Molybdenum purple bronze Li0.9Mo6O17 is an exceptional material known to exhibit one-dimensional (1D) properties for energies down to a few meV. This fact seems to be well established both in experiments and in band structure theory. We use the unusual, very 1D band dispersion obtained in ab initio DFT-LMTO band calculations as our starting point to study the physics emerging below 300 meV. A dispersion perpendicular to the main dispersive direction is obtained and investigated in detail. Based on this, we derive an effective low-energy theory within the Tomonaga-Luttinger liquid (TLL) framework. We estimate the strength of the possible interactions and from this deduce the values of the TLL parameters for charge modes. Finally, we investigate possible instabilities of TLL by deriving renormalization group equations which allow us to predict the size of potential gaps in the spectrum. While 2kF instabilities strongly suppress each other, the 4kF instabilities cooperate, which paves the way for a possible charge-density wave at the lowest energies. The aim of this work is to understand the experimental findings, in particular the ones which are certainly lying within the 1D regime. We discuss the validity of our 1D approach and further perspectives for the lower-energy phases.