  Let $\Gamma$ be a discrete group. A $C^*$-algebra $A$ is an exotic
$C^*$-algebra (associated to $\Gamma$) if there exist proper surjective
$C^*$-quotients $C^*(\Gamma)\to A\to C^*_r(\Gamma)$. In this paper, we show
that a large class of exotic $C^*$-algebras have poor local properties. More
precisely, we demonstrate the failure of local reflexitity, exactness, and
local lifting property. Additionally, $A$ does not admit an amenable trace and,
hence, is not quasidiagonal and does not have the WEP when $A$ is from the
class of exotic $C^*$-algebras defined by Brown and Guentner. In order to
achieve the main results of this paper, we prove a result which implies the
factorization property for the class of discrete groups which are algebraic
subgroups of locally compact amenable groups.
