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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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References: PDF file   HTML file

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, “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
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