  We construct a Chern character map from the K-theory of the reduced C^*
algebra of the p-adic GL(n) with values in the periodic cyclic homology of the
Schwartz algebra of this group. We prove that this map is an isomorphism after
tensoring with C by comparing an explicit formula, stated in the algebraic case
by Cuntz and Quillen, with the classical Chern character. This Chern character
is a crucial ingredient in the proof of the Baum-Connes conjecture for the
p-adic GL(n) due to Baum, Higson and Plymen.
