Turbines can run on all different kinds of fuels, including gasoline/avgas. It doesn't mean it will produce its rated power, or will be as reliable as when running jet fuel, but it will run. Avgas/gasoline will definitely ignite from compression alone. Remember, Avgas is leaded so it won't ignite from the heat of compression in a piston engine at its designed compression ratios.

C-130s for example have a section in the flight manual about alternate fuels. They list things like diesel, kerosene, gasoline, avgas, etc, but they're all intended for extreme circumstances (stuck somewhere without jet fuel and need to get out now) and all have maintenance requirements after using them.

That being said, I highly doubt modern commercial turbofans are certified to run these alternate fuels. But from a technical standpoint, they would most likely run.