Plutonium fuel on Galileo reacts to the high pressure, and like a small match starting a big forest fire, ignites Jupiter!
Possible if the ignition explosion is deep enough.
It would not be a steady state sun, becuse Jupiter is not dense enough. It would heat up and expand as it did. Eventually, the density would become too light to sustain fusion and would "go out". The Jupiter Sun would then begin to collapse, releasing gravitational energy as heat and light. It is possible that it would re-ignite spontaneously when it had fully collapsed.