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 \(\ell^ 2(G)\).

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 |