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Flows on invariant subsets and compactifications of a locally compact group. (English) Zbl 0920.43005

The authors consider the action of a locally compact group \(G\) on an invariant compact subset \(V\) by inner automorphism \(x\mapsto gxg^{-1}\) and investigate relationships between these inner automorphic flows and the \({\mathcal LC}\)-compactification (a semigroup compactification determined by the left uniformly continuous functions). The authors demonstrate that such flows are to a large extent independent of \(V\) and derive conditions for such flows to be distal (for example \(V\) is normal in the \({\mathcal LC}\)-compactification). They show for normal subgroups \(N\) of \(G\) for which each element has a precompact conjugacy class that the idempotents in the \({\mathcal LC}\)-compactification give rise to a natural semidirect decomposition of \(N\), although topologically the situation is not so simple. In the last part of the paper the authors apply their considerations to groups which are semidirect products.

MSC:

43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
22D05 General properties and structure of locally compact groups
54H20 Topological dynamics (MSC2010)
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