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Graphs and convolution operators. (English) Zbl 0537.43006
Topics in modern harmonic analysis, Proc. Semin., Torino and Milano 1982, Vol. I, 187-208 (1983).
[For the entire collection see Zbl 0527.00011.]
This paper is concerned with obtaining norm estimates for convolution operators on groups G acting simply transitively on polygonal graphs $$\Gamma$$. There are two natural equivalent metrics on $$\Gamma$$ termed the step length, $$\| v\|$$, and the block length, $$| v|$$, for each vertex v of $$\Gamma$$, and defined with reference to a fixed origin e in $$\Gamma$$. Then if G acts isometrically on $$\Gamma$$ with respect to one of these metrics, it follows that G is the free product of finitely many cyclic groups, and in fact acts isometrically with respect to the other metric. By identifying the elements of G with the vertices of $$\Gamma$$, both lengths can be formulated in terms of the reduced word expressions of the elements of G. Denote by $$W_ n$$ (resp. $$E_ n)$$ the set of words of block length (resp. step length) n.
The main result of this paper is that there exist constants $$K_ n$$ such that for every f with $$\sup p(f)\subseteq W_ n$$ or $$E_ n \| \lambda(f)\| \leq K_ n\| f\|_ 2$$, where $$\lambda$$ (f) denotes the convolution operator $$g\to f*g$$ on $$\ell^ 2(G)$$. This result is closely related to Haagerup’s inequality $$\| \lambda(f)\| \leq(n+1)\| f\|_ 2$$ for free groups G (here f is supported on words of length n), see [U. Haagerup, Invent. Math. 50, 279-293 (1979; Zbl 0408.46046)].
The authors briefly apply their results to the harmonic analysis of G. For example, it is shown that if $$\sup_{x\in G}| x|^ 2| h(x)|<\infty,$$ then h is a multiplier of the $$C^*$$-algebra of left convolution operators $$\ell^ 2(G)$$.
Reviewer: A.K.Seda

##### MSC:
 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations