First module cohomology group of induced semigroup algebras
Let $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$. A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $. In this paper, we show that if $T$ is bijective, then the first module cohomology groups $ \HH_{\ell^1(E)}^{1}(\ell^1(S), \ell^{\in...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Sociedade Brasileira de Matemática
2022-12-01
|
| Series: | Boletim da Sociedade Paranaense de Matemática |
| Online Access: | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/51414 |
| _version_ | 1797634068725104640 |
|---|---|
| author | Mohammad Reza Miri Ebrahim Nasrabadi Kianoush Kazemi |
| author_facet | Mohammad Reza Miri Ebrahim Nasrabadi Kianoush Kazemi |
| author_sort | Mohammad Reza Miri |
| collection | DOAJ |
| description |
Let $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$. A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $. In this paper, we show that if $T$ is bijective, then the first module cohomology groups $ \HH_{\ell^1(E)}^{1}(\ell^1(S), \ell^{\infty}(S))$ and $ \HH_{\ell^1(E_{T})}^{1}(\ell^1({S_{T}}), \ell^{\infty}(S_{T})) $ are equal, where $E$ and $E_{T}$ are sets of idempotent elements in $S$ and $S _{T}$, respectively. Which in particular means that $\ell^1(S)$ is weak $\ell^1(E)$-module amenable if and only if $\ell^1(S_T)$ is weak $\ell^1(E_T)$-module amenable. Finally, by giving an example, we show that the condition of bijectivity for $T$, is necessary.
|
| first_indexed | 2024-03-11T12:02:36Z |
| format | Article |
| id | doaj.art-577e36941cc449228d008580d5c8d4f8 |
| institution | Directory Open Access Journal |
| issn | 0037-8712 2175-1188 |
| language | English |
| last_indexed | 2024-03-11T12:02:36Z |
| publishDate | 2022-12-01 |
| publisher | Sociedade Brasileira de Matemática |
| record_format | Article |
| series | Boletim da Sociedade Paranaense de Matemática |
| spelling | doaj.art-577e36941cc449228d008580d5c8d4f82023-11-07T20:12:18ZengSociedade Brasileira de MatemáticaBoletim da Sociedade Paranaense de Matemática0037-87122175-11882022-12-014110.5269/bspm.51414First module cohomology group of induced semigroup algebrasMohammad Reza Miri0Ebrahim Nasrabadi1Kianoush Kazemi2University of BirjandUniversity of BirjandUniversity of Birjand Let $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$. A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $. In this paper, we show that if $T$ is bijective, then the first module cohomology groups $ \HH_{\ell^1(E)}^{1}(\ell^1(S), \ell^{\infty}(S))$ and $ \HH_{\ell^1(E_{T})}^{1}(\ell^1({S_{T}}), \ell^{\infty}(S_{T})) $ are equal, where $E$ and $E_{T}$ are sets of idempotent elements in $S$ and $S _{T}$, respectively. Which in particular means that $\ell^1(S)$ is weak $\ell^1(E)$-module amenable if and only if $\ell^1(S_T)$ is weak $\ell^1(E_T)$-module amenable. Finally, by giving an example, we show that the condition of bijectivity for $T$, is necessary. https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/51414 |
| spellingShingle | Mohammad Reza Miri Ebrahim Nasrabadi Kianoush Kazemi First module cohomology group of induced semigroup algebras Boletim da Sociedade Paranaense de Matemática |
| title | First module cohomology group of induced semigroup algebras |
| title_full | First module cohomology group of induced semigroup algebras |
| title_fullStr | First module cohomology group of induced semigroup algebras |
| title_full_unstemmed | First module cohomology group of induced semigroup algebras |
| title_short | First module cohomology group of induced semigroup algebras |
| title_sort | first module cohomology group of induced semigroup algebras |
| url | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/51414 |
| work_keys_str_mv | AT mohammadrezamiri firstmodulecohomologygroupofinducedsemigroupalgebras AT ebrahimnasrabadi firstmodulecohomologygroupofinducedsemigroupalgebras AT kianoushkazemi firstmodulecohomologygroupofinducedsemigroupalgebras |