  The concept of boundary values of holomorphic semigroups in a general Banach
space is studied. As an application, we consider the Riemann-Liouville
semigroup of integration operator in the little H\"older spaces
$\rm{lip}_0^\alpha[0,\, 1] , \, 0<\alpha<1$ and prove that it admits a strongly
continuous boundary group, which is the group of fractional integration of
purely imaginary order. The corresponding result for the $L^p$-spaces
($1<p<\infty$) has been known for some time, the case $p=2$ dating back to the
monograph by Hille and Phillips. In the context of $L^p$ spaces, we establish
the existence of the boundary group of the Hadamard fractional integration
operators using semigroup methods. In the general framework, using a suitable
spectral decomposition,we give a partial treatment of the inverse problem,
namely: Which $C_0$-groups are boundary values of some holomorphic semigroup of
angle $\pi/2$?
