Ergodic cocycles for Gaussian actions (Q933134)

In this paper, \(G\) is a countable Abelian group equipped with the discrete topology, \((X,\mathcal{B},\mu)\) is a standard probability space and \(\mathcal{T}:G\times X\to X\) is a free \(G\)-action on \((X,\mathcal{B},\mu)\). The action \(\mathcal{T}\) is called Gaussian if there exists a \(\mathcal{T}\)-invariant closed subspace \(\mathcal{H}\subset L^2(X,\mathcal{B},\mu)\) of the zero-mean real functions such that each nonzero \(h\) in \(\mathcal{H}\) is a Gaussian variable and the smallest \(\sigma\)-algebra which makes all variables of \(\mathcal{H}\) measurable is equal to \(\mathcal{B}\). The main result in this paper is a construction of ergodic cocycles for rigid Gaussian actions. The result is related to a conjecture made in 1999 by M. Lemańczyk. The author also shows that any isomorphism between Gaussian actions is Gaussian.