  The hypothesized supermassive black hole in the nucleus of M31 (which we
shall hereafter call M31*) has many features in common with Sgr A* at the
Galactic Center, yet they differ in several significant and important ways.
Though M31* is probably ten times heavier, its radio luminosity at 3.6 cm is
only one third that of Sgr A*. At the same time, M31* is apparently thousands
of times more luminous in X-rays than its Galactic Center counterpart. Thus, a
comparative study of these objects can be valuable in helping us to understand
the underlying physical basis for their activity. We show here that the
accretion model being developed for Sgr A* comprises two branches of solutions,
distinguished by the relative importance of cooling compared to compressional
heating at the radius $r_C$ where the ambient gas is captured by the black
hole. For typical conditions in the ISM, the initial temperature ($T[r_C]\sim
10^6-10^7$ K) sits on the unstable branch of the cooling function.
  Depending on the actual value of $T(r_C)$ and the accretion rate, the plasma
settles either onto a hot branch (attaining a temperature as high as $10^{10}
K$ or so at small radii) or a cold branch, in which $T$ drops to $\sim 10^4$ K.
Sgr A* is presumably a `hot' black hole. We show here that the VLA, UV and {\it
Chandra} observations of M31* reveal it to be a member of the `cold' black hole
family. We discuss several predicted features in the spectrum of M31* that may
be testable by future multi-wavelength observations, including the presence of
a prominent UV spike (from hydrogen line emission) that would be absent on the
hot branch.
