Noncommutative gauge theories in matrix theory

We present a general framework for matrix theory compactified on a quotient space ${\mathbf{R}}^n/\Gamma ,$ with Γ a discrete group of Euclidean motions in ${\mathbf{R}}^n.$ The general solution to the quotient conditions gives a gauge theory on a noncommutative space. We characterize the resulting noncommutative gauge theory in terms of the twisted group algebra of Γ associated with a projective regular representation. Also we show how to extend our treatments to incorporate orientifolds.