Motivated by a general dilation theory for operator-valued measures, framings
and bounded linear maps on operator algebras, we consider the dilation theory
of the above objects with special structures. We show that every
operator-valued system of imprimitivity has a dilation to a probability
spectral system of imprimitivity acting on a Banach space. This completely
generalizes a well-kown result which states that every frame representation of
a countable group on a Hilbert space is unitarily equivalent to a
subrepresentation of the left regular representation of the group. The dilated
space in general can not be taken as a Hilbert space. However, it can be taken
as a Hilbert space for positive operator valued systems of imprimitivity. We
also prove that isometric group representation induced framings on a Banach
space can be dilated to unconditional bases with the same structure for a
larger Banach space This extends several known results on the dilations of
frames induced by unitary group representations on Hilbert spaces.Comment: 21 page
