Have we see the event horizon of the Galactic black hole?
Recent observations with mm radio waves have determined Sgr A*'s mass , distance and diameter (Nature 455, p.78), which make it "very likely" to be a black hole, unless there is some sort of exotic physics leading to a compact, but still extended and visible, object at that location.
This new paper uses very general arguments from energy conservation and radiative efficiency to demonstrate that it is "impossible" for the object at Sgr A* to have a visible surface onto which accreting material collects and radiates. While the authors go through calculations in standard General Relativity for demonstration, their results hold much more broadly than that, covering any gravity theory which includes stationary solutions (i.e., horizons).
If there is an object with a real surface at Sgr A*, then all the material falling in must either give up it's full potential energy into radiation as it is falling, or we should see a separate, thermal spectrum from the surface of the object itself. The Sgr A* spectrum is extremely non-thermal; any thermal component is less than 0.4% of the total luminosity we see. That is equivalent to saying that the accretion efficiency (how much radiation you get from the falling material) must be at least 99.6%, compared to the actual efficiency estimated at 0.01 to 1% for Sgr A*.
On the other hand, a "horizon" by definition is a position in space "below" which photons just can't get out. Material can fall through a horizon with no difficulty, but we will never see it, or any radiation from it, again. Whether there is a black hole or some other exotic object inside the horizon, it is invisible. The accreting material's luminosity may well be "low" (as observed); the rest of its potential energy is "lost" to our observation when it falls through the horizon.