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Pentagonal transformations. (Transformations pentagonales.) (French. Abridged English version) Zbl 1040.37500

Summary: We prove that every bimeasurable transformation \(\upsilon : X \times X \to X \times X\), where \(X\) is a measured space satisfying the pentagon identity \(\upsilon_{23} \circ\upsilon_{13}\circ\upsilon_{12} = \upsilon_{12}\circ \upsilon_{23}\) (and sufficiently regular) comes from a matched pair of locally compact groups, i.e., a locally compact group \(G\) and two closed subgroups, \(G_1\) and \(G_2\) such that the map \((g_1, g_2) \mapsto g_1g_2\) is a homeomorphism from \(G_1 \times G_2\) onto a dense open subset of \(G\).

MSC:

37A05 Dynamical aspects of measure-preserving transformations
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
28D05 Measure-preserving transformations
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