Abstract
We present a brief review of the theory of quasi-characters and quasi-representations and prove a necessary and sufficient condition that the second real continuous bounded cohomology of a locally compact group to be finite-dimensional. This criterion is established by using the properties of continuous pseudocharacters on a locally compact group.
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Baker, J., Lawrence, J. and Zorzitto, F.: The stability of equation f (xy) = f (x)f (y), Proc. Amer. Math. Soc. 74(2) (1979), 242-246.
Bargmann, V.: Irreducible unitary representations of the Lorentz group, Ann. Math. 48 (1947), 568-640.
Bavard, C.: Longueur stable des commutateurs, Enseign. Math. 37 (1991), 109-150.
Besson, G.: Séminaire cohomologie bornée, École Norm. Sup. Lyon Report (Février 1988).
Blanc, P.: Sur la cohomologie continue des groupes localement compacts, Ann. Sci. École Norm. Sup. 12 (1979), 137-168.
Bouarich, A.: Suites exactes en cohomologie bornée réelle des groups discrets, C. R. Acad. Sci. Paris 320 (1995), 1355-1359.
Censer, D.: The stability problem for transformations of the circle, Proc. Roy. Soc. Edinburgh. Sect. A 84 (1979), 279-281.
Faiziev, V. A.: Pseudocharacters on free products of semigroups, Funktsional. Anal. i Prilozhen. 21(1) (1987), 86-87.
Faiziev, V. A.: Pseudocharacters on free groups, free semigroups, and some group constructions, Uspekhi Mat. Nauk 43(5) (1988), 225-226.
Forti, G. L.: The stability of homomorphisms and amenability, with applications to functional equations, Abh. Math. Sem. Univ. Hamburg 57 (1987), 215-226.
Fuks, D. B.: Continuous cohomology of topological groups and characteristic classes, addendum to the book K. S. Brown, Cohomology of Groups (Russian translation), 1987, pp. 342-368.
Gelfand, I.: Zur Theorie der Charactere der Abelschen topologischen Gruppen, Mat. Sb. 9(1) (51) (1941), 49-50.
Grosser, M.: 95g#22008, Review of [28], Math. Rev. (1995).
Grove, K., Karcher, H. and Roh, E. A.: Jacobi fields and Finsler metrics on a compact Lie groups with an application to differential pinching problems, Math. Ann. 211(1) (1974), 7-21.
Guichardet, A.: Cohomologie des groupes topologiques et des algèbres de Lie, Cedic/Fernand Nathan, Paris, 1980.
Guichardet, A. and Wigner, D.: Sur la cohomologie réele des groupes de Lie simples réels, Ann. Sci. École Norm. Sup. 11 (1978), 277-292.
de la Harpe, P. and Karoubi, M.: Represéntations approchées d'un groupe dans une algébre de Banach, Manuscripta Math. 22(3) (1977), 297-310.
Hyers, D. H.: On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27(2) (1941), 222-224.
Hyers, D. H. and Ulam, S. M.: On approximate isometry, Bull. Amer. Math. Soc. 51 (1945), 288-292.
Johnson, B. E.: Cohomology in Banach Algebras, Mem. Amer. Math. Soc. 127, Amer. Math. Soc., Providence, 1972.
Johnson, B. E.: Approximately multiplicative functionals, J. London Math. Soc. 34(3) (1986), 489-510.
Johnson, B. E.: Continuity of generalized homomorphisms, Bull. London Math. Soc. 19(1) (1987), 67-71.
Johnson, B. E.: Approximately multiplicative maps between Banach algebras, J. London Math. Soc. 37(2) (1988), 294-316.
Kazhdan, D.: On ∈-representations, Israel J. Math. 43(4) (1982), 315-323.
Lawrence, J. W.: The stability of multiplicative semigroup homomorphisms to real normed algebras, Aequationes Math. 28(1-2) (1985), 94-101.
Paterson, A. L. T.: Amenability, Amer. Math. Soc., Providence, RI, 1988.
Shtern, A. I.: Stability of representations and pseudocharacters, Report on Lomonosov Readings 1983, Moscow State University, Moscow, 1983.
Shtern, A. I.: Continuous pseudocharacters on connected locally compact groups are characters, Funktsional. Anal. i Prilozhen. 27(4) (1993), 94-96.
Shtern, A. I.: Quasi-symmetry. I, Russian J. Math. Phys. 2(3) (1994), 353-382.
Shtern, A. I.: Triviality and continuity of pseudocharacters and pseudorepresentations, Russian J. Math. Phys. 5(1) (1997), 135-138.
Shtern, A. I.: Remarks on pseudocharacters and the real continuous bounded cohomology of connected locally compact groups, Sfb 288, Preprint No. 289, Berlin, 1997.
Shtern, A. I.: Roughness and approximation of quasi-representations of amenable groups, Mat. Zam. 65(6) (1999), 908-920.
Shtern, A. I.: Remarks on pseudocharacters, and the real continuous bounded cohomology of connected locally compact groups (to appear).
Sund, T.: Remarks on locally compact group extensions, Math. Scand. 69 (1991), 199-210.
Ulam, C.: A Collection of Mathematical Problems, Tracts on Pure Appl. Math., Interscience, New York, 1960.
Warner, G.: Harmonic Analysis on Semi-Simple Lie Groups, Marcel Dekker, New York, 1972.
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Shtern, A.I. A Criterion for the Second Real Continuous Bounded Cohomology of a Locally Compact Group to be Finite-Dimensional. Acta Applicandae Mathematicae 68, 105–121 (2001). https://doi.org/10.1023/A:1012295625631
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DOI: https://doi.org/10.1023/A:1012295625631