A Note on the Topological Group <i>c</i><sub>0</sub>
A well-known result of Ferri and Galindo asserts that the topological group <inline-formula> <math display="inline"> <semantics> <msub> <mi>c</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> is not reflex...
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| Language: | English |
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MDPI AG
2018-10-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/7/4/77 |
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| author | Michael Megrelishvili |
| author_facet | Michael Megrelishvili |
| author_sort | Michael Megrelishvili |
| collection | DOAJ |
| description | A well-known result of Ferri and Galindo asserts that the topological group <inline-formula> <math display="inline"> <semantics> <msub> <mi>c</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> is not reflexively representable and the algebra WAP<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> 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<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> of tame functions. Respectively, it is an open question if <inline-formula> <math display="inline"> <semantics> <msub> <mi>c</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> is representable on a Rosenthal Banach space. In the present work we show that Tame<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is small in a sense that the unit sphere <i>S</i> and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>S</mi> </mrow> </semantics> </math> </inline-formula> cannot be separated by a tame function <i>f</i> ∈ Tame<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. As an application we show that the Gromov’s compactification of <inline-formula> <math display="inline"> <semantics> <msub> <mi>c</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> is not a semigroup compactification. We discuss some questions. |
| first_indexed | 2024-12-21T19:39:28Z |
| format | Article |
| id | doaj.art-18808a22d1e74ac193881a9c6eafc4ee |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-12-21T19:39:28Z |
| publishDate | 2018-10-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj.art-18808a22d1e74ac193881a9c6eafc4ee2022-12-21T18:52:30ZengMDPI AGAxioms2075-16802018-10-01747710.3390/axioms7040077axioms7040077A Note on the Topological Group <i>c</i><sub>0</sub>Michael Megrelishvili0Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, IsraelA well-known result of Ferri and Galindo asserts that the topological group <inline-formula> <math display="inline"> <semantics> <msub> <mi>c</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> is not reflexively representable and the algebra WAP<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> 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<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> of tame functions. Respectively, it is an open question if <inline-formula> <math display="inline"> <semantics> <msub> <mi>c</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> is representable on a Rosenthal Banach space. In the present work we show that Tame<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is small in a sense that the unit sphere <i>S</i> and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <mi>S</mi> </mrow> </semantics> </math> </inline-formula> cannot be separated by a tame function <i>f</i> ∈ Tame<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. As an application we show that the Gromov’s compactification of <inline-formula> <math display="inline"> <semantics> <msub> <mi>c</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> is not a semigroup compactification. We discuss some questions.https://www.mdpi.com/2075-1680/7/4/77Gromov’s compactificationgroup representationmatrix coefficientsemigroup compactificationtame function |
| spellingShingle | Michael Megrelishvili A Note on the Topological Group <i>c</i><sub>0</sub> Axioms Gromov’s compactification group representation matrix coefficient semigroup compactification tame function |
| title | A Note on the Topological Group <i>c</i><sub>0</sub> |
| title_full | A Note on the Topological Group <i>c</i><sub>0</sub> |
| title_fullStr | A Note on the Topological Group <i>c</i><sub>0</sub> |
| title_full_unstemmed | A Note on the Topological Group <i>c</i><sub>0</sub> |
| title_short | A Note on the Topological Group <i>c</i><sub>0</sub> |
| title_sort | note on the topological group i c i sub 0 sub |
| topic | Gromov’s compactification group representation matrix coefficient semigroup compactification tame function |
| url | https://www.mdpi.com/2075-1680/7/4/77 |
| work_keys_str_mv | AT michaelmegrelishvili anoteonthetopologicalgroupicisub0sub AT michaelmegrelishvili noteonthetopologicalgroupicisub0sub |