## An analytic family of uniformly bounded representations of free groups.(English)Zbl 0681.43011

In 1979 U. Haagerup showed that on the free group $$F=F_ N$$, $$N=\infty,1,2$$ the function $$\phi_ r(x)=r^{| x|}$$ is positive definite, where $$| x|$$ is the natural length of the word $$x\in F$$ [see Invent. Math. 50, 279-293 (1979; Zbl 0408.46046)]. In the paper under review the authors extend that result in a very nice and constructive way. They explicitly present a construction of a uniformly bounded representation $$\pi_ z$$ on $$\ell_ 2(F)$$ such that $$<\pi_ z(x)\delta_ e,\delta_ e>=z^{| x|}$$ for $$| z| <1$$ and $(i)\quad \sup_{x}\| \pi_ z(x)\| \leq 2(| 1-z^ 2|)(1-| z|)^{-1}\quad (ii)\quad \pi_ z(x)^*=\pi_{\bar z}(x^{-1}).$ Moreover (iii) $$\pi_ z-\lambda (x)$$ is a finite dimensional operator for every $$x\in F$$, where $$\lambda$$ is the left regular representation of F. (iv) The map $$z\to \pi_ z(x)$$ is holomorphic for $$| z| <1$$ and $$x\in F$$. (v) If F has infinitely many generators, then the representations $$\pi_ z$$, $$z\neq 0$$ have no nontrivial invariant subspaces and any two different $$\pi_ z's$$ are topologically inequivalent.
In the proof the authors use the geometry of free groups i.e. in the language of J.-P. Serre this means trees. For similar ideas see also the paper of P. Julg and A. Valette [J. Funct. Anal. 58, 194-215 (1984; Zbl 0559.46030)].
In the second part of their paper the authors apply their results to the construction of Herz-Schur multipliers $$B_ 2(G)$$, G is a discrete group or the same to completely bounded multipliers of the Fourier-Eymard algebra $$A(G)=l_ 2(G)*l_ 2(G)$$. For a description and other facts about $$B_ 2(G)$$ see the paper of J. de Cannière and U. Haagerup [Am. J. Math. 107, 455-500 (1984; Zbl 0577.43002)], M. Bożejko and G. Fendler [Boll. Unione Mat. Ital., VI. Ser. A 3, 297-302 (1984; Zbl 0564.43004)].
Among other interesting results the authors prove that if $$\alpha_ n$$ is a decreasing sequence of positive numbers such that $$\sum_{n}\alpha_ n<\infty$$, then the function $$\sum^{\infty}_{m=0}\alpha_ m\chi_ m$$ belongs to $$B_ 2(F)$$, where $$\chi_ m$$ is the characteristic function of the set $$E_ m=\{x\in F:$$ $$| x| =m\}$$. Hence they get that the Fourier- Stieltjes algebra $$B(F)\varsubsetneq B_ 2(F)$$, which extends the result of J. de Canniere and U. Haagerup in the paper mentioned above and the result of G. Fendler of that same type which was obtained by G. Fendler in 1981 but will be published in Colloq. Math. That result was also extended by M. Bożejko who showed that $$B(G)=B_ 2(G)$$ if and only if the discrete group G is amenable [see Proc. Am. Math. Soc. 95, 357-360 (1985; Zbl 0593.43003)].
The paper under review has many other interesting ideas and I recomment it to everybody who wants to work in harmonic analysis on discrete groups. The paper of the authors has a natural continuation in the paper of R. Szwarc [Ann. Inst. Fourier 38, 87-110 (1988; Zbl 0634.22003)]. Another technique of the construction of a uniformly bounded representation was presented by M. Bożejko [J. Reine Angew. Math. 377, 170-186 (1987; Zbl 0604.43004)]. For other extensions of the paper under review see a paper of R. Szwarc about the isometry groups acting on trees [to appear in J. Funct. Anal.] and a paper of A. Valette which appears soon.
Reviewer: M.Bożejko

### MSC:

 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 43A35 Positive definite functions on groups, semigroups, etc. 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 22D20 Representations of group algebras 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations
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