Let G be a non-compact, locally compact group. The minimal ideals of the group algebra L1(G), the measure algebra M(G), and other Banach algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arens-type product, are studied in the paper. Using integrable representations, which exist on some semisimple Lie groups, it is seen that the minimal left ideals can be of infinite dimension, and that the compactness of G is not necessary for these ideals to exist in L1(G) and M(G). It is shown also that, although the coefficients of integrable representations are minimal idempotents in LUC(G)* and L1(G)** they do not generate minimal right ideals in these algebras, and that for a large class of groups, they do not generate minimal left ideals either.