It is possible to find numerous studies of space elevators, for Earth, Luna, Mars and even Phobos. But would a 'space elevator' work on an asteroid? I use scare quotes because typically the point of a space elevator is to escape from the gravity well of the planet, but in this case the second function of a space elevator is relevant - the ability to launch (or catch) objects at high velocities, acting as a momentum exchanger between the planet and spaceships.
The concept is simple enough, find a rapidly rotating asteroid, one which rotates in hours or minutes would be ideal, it doesn't matter too much how big it is as long as it's not so large that it has significant gravity. It also doesn't matter how small it is, as long as it's big enough to make the venture worthwhile.
A cable is anchored to the asteroid (ideally to a solid iron core) and a counterweight is flung or launched off into space, so that the rotation of the asteroid causes the cable to maintain it's extension.
The cable could then be extended further, until it reaches off as far into space as is desirable. Crawlers would bring objects up the cable and release them from the top. If the cable is long enough then the objects could be released with sufficient velocity to reach Earth.
If this scheme worked, then packets of material mined from the asteroid could be launched without the use of rocket fuel, requiring only the electricity for climbing the cable (which would not be great, as there is no real gravity to climb against - in fact mainly it would be braking which would be required as the cable would try to fling the crawlers off prematurely). Essentially the cable would transfer angular momentum from the asteroid to packet launches. The asteroid would slow down over time, but the cable could be extended to maintain launch velocities.
Furthermore, higher up the cable there would be increasing apparent gravity due to centripetal force, this would permit the construction of a station with comfortable gravity for human habitation. The cable could then also be used to catch incoming spaceships.
Are there problems with space elevator physics that would make the system hopelessly unstable in the absence of gravity? If it did work, how would the construction difficulty/cost compare with a space elevator on Luna or Mars?