We present the results of a combined study by band theory and angle resolved photoemission spectroscopy (ARPES) of the purple bronze, Li$_{1-x}$Mo$_{6}$O$_{17}$. Structural and electronic origins of its unusually robust quasi-one dimensional (quasi-1D) behavior are investigated in detail. The band structure, in a large energy window around the Fermi energy, is basically 2D and formed by three Mo $t_{2g}$-like extended Wannier orbitals, each one giving rise to a 1D band running at a 120$^circ$ angle to the two others. A structural dimerization from $mathbf{c}/2$ to $mathbf{c}$ gaps the $xz$ and $yz$ bands while leaving the $xy$ bands metallic in the gap, but resonantly coupled to the gap edges and, hence, to the other directions. The resulting complex shape of the quasi-1D Fermi surface (FS), verified by our ARPES, thus depends strongly on the Fermi energy position in the gap, implying a great sensitivity to Li stoichiometry of properties dependent on the FS, such as FS nesting or superconductivity. The strong resonances prevent either a two-band tight-binding model or a related real-space ladder picture from giving a valid description of the low-energy electronic structure. We use our extended knowledge of the electronic structure to newly advocate for framing LiMo$_{6}$O$_{17}$ as a weak-coupling material and in that framework can rationalize both the robustness of its quasi-1D behavior and the rather large value of its Luttinger liquid (LL) exponent $alpha$. Down to a temperature of 6$,$K we find no evidence for a theoretically expected downward renormalization of perpendicular single particle hopping due to LL fluctuations in the quasi-1D chains.
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