  We will give an outline of the main results in our recent AMS Memoir, and
include some new results, exposition and open problems. In that memoir we
developed a general dilation theory for operator valued measures acting on
Banach spaces where operator-valued measures (or maps) are not necessarily
completely bounded. The main results state that any operator-valued measure,
not necessarily completely bounded, always has a dilation to a
projection-valued measure acting on a Banach space, and every bounded linear
map, again not necessarily completely bounded, on a Banach algebra has a
bounded homomorphism dilation acting on a Banach space. Here the dilation space
often needs to be a Banach space even if the underlying space is a Hilbert
space, and the projections are idempotents that are not necessarily
self-adjoint. These results lead to some new connections between frame theory
and operator algebras, and some of them can be considered as part of the
investigation about "noncommutative" frame theory.
