Molybdenum purple bronze Li$_{0.9}$Mo$_{6}$O$_{17}$ 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 1-dimensional band dispersion obtained in emph{ab-initio} DFT-LMTO band calculations as our starting point to study the physics emerging below 300meV. 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 (RG) equations which allow us to predict the size of potential gaps in the spectrum. While $2k_F$ instabilities strongly suppress each other, the $4k_F$ instabilities cooperate, which paves the way for a possible CDW 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.
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