TY - JOUR

T1 - On metrizable enveloping semigroups

AU - Glasner, Eli

AU - Megrelishvili, Michael

AU - Uspenskij, Vladimir V.

PY - 2008/3

Y1 - 2008/3

N2 - When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g G, in the compact space X X . Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily nonsensitive; (3) for any compatible metric d on X the metric d G (x, y) := sup{d(gx, gy): g G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.

AB - When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g G, in the compact space X X . Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily nonsensitive; (3) for any compatible metric d on X the metric d G (x, y) := sup{d(gx, gy): g G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.

UR - http://www.scopus.com/inward/record.url?scp=58449115114&partnerID=8YFLogxK

U2 - 10.1007/s11856-008-0032-3

DO - 10.1007/s11856-008-0032-3

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AN - SCOPUS:58449115114

SN - 0021-2172

VL - 164

SP - 317

EP - 332

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -