Positive definite functionals and \(L^ p\) convolution operators on amalgams (Q1074809)

This paper extends some known harmonic analytic results about free groups to amalgamated products of compact groups. More precisely, let \(K_ i\) \((i=1,2,3,...)\) be compact groups with a common open subgroup H of (necessarily finite) index \(k_ i\). The amalgamated product \(G=*_ H K_ i\) is the group whose elements are strings \(x_ 1...x_ n\) of elements of \(\cup_ i K_ i\), with the string identifications \(S_ 1S_ 2S_ 3=S_ 1S'\!_ 2S_ 3\) if \(S_ 2=S'\!_ 2\), and the identifications \(x_ 1x_ 2=x_ 3\) if \(x_ 1,x_ 2,x_ 3\in K_ i\) and \(x_ 1x_ 2=x_ 3\) in \(K_ i\), and \(x_ 1x_ 2=x'\!_ 1x'\!_ 2\) if \(x_ 1x'\!_ 1\in K_ j\), and \(x_ 1^{'-1}x_ 1=x'\!_ 1x_ 2^{-1}\in H.\) There is a well-defined notion of length of elements of G, with similar properties to the length in free groups. This permits the extension to amalgamated products of some estimates of \textit{U. Haagerup} [Invent. Math. 50, 279-293 (1979; Zbl 0408.46046)]. Various results which depend on these estimates - characterisation of matrix coefficients of representations weakly contained in the regular representation, and approximation properties of convolution operators - are then established.