On almost representations of groups
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Abstract:
We say that a group $G$ belongs to the class $\mathcal {K}$ if every nonunit quotient group of $G$ has an element of order two. Let $H$ be a Hilbert space and let $U(H)$ be its group of unitary operators. Suppose that groups $A$ and $B$ belong to the class $\mathcal {K}$ and the order of $B$ is more than two. Then the free product $G=A\ast B$ has the following property. For any $\varepsilon >0$ there exists a mapping $T:G \to U(H)$ satisfying the following conditions : 1) $\Vert T(xy) - T(x)\cdot T(y) \Vert \le \varepsilon , \quad \forall x, \forall y \in G;$ 2) for any representation $\pi : G\to U(H)$ the relation \begin{equation*}\sup \{\Vert T(x) - \pi (x) \Vert ,x\in G\} =2\end{equation*} holds.References
- John Baker, J. Lawrence, and F. Zorzitto, The stability of the equation $f(x+y)=f(x)f(y)$, Proc. Amer. Math. Soc. 74 (1979), no. 2, 242–246. MR 524294, DOI 10.1090/S0002-9939-1979-0524294-6
- V. A. Faĭziev, Spaces of pseudocharacters of the free product of semigroups, Mat. Zametki 52 (1992), no. 6, 119–130, 160 (Russian, with Russian summary); English transl., Math. Notes 52 (1992), no. 5-6, 1255–1264 (1993). MR 1208010, DOI 10.1007/BF01209380
- V. A. Faĭziev, Pseudocharacters on semidirect products of semigroups, Mat. Zametki 53 (1993), no. 2, 132–139 (Russian); English transl., Math. Notes 53 (1993), no. 1-2, 208–213. MR 1220820, DOI 10.1007/BF01208329
- Karsten Grove, Hermann Karcher, and Ernst A. Ruh, Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems, Math. Ann. 211 (1974), 7–21. MR 355917, DOI 10.1007/BF01344138
- Pierre de la Harpe and Max Karoubi, Représentations approchées d’un groupe dans une algèbre de Banach, Manuscripta Math. 22 (1977), no. 3, 293–310 (French, with English summary). MR 498959, DOI 10.1007/BF01172669
- Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
- D. Kazhdan, On $\varepsilon$-representations, Israel J. Math. 43 (1982), no. 4, 315–323. MR 693352, DOI 10.1007/BF02761236
- A. I. Shtern, Stability of homomorphisms into the group $\textbf {R}^{\ast }$, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (1982), 29–32, 110 (Russian, with English summary). MR 671054
- S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. MR 0120127
Additional Information
- Valeriĭ Faĭziev
- Affiliation: Institute for Mathematics with Computational Center, Tadzhikistan Academy of Sciences, Dushanbe, Tadzhikistan
- Address at time of publication: Shirokaia St. 7-3-137, 129282 Moscow, Russia
- Received by editor(s): November 25, 1996
- Received by editor(s) in revised form: May 13, 1997
- Communicated by: Dale E. Alspach
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 57-61
- MSC (1991): Primary 20C99
- DOI: https://doi.org/10.1090/S0002-9939-99-04539-6
- MathSciNet review: 1468189