  Two main problems face the construction of noncommutative actions for gravity
with star products: the complex metric and finding an invariant measure. The
only gauge groups that could be used with star products are the unitary groups.
I propose an invariant gravitational action in D=4 dimensions based on the
constrained gauge group U(2,2) broken to $U(1,1)\times U(1,1).$ No metric is
used, thus giving a naturally invariant measure. This action is generalized to
the noncommutative case by replacing ordinary products with star products. The
four dimensional noncommutative action is studied and the deformed action to
first order in deformation parameter is computed.
