The low-energy single-particle excitations of highly ordered C60 chains adsorbed on a vicinal copper substrate are investigated by angle-resolved photoemission spectroscopy. The interface state previously identified on C60/Cu(111) shows a one-dimensional dispersion on Cu(553). In contrast, no significant momentum dependence is detected for emission from the lowest unoccupied molecular orbital (LUMO). The LUMO displays similar phonon features as in C60/Cu(111) but it does not peak toward the Fermi level for all considered potassium dopings and its photoemission lineshape is broader than in any other monolayer system investigated so far. This behavior is not easily reconciled with existing theory and indicates that the one-dimensional character of the chains affects the electronic structure of the monolayer in an intricate way.