Diffuse $\gamma$-ray emission from unresolved BL Lac objects
  Blazars represent the most abundant class of high-energy extragalactic
$\gamma$-ray sources. The subset of blazars known as BL Lac objects is on
average closer to Earth and characterized by harder spectra at high energy than
the whole sample. The fraction of BL Lacs that is too dim to be detected and
resolved by current $\gamma$-ray telescopes is therefore expected to contribute
to the high-energy isotropic diffuse $\gamma$-ray background (IGRB). The IGRB
has been recently measured over a wide energy range by the Large Area Telescope
(LAT) on board the Gamma-ray Space Telescope ({\it Fermi}). We present a new
prediction of the diffuse $\gamma$-ray flux due to the unresolved BL Lac blazar
population. The model is built upon the spectral energy distribution and the
luminosity function derived from the fraction of BL Lacs detected (and
spectrally characterized) in the $\gamma$-ray energy range. We focus our
attention on the ${\cal O}(100)$ GeV energy range, predicting the emission up
to the TeV scale and taking into account the absorption on the extragalactic
background light. In order to better shape the BL Lac spectral energy
distribution, we combine the {\it Fermi}-LAT data with Imaging Atmospheric
Cerenkov Telescopes measurements of the most energetic sources. Our analysis is
carried on separately for low- and intermediate-synchrotron-peaked BL Lacs on
one hand, and high-synchrotron-peaked BL Lacs on the other one: we find in fact
statistically different features for the two. The diffuse emission from the sum
of both BL Lac classes increases from about 10$\%$ of the measured IGRB at 100
MeV to $\sim$100$\%$ of the data level at 100 GeV. At energies greater than 100
GeV, our predictions naturally explain the IGRB data, accommodating their
softening with increasing energy. Uncertainties are estimated to be within of a
factor of two of the best-fit flux up to 500 GeV.
