Lifting mixing properties by Rokhlin cocycles

We study the problem of lifting various mixing properties from a base automorphism \(T\in {\rm Aut}\xbm\) to skew products of the form \(\tfs\), where \(\va:X\to G\) is a cocycle with values in a locally compact Abelian group \(G\), \(\cs=(S_g)_{g\in G}\) is a measurable representation of \(G\) in \({\rm Aut}\ycn\) and \(\tfs\) acts on the product space \((X\times Y,\cb\ot\cc,\mu\ot\nu)\) by $$\tfs(x,y)=(Tx,S_{\va(x)}(y)).$$ It is also shown that whenever \(T\) is ergodic (mildly mixing, mixing) but \(\tfs\) is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor \(\ca\subset\cc\) of \(\cs\) the corresponding Rokhlin cocycle \(x\mapsto S_{\va(x)}|_{\ca}\) is a coboundary (a quasi-coboundary).