  In this paper, we initiate the study of C*-algebras endowed with a twisted
action of a locally compact Abelian Lie group, and we construct a twisted
crossed product, which is in general a nonassociative, noncommutative, algebra.
The properties of this twisted crossed product algebra are studied in detail,
and are applied to T-duality in Type II string theory to obtain the T-dual of a
general principal torus bundle with general H-flux, which we will argue to be a
bundle of noncommutative, nonassociative tori. We also show that this
construction of the T-dual includes all of the special cases that were
previously analysed.
