First module cohomology group of induced semigroup algebras

IF 0.4 Q4 MATHEMATICS
M. Miri, E. Nasrabadi, Kianoush Kazemi
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引用次数: 0

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.
诱导半群代数的第一模上同调群
设$S$为离散半群,$T$为$S$上的左乘子算子。在$S$上由$T$定义的一个新产品创建了一个新的诱导半群$S _{T} $。在本文中,我们证明了如果$T$是双射,那么第一模上同群$ \HH_{\ell^1(E)}^{1}(\ell^1(S)‎, ‎\ell^{\infty}(S))$和$ \HH_{\ell^1(E_{T})}^{1}(\ell^1({S_{T}})‎, ‎\ell^{\infty}(S_{T})) $是相等的,其中$E$和$E_{T}$分别是$S$和$S _{T}$中幂等元的集合。这特别意味着$\ell^1(S)$是弱的$\ell^1(E)$ -模块可接受的当且仅当$\ell^1(S_T)$是弱的$\ell^1(E_T)$ -模块可接受。最后,通过一个例子,我们证明了$T$的双射性条件是必要的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
140
审稿时长
25 weeks
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