Bouwknegt, Pier (Peter)Hannabuss, KeithMathai, Varghese2015-12-130010-3616http://hdl.handle.net/1885/76776In this paper, we initiate the study of C *-algebras [InlineMediaObject not available: see fulltext.] endowed with a twisted action of a locally compact abelian Lie group [InlineMediaObject not available: see fulltext.], and we construct a twisted crossed product [InlineMediaObject not available: see fulltext.], which is in general a nonassociative, noncommutative, algebra. The duality 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. Nonassociativity is interpreted in the context of monoidal categories of modules. We also show that this construction of the T-dual includes the other special cases already analysed in a series of papers.200610.1007/s00220-005-1501-82015-12-11