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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On almost representations of groups


Author: Valerii Faiziev
Journal: Proc. Amer. Math. Soc. 127 (1999), 57-61
MSC (1991): Primary 20C99
MathSciNet review: 1468189
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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.


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Additional Information

Valerii Faiziev
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

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04539-6
PII: S 0002-9939(99)04539-6
Keywords: Representation, $\varepsilon $-representation, pseudocharacter
Received by editor(s): November 25, 1996
Received by editor(s) in revised form: May 13, 1997
Communicated by: Dale E. Alspach
Article copyright: © Copyright 1999 American Mathematical Society