The Group Aut (μ) is Roelcke Precompact

2012 ◽  
Vol 55 (2) ◽  
pp. 297-302 ◽  
Author(s):  
Eli Glasner
Keyword(s):  
Probability Space ◽  
Separable Hilbert Space ◽  
Continuous Functions ◽  
Uniform Structure ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Markov Operators ◽  
Weakly Almost Periodic ◽  
Borel Probability ◽  
Uniformly Continuous

AbstractFollowing a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group G = Aut(μ) of automorphisms of an atomless standard Borel probability space (X, μ) is precompact. We identify the corresponding compactification as the space of Markov operators on L2(μ) and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on G, i.e., functions on G arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that G is totally minimal.

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

scholarly journals A Note on the Topological Group c0

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 77
Author(s):  
Michael Megrelishvili
Keyword(s):  
Banach Space ◽  
Topological Group ◽  
Unit Sphere ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Semigroup Compactification ◽  
Weakly Almost Periodic ◽  
Open Question

A well-known result of Ferri and Galindo asserts that the topological group c 0 is not reflexively representable and the algebra WAP ( c 0 ) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame ( c 0 ) of tame functions. Respectively, it is an open question if c 0 is representable on a Rosenthal Banach space. In the present work we show that Tame ( c 0 ) is small in a sense that the unit sphere S and 2 S cannot be separated by a tame function f ∈ Tame ( c 0 ) . As an application we show that the Gromov’s compactification of c 0 is not a semigroup compactification. We discuss some questions.

V.—Les transformations asymptotiquement presque périodiques discontinues et le lemme ergodique. (Première Note.)

1950 ◽  
Vol 63 (1) ◽  
pp. 61-68
Author(s):  
Maurice Fréchet
Keyword(s):  
Ergodic Theorem ◽  
Continuous Functions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Discontinuous Functions ◽  
Extended Class ◽  
Class Of Functions ◽  
Asymptotically Almost Periodic

SynopsisWith the aim of establishing, under wide conditions, the ergodic theorem of G. D. Birkhoff, the author extends the class of asymptotically almost-periodic functions, considering now not only continuous functions, as he had already done in 1943, but discontinuous functions. Definitions and properties of the extended class of functions are set out, some comparisons being made with almost-periodic functions in the sense of Bohr, Stepanoff, Weyl and Besicovitch. Applications to the ergodic theorem are adumbrated.

On the actions of a locally compact group on some of its semigroup compactifications

2007 ◽  
Vol 143 (1) ◽  
pp. 25-39 ◽  
Author(s):  
M. FILALI
Keyword(s):  
Compact Group ◽  
Local Structure ◽  
Structure Theorem ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Locally Compact ◽  
Semigroup Compactifications ◽  
Weakly Almost Periodic

AbstractLetGbe a locally compact group and$\mathcal{U}G$be its largest semigroup compactification. Thensx≠xin$\mathcal{U}G$wheneversis an element inGother thane. This result was proved by Ellis in 1960 for the caseGdiscrete (and so$\mathcal{U}G$is the Stone–Čech compactification βGofG), and by Veech in 1977 for any locally compact group. We study this property in theWAP– compactification$\mathcal{W}G$ofG; and in$\mathcal{U}G$, we look at the situation whenxs≠x. The points are separated by some weakly almost periodic functions which we are able to construct on a class of locally compact groups, which includes the so-called E-groups introduced by C. Chou and which is much larger than the class of SIN groups. The other consequences deduced with these functions are: a generalization of some theorems on the regularity of$\mathcal{W}G$due to Ruppert and Bouziad, an analogue in$\mathcal{W}G$of the “local structure theorem” proved by J. Pym in$\mathcal{U}G$, and an improvement of some earlier results proved by the author and J. Baker on$\mathcal{W}G$.

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