arXiv 1301.32372013-01-15math.FA

三角巴拿赫代数的n-弱模顺从性

n-Weak Module Amenability of Triangular Banach Algebras

Abasalt Bodaghi, Ali Jabbari

摘要：

设$\mathcal A$，$\mathcal B$是具有相容作用的Banach $\mathfrak A$ -模，$\mathcal M$是左Banach $\mathcal A$ - $\mathfrak A$ -模和右Banach $\mathcal B$ - $\mathfrak A$ -模.本文研究了三角Banach代数$\mathcal T=[ {cc} \mathcal A & \mathcal M & \mathcal B ]$（作为$\mathfrak T:=\Big{[ {cc} α& & α ] | α\in\mathfrak A\Big}$ -模）的模顺从性、$n$ -弱模顺从性和模Arens正则性.我们利用这些结果证明了：对于一个逆半群$S$，其子半群$E$是幂等元的，三角Banach代数$\mathcal T_0=[ {cc} \ell^1(S)& \ell^1(S) & \ell^1(S) ]$是永久弱模顺从的（作为$\mathfrak T_0=[ {cc} \ell^1(E)& & \ell^1(E) ]$ -模）.作为例子，我们证明了$\mathcal T_0$是$\mathfrak T_0$ -模Arens正则的当且仅当$S$的极大群同态象$G_S$是有限的.

英文摘要：

Let $\mathcal A$, $\mathcal B$ be Banach $\mathfrak A$-modules with compatible actions and $\mathcal M$ be a left Banach $\mathcal A$-$\mathfrak A$-module and a right Banach $\mathcal B$-$\mathfrak A$-module. In the current paper, we study module amenability, $n$-weak module amenability and module Arens regularity of the triangular Banach algebra $\mathcal T=[ {cc} \mathcal A & \mathcal M & \mathcal B ]$ (as an $\mathfrak T:=\Big{[ {cc} α& & α ] | α\in\mathfrak A\Big}$-module). We employ these results to prove that for an inverse semigroup $S$ with subsemigroup $E$ of idempotents, the triangular Banach algebra $\mathcal T_0=[ {cc} \ell^1(S)& \ell^1(S) & \ell^1(S) ]$ is permanently weakly module amenable (as an $\mathfrak T_0=[ {cc} \ell^1(E)& & \ell^1(E) ]$-module). As an example, we show that $\mathcal T_0$ is $\mathfrak T_0$-module Arens regular if and only if the maximal group homomorphic image $G_S$ of $S$ is finite.

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