  The bulk of the diffuse galactic gamma-ray emission above a few tens of GeV
has been conventionally ascribed to the decay of neutral pions produced in
cosmic-ray interactions with interstellar matter. Cosmic-ray electrons may,
however, make a significant contribution to the gamma-ray spectrum at high
energies, and even dominate at TeV-PeV energies depending on their injection
spectral index and acceleration cut-off energy. If the injection spectrum is
flat, the highest energy electrons will also contribute a diffuse hard
X-ray/soft gamma-ray flux via synchrotron emission, and this may offer an
explanation for the OSSE observation of a steep spectrum below a few MeV from
the inner Galaxy. We perform a propagation calculation for cosmic-ray
electrons, and use the resulting interstellar electron spectrum to obtain the
gamma-ray spectrum due to inverse Compton, synchrotron and bremsstrahlung
interactions consistently from MeV to PeV energies. We compare our results with
available observations from satellite-borne telescopes, optical Cherenkov
telescopes and air shower arrays and place constraints on the injection
spectrum of cosmic-ray electrons. With future observations at TeV-PeV energies
it should be possible to determine the average interstellar spectrum of
cosmic-ray electrons, and hence estimate their spectrum on acceleration.
