  Let $\Omega\subset\RR^n$ ($n\ge 1$) be a bounded open set with a Lipschitz
continuous boundary. In the first part of the paper, using the method of
bilinear forms, we give a rigorous characterization of the realization in
$L^2(\Omega)$ of the fractional Laplace operator $(-\Delta)^s$ ($0<s<1$) with
the nonlocal Neumann and Robin exterior conditions. Contrarily to the classical
local case $s=1$, it turns out that the nonlocal (Robin and Neumann) exterior
conditions are incorporated in the form domain. We show that each of the above
operators generates a strongly continuous submarkovian semigroup which is also
ultracontractive. In the second part, we show that the semigroup corresponding
to the nonlocal Robin exterior condition is always sandwiched between the
fractional Dirichlet semigroup and the fractional Neumann semigroup.
