Are there female threads down inside the hole in the middle of the handle? Usually there's a screw holding the handle onto the stem. After a while, the screw isn't doing much of anything, 'cause the handle is firmly fastened to the splined stem by corrosion, etc. Looks like maybe somebody took out the screw, failed to get the handle off, and lost the screw.
If that's the case, I'd soak the thing in PB Blaster, then go back the next day and use a puller to try to get the handle -- or what's left of it -- off. If that fails, a Sawzall or nut splitter can be used to split the handle, but that's a last-ditch maneuver.