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The spectrum of dynamical systems and Riesz products. (English. Russian original) Zbl 0704.46046
Math. USSR, Sb. 67, No. 2, 341-366 (1990); translation from Mat. Sb. 180, No. 7, 888-912 (1989).
A dynamical system (X,\(\mu\) ;G) is considered, where (X,\(\mu\)) is a measure space and G is a group whose action on X (x\(\to x.g)\) leaves quasi-invariant the measure \(\mu\). It appears a unitary representation \(g\to T(g)\), \(g\in G:\) for \(f\in L_ 2(X)\), \[ (1)\quad (T(g)f)(x)=a(g,x)(d\mu (x.g)/d\mu (x)^{1/2}f(x.g), \] where the function a(g,x), \(x\in X mod o\) is measurable in x and \(| a(g,x)| =1\), \(a(g_ 1g_ 2,x)=a(g_ 1,x)a(g_ 2,x.g_ 1).\)
The problem of irreducible factorization of (1) is considered when (X,\(\mu\) ;G) is a particular Mackey action and the problem reduces to an infinite product of operator-valued functions, defined on the dual space \(\hat G.\) If \(G={\mathbb{R}}\) this product is the classical Riesz product; in this case the Mackey action is realized as a special flow and an existence criterion for the finite invariant measure for Mackey action is derived, as well the convergence of the Riesz product in more general condition than the classical ones.
For the case \(G=N_ 3\) (the group of the upper-triangular \(3\times 3\) matrices) it is shown that the representation (1) is simple and singular.
The connection between the Mackey action and the residual \(\sigma\)- algebra of the wandering on G is emphasized.
Reviewer: C.Udrişte
MSC:
46L55 Noncommutative dynamical systems
22D10 Unitary representations of locally compact groups
46L45 Decomposition theory for \(C^*\)-algebras
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