We examine the classification problem of the fixed point algebra O(lam
bda)(G) of the Cuntz algebra O(L2(G)) under the regular action a of co
mpact and metrizable group G. We determine the K-theory of O(lambda(G)
) and the order structure of the K0-group of a canonical AF C-subalge
bra. (C) 1994 Academic Press, Inc.