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Uniformly bounded representations of free groups. (English) Zbl 0604.43004
In the paper the construction of a large class of uniformly bounded representations on free products of groups - regular free product representations - is presented. The main theorem 4.2 says that if \(\pi_ 1\) and \(\pi_ 2\) are uniformly bounded representations of the groups A and B, respectively, satisfying some natural assumptions, then the regular free product representation \(\pi =\pi_ 1*\pi_ 2\) is a uniformly bounded representation of the free product group A*B.
As a consequence we get some of the results of T. Pytlik and R. Szwarc [Acta Math. 157, 287-309 (1986)] and of A. M. Mantero and A. Zappa [J. Funct. Anal. 51, 372-399 (1983; Zbl 0532.43006)].
We also obtain an analytic family \(\pi_ z\) of representations of the free product group \(F_ 2\) satisfying the following properties \(<\pi_ z(x)\xi,\quad \xi >=z^{\| x\|}\) and \(\sup_{x\in F_ 2}\| \pi_ z(x)\| \leq 43(1-| z |)^{-1},\) where \(\| x\|\) is the block length function of the word \(x\in F_ 2.\)
There are some applications to the Herz-Schur multipliers of the free group \((= completely\) bounded multipliers of the Fourier algebra \(A(F_ 2))\) [see the author and G. Fendler, Boll. Unione Mat. Ital., VI. Ser., A 3, 297-302 (1984; Zbl 0564.43004)].

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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