Enveloping Ellis semigroups as compactifications of transformations groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911078853640192 |
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| author | Kozlov, K. L. Sorin, B. V. |
| author_facet | Kozlov, K. L. Sorin, B. V. |
| contents | The notion of a proper Ellis semigroup compactification is introduced. Ellis's functional approach shows how to obtain them from totally bounded equiuniformities on a phase space $X$ when the acting group $G$ is with the topology of pointwise convergence and the $G$-space $(G, X, \curvearrowright)$ is $G$-Tychonoff.
The correspondence between proper Ellis semigroup compactifications of a topological group and special totally bounded equiuniformities (called Ellis equiuniformities) on a topological group is established. The Ellis equiuniformity on a topological transformation group $G$ from the maximal equiuniformity on a phase space $G/H$ in the case of its uniformly equicontinuous action is compared with Roelcke uniformity on $G$.
Proper Ellis semigroup compactifications are described for groups $S\,(X)$ (the permutation group of a discrete space $X$) and $Aut\,(X)$ (automorphism group of an ultrahomogeneous chain $X$) in the permutation topology. It is shown that this approach can be applied to the unitary group of a Hilbert space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04281 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enveloping Ellis semigroups as compactifications of transformations groups Kozlov, K. L. Sorin, B. V. General Topology Group Theory Primary 57S05, 20E22 Secondary 22F05, 22F50, 54E05, 54D35, 54H15, 47B02 The notion of a proper Ellis semigroup compactification is introduced. Ellis's functional approach shows how to obtain them from totally bounded equiuniformities on a phase space $X$ when the acting group $G$ is with the topology of pointwise convergence and the $G$-space $(G, X, \curvearrowright)$ is $G$-Tychonoff. The correspondence between proper Ellis semigroup compactifications of a topological group and special totally bounded equiuniformities (called Ellis equiuniformities) on a topological group is established. The Ellis equiuniformity on a topological transformation group $G$ from the maximal equiuniformity on a phase space $G/H$ in the case of its uniformly equicontinuous action is compared with Roelcke uniformity on $G$. Proper Ellis semigroup compactifications are described for groups $S\,(X)$ (the permutation group of a discrete space $X$) and $Aut\,(X)$ (automorphism group of an ultrahomogeneous chain $X$) in the permutation topology. It is shown that this approach can be applied to the unitary group of a Hilbert space. |
| title | Enveloping Ellis semigroups as compactifications of transformations groups |
| topic | General Topology Group Theory Primary 57S05, 20E22 Secondary 22F05, 22F50, 54E05, 54D35, 54H15, 47B02 |
| url | https://arxiv.org/abs/2412.04281 |