The maximal abelian subextension determines weak mixing for group extensions

The Maximal Abelian Sub-Extension Determines Weak Mixing for Group Extensions

Weak mixing is an important property for group extensions because its absence is the principal obstruction to lifting a large number of stronger mixing properties. Whether or not an extension is weakly mixing can be determined by studying the sub-extension corresponding to the quotient by the commutator subgroup. Several conditions equivalent to weak mixing are given. In particular, an extensio...

Fast Fourier Analysis for Abelian Group Extensions

Let G be a nite group and f any complex-valued function deened on G and an irreducible complex matrix representation of G. The Fourier transform of f at is deened to be the matrix P s2G f(s)(s). The Fourier transforms of f at all the irreducible representations of G determine f via the Fourier inversion formula f(s) = 1 j Gj P d trace(b f()(s ?1)): Direct computation of all Fourier transforms o...

On Galois Groups of Abelian Extensions over Maximal Cyclotomic Fields

Let k0 be a finite algebraic number field in a fixed algebraic closure Ω and ζn denote a primitive n-th root of unity ( n ≥ 1). Let k∞ be the maximal cyclotomic extension of k0, i.e. the field obtained by adjoining to k0 all ζn ( n = 1, 2, ...). Let M and L be the maximal abelian extension of k∞ and the maximal unramified abelian extension of k∞ respectively. The Galois groups Gal(M/k∞) and Gal...

The weak Paley-Wiener property for group extensions

The paper studies weak Paley-Wiener properties for group extensions by use of Mackey’s theory. The main theorem establishes sufficient conditions on the dual action to ensure that the group has the weak Paley-Wiener property. The theorem applies to yield the weak Paley-Wiener property for large classes of simply connected, connected solvable Lie groups (including exponential Lie groups), but al...

Weak Compactness of the Set of ε-Extensions