Stellar kinematic studies indicate the presence of a concentrated central mass at just under 2 × 106M⊙, in close agreement with the mass deduced from gas velocities measured with the [Ne II] line. Although this mass is most likely a black hole, it may be dominated by a tightly concentrated cluster of stellar remnants. If Sgr A*, a point radio source coincident with this central mass, is a massive black hole embedded in a region with strong gaseous outflows, as suggested by the observation of He I, Brα and Brγ line emission, it is accreting from its environment via the Bondi-Hoyle process. We discuss the consequences of this activity, including the expected mass and angular momentum accretion rate onto the black hole, and the resulting observable characteristics. The latest infrared images of this region appear to rule out the possibility that this large scale flow settles down into a standard α-disk at small radii. We discuss some possible scenarios that might account for this, including strong advection in the disk or the presence of a massive, fossilized disk. Not all of the gas affected in this way by Sgr A*’s strong gravitational field becomes bound. Some of it is redirected into a focused flow that in turn interacts with other coherent gas structures near the black hole. We suggest that the mini-cavity (to the south-west of Sgr A*) may be formed as a result of this activity, and argue that the characteristics of the mini-cavity lend some observational support for the presence of a concentrated mass near Sgr A*. We show, however, that as far as the mini-cavity is concerned, this concentrated mass need not be in the form of a point mass, but may instead be a highly concentrated cluster of stellar remnants.