funkts. analiz i ego pril., 54:2 (2020), 3–24  mathnet  crossref
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