  Supermassive black holes in giant elliptical galaxies are remarkably faint
given their expected accretion rates. This motivates models of radiatively
inefficient accretion, due to either ion-electron thermal decoupling,
generation of outflows that inhibit accretion, or settling of gas to a
gravitationally unstable disk that forms stars in preference to feeding the
black hole. The latter model predicts the presence of cold molecular gas in a
thin disk around the black hole. Here we report Submillimeter Array
observations of the nucleus of the giant elliptical galaxy M87 that probe 230
GHz continuum and CO(J=2--1) line emission. Continuum emission is detected from
the nucleus and several knots in the jet, including one that has been
undergoing flaring behavior. We estimate a conservative upper limit on the mass
of molecular gas within ~100pc and +-400km/s line of sight velocity of the
central black hole of ~8x10^6Msun, which includes an allowance for possible
systematic errors associated with subtraction of the continuum. Ignoring such
errors, we have a 3 sigma sensitivity to about 3x10^6Msun. In fact, the
continuum-subtracted spectrum shows weak emission features extending up to 4
sigma above the RMS dispersion of the line-free channels. These may be
artifacts of the continuum subtraction process. Alternatively, if they are
interpreted as CO emission, then the implied molecular gas mass is ~5x10^6Msun
spread out over a velocity range of 700km/s. These constraints on molecular gas
mass are close to the predictions of the model of self-gravitating,
star-forming accretion disks fed by Bondi accretion (Tan & Blackman 2005).
