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New algebras of functions on topological groups arising from \(G\)-spaces. (English) Zbl 1161.37014

Let \(G\) be a topological group. It is shown that the algebra \(SUC(G)\) of strongly uniformly continuous functions contains the algebra \(WAP(G)\) of weakly almost periodic functions as well as the algebras \(LE(G)\) of locally equicontinuous functions and the algebra \(\text{Asp}(G)\) of Asplund functions.
The Roelcke algebra \(UC(G)\) of right and left uniformly continuous functions coincides with \(SUC(G)\) in the case \(G\) is discrete or Abelian, but for general non-locally compact large groups \(SUC(G)\subset UC(G)\) is a proper uniformly closed \(G\)-invariant subalgebra.
To understand dynamical complexity of a function \(f\in RUC(G)\subset UC(G)\) the topological complexity of the cyclic \(G\)-flow \(X_f\) (=pointwise closure of the left \(G\)-orbit of \(f\)) is studied. \(X_f\subset UC(G)\) iff \(f\in SUC(G)\).
For the Polish groups \(G\) of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1 it is shown that \(SUC(G)\) is trivial.
For the group \(G=S(\mathbb N)\) of permutations of a countable set it is shown that \(WAP(G)=SUC(G)=UC(G)\) and the corresponding metrizable Cantor semitopological semigroup compactification is described.
For the group \(G=H(C)\) of homeomorphisms of the Cantor set it is shown that \(SUC(G)\) is properly contained in \(UC(G)\). Moreover for this group \(UC(G)\) does not yield a right topological semigroup compactification.
Some other groups \(G\) are considered as well. Though results are proved for real-valued functions it is expected that they extend to the algebras of complex-valued function.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
54H20 Topological dynamics (MSC2010)
54H15 Transformation groups and semigroups (topological aspects)
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