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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)].

MSC:
 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
Citations:
Zbl 0532.43006; Zbl 0564.43004
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