  The central parsec of the Galaxy contains dozens of massive stars with a
cumulative mass loss rate of ~ 10^{-3} solar masses per year. Shocks among
these stellar winds produce the hot plasma that pervades the central part of
the galaxy. We argue that these stellar wind shocks also efficiently accelerate
electrons and protons to relativistic energies. The relativistic electrons
inverse Compton scatter the ambient ultraviolet and far infrared radiation
field, producing high energy gamma-rays with a roughly constant luminosity from
\~ GeV to ~ 10 TeV. This can account for the TeV source seen by HESS in the
Galactic Center. Our model predicts a GLAST counterpart to the HESS source with
a luminosity of ~ 10^{35} ergs/s and cooling break at ~ 4 GeV. Synchrotron
radiation from the same relativistic electrons should produce detectable
emission at lower energies, with a surface brightness ~ 10^{32} B^2_{-3}
ergs/s/arcsec^2 from ~ THz to ~ keV, where B_{-3} is the magnetic field
strength in units of mG. The observed level of diffuse thermal X-ray emission
in the central parsec requires B < 300 micro-G in our models. Future detection
of the diffuse synchrotron background in the central parsec can directly
constrain the magnetic field strength, providing an important boundary
condition for models of accretion onto Sgr A*.
