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 Tr. Mat. Inst. Steklova, 2000, Volume 231, Pages 249–283 (Mi tm518)

On Ulam's Problem of Stability of Non-exact Homomorphisms

V. G. Kanoveia, M. Reekenb

a Moscow Center for Continuous Mathematical Education
b University of Wuppertal

Abstract: The paper is mainly devoted to Ulam's problem of stability of approximate homomorphisms, i.e., the problem of approximation of approximate group homomorphisms by exact homomorphisms. We consider the case when one of the groups is equipped with an invariant probability measure while the other one is a countable product of groups equipped with a (pseudo)metric induced by a submeasure on the index set. We demonstrate that, for submeasures satisfying a certain form of the Fubini theorem for the product with probability measures, the stability holds for all measurable homomorphisms. Special attention is given to the case of dyadic submeasures, or, equivalently, approximations modulo an ideal on the index set. Those ideals which give rise to Ulam–stable approximate homomorphisms have recently been distinguished as Radon–Nikodým (or RN) ideals. We prove that this class of ideals contains all Fatou ideals; in particular, all indecomposable ideals and Weiss ideals. Some counterexamples are also considered. In the last part of the paper we study the structure of Borel cohomologies of some groups; in particular, we show that the group $\mathrm H_{\mathrm {Bor}}^2(\mathbb R,G)$ is trivial for any at most countable group $G$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2000, 231, 238–270

Bibliographic databases:
UDC: 510.225+517.518.2+517.987.1+512.662

Citation: V. G. Kanovei, M. Reeken, “On Ulam's Problem of Stability of Non-exact Homomorphisms”, Dynamical systems, automata, and infinite groups, Collected papers, Tr. Mat. Inst. Steklova, 231, Nauka, MAIK «Nauka/Inteperiodika», M., 2000, 249–283; Proc. Steklov Inst. Math., 231 (2000), 238–270

Citation in format AMSBIB
\Bibitem{KanRee00} \by V.~G.~Kanovei, M.~Reeken \paper On Ulam's Problem of Stability of Non-exact Homomorphisms \inbook Dynamical systems, automata, and infinite groups \bookinfo Collected papers \serial Tr. Mat. Inst. Steklova \yr 2000 \vol 231 \pages 249--283 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm518} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1841758} \zmath{https://zbmath.org/?q=an:0995.03036} \transl \jour Proc. Steklov Inst. Math. \yr 2000 \vol 231 \pages 238--270 

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This publication is cited in the following articles:
1. Kanovei V., Reeken M., “New Radon–Nikodym ideals”, Mathematika, 47:93–94, Part 1-2 (2000), 219–227
2. Farah I., “How many Boolean algebras –P(N)I are there?”, Illinois Journal of Mathematics, 46:4 (2002), 999–1033
3. Farah I., “Luzin gaps”, Transactions of the American Mathematical Society, 356:6 (2004), 2197–2239
4. Spakula J., Zlatos P., “Almost homomorphisms of compact groups”, Illinois Journal of Mathematics, 48:4 (2004), 1183–1189
5. Kanovei V., Lyubetsky V., “Reasonable non–Radon–Nikodym ideals”, Topology and Its Applications, 156:5 (2009), 911–914
6. Farah I., “All automorphisms of the Calkin algebra are inner”, Ann of Math (2), 173:2 (2011), 619–661
7. Todorcevic S., “Combinatorial Dichotomies in Set Theory”, Bull Symbolic Logic, 17:1 (2011), 1–72
8. Hrusak M., “Combinatorics of filters and ideals”, Set Theory and its Applications, Contemporary Mathematics, 533, 2011, 29–69
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