Abstract
In this chapter, we introduce the space B(G) of coefficients of unitary representations on a discrete group G and a related space \(T_p(G), (1 \leq p \leq \infty)\) of complex valued functions on G. We show that if \(G = \mathbb{F}_{N}\), the free group on \(N \geq 2\) generators, there are non-unitarizable uniformly bounded representations on G. More generally, this holds whenever G contains a subgroup isomorphic to \(\mathbb{F}_{2}\). We give several related characterizations of amenable groups. Then we extend the method to the case of semi-groups. This allows us to produce (this time for \(G = \mathbb{N}\)) examples of power bounded operators which are not polynomially bounded.
Mathematics Subject Classification (2000):
- primary 47AO5
- 46LO5
- 43A65 secondary 47A20
- 47B 10
- 42B30
- 46E40
- 46L57
- 47C 15
- 47L20
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© 2001 Springer-Verlag Berlin/Heidelberg
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Pisier, G. (2001). 2. Non-unitarizable uniformly bounded group representations. In: Similarity Problems and Completely Bounded Maps. Lecture Notes in Mathematics, vol 1618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44563-0_3
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DOI: https://doi.org/10.1007/978-3-540-44563-0_3
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41524-4
Online ISBN: 978-3-540-44563-0
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