  The Koslowski-Sahlmann (KS) representation is a generalization of the
representation underlying the discrete spatial geometry of Loop Quantum Gravity
(LQG), to accommodate states labelled by smooth spatial geometries. As shown
recently, the KS representation supports, in addition to the action of the
holonomy and flux operators, the action of operators which are the quantum
counterparts of certain connection dependent functions known as "background
exponentials".
  Here we show that the KS representation displays the following properties
which are the exact counterparts of LQG ones: (i) the abelian $*$ algebra of
$SU(2)$ holonomies and `$U(1)$' background exponentials can be completed to a
$C^*$ algebra (ii) the space of semianalytic $SU(2)$ connections is
topologically dense in the spectrum of this algebra (iii) there exists a
measure on this spectrum for which the KS Hilbert space is realised as the
space of square integrable functions on the spectrum (iv) the spectrum admits a
characterization as a projective limit of finite numbers of copies of $SU(2)$
and $U(1)$ (v) the algebra underlying the KS representation is constructed from
cylindrical functions and their derivations in exactly the same way as the LQG
(holonomy-flux) algebra except that the KS cylindrical functions depend on the
holonomies and the background exponentials, this extra dependence being
responsible for the differences between the KS and LQG algebras.
  While these results are obtained for compact spaces, they are expected to be
of use for the construction of the KS representation in the asymptotically flat
case.
