  Our approach to higher order Fourier analysis is to study the ultra product
of finite (or compact) Abelian groups on which a new algebraic theory appears.
This theory has consequences on finite (or compact) groups usually in the form
of approximative statements. The present paper is the second part of a paper in
which higher order characters and decompositions were introduced. We generalize
the concept of the Pontrjagin dual group and introduce higher order versions of
it. We study the algebraic structure of the higher order dual groups. We prove
a simple formula for the Gowers uniformity norms in terms of higher order
decompositions. We present a simple spectral algorithm to produce higher order
decompositions. We briefly study a multi linear version of Fourier analysis.
Along these lines we obtain new inverse theorems for Gowers's norms.
