{"title":"诱导半群代数的第一模上同调群","authors":"M. Miri, E. Nasrabadi, Kianoush Kazemi","doi":"10.5269/bspm.51414","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"First module cohomology group of induced semigroup algebras\",\"authors\":\"M. Miri, E. Nasrabadi, Kianoush Kazemi\",\"doi\":\"10.5269/bspm.51414\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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.\",\"PeriodicalId\":44941,\"journal\":{\"name\":\"Boletim Sociedade Paranaense de Matematica\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-12-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Boletim Sociedade Paranaense de Matematica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5269/bspm.51414\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boletim Sociedade Paranaense de Matematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5269/bspm.51414","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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^{\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.