Preliminaries on Crossed Products - Schulz-Baldes, Hermann; Stoiber, Tom (Springer, 2022)
This chapter reviews several facts about crossed products. Many aspects are covered in the standard reference [1], but some are taken from other works like [2–7]. Many results hold for general locally compact abelian group actions, but for sake of concreteness we will only restrict to the case of abelian n-parameter groups which are relevant for the applications that we have in mind. Hence throughout G=Tn0⊕Rn1 where n= n0+ n1 and we make the identification T= R Z= [ 0, 1 ] / ∼. Unless there is an explicitly different choice of normalization (which only applies in Chap. 5), we fix the Haar measures on Td and Rd to be the usual Lebesgue measure and on Zd the counting measure. Combined with the conventions for the Fourier transform as given below, one can then identify the dual groups T^ = Z, Z^ = T and R^ = R in such a way that the Plancherel theorem holds without proportionality constants.
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