Let Ki be a countable
collection of compact groups, and assume that H =⋂iKi is an open subgroup of Ki
for every i. In this paper we consider positive definite functions and convolution
operators on the amalgamated product G = ∗HKi, and we study their properties in
relation with the notion of length of reduced words. In particular, if supiki< ∞, we
show that there exist unbounded approximate identities in A(G), that the space of
bounded convolution operators on Lp(G) is the dual space of the algebra Ap(G), and,
under the additional assumption that H be finite, that there exist unbounded
approximate identities in A(G).
