论文标题
BANACH代数的共同体特性和Arens的规律性
Cohomological properties and Arens regularity of Banach algebras
论文作者
论文摘要
在本文中，我们研究了Banach代数的一些同时性特性。对于Banach代数$ A $和Banach $ A $ -BIMODULE $ B $，我们研究了第一个Hochschild cohomology群体的消失，$ h^1（a^n，b^m）$和$ h_ {w^*}^1（a^n，b^m）$，其中$ 0 \ leq m，n \ n \ n \ leq 3 $。对于Amenable Banach代数$ a $，我们表明有Banach $ a $ a-bimodules $ c $，$ d $和elements $ \ mathfrak {a}，\ mathfrak {b} \ in a^{**} $ $ z^1（a，c^*）= \ {r_ {d^{\ prime \ prime}（\ mathfrak {a}）}：〜d \ in z^1（a，c^*）= \} = \ {l_ z^1（a，d^*）\}。$$，对于b $中的每个$ b \，$ l_ {b}（a）= ba $和$ r_ $ and $ r_ {b}（a）= a b，每$ a \ in $ in $。此外，在某种情况下，我们表明，如果从Banach代数$ a $连续推导的第二次转介到$ a^*$中，即从$ a^{**} $中的连续线性映射到$ a^{***} $，就是一个衍生，那么$ a $是$ a $是普通的。最后，我们表明，如果$ a $是双重不规则的Banach代数，以使其第二个双重二元相吻合，那么$ a $是反射性的。
In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$ where, for every $b\in B$, $L_{b}(a)=ba$ and $R_{b}(a)=a b,$ for every $a\in A$. Moreover, under a condition, we show that if the second transpose of a continuous derivation from the Banach algebra $A$ into $A^*$ i.e., a continuous linear map from $A^{**}$ into $A^{***}$, is a derivation, then $A$ is Arens regular. Finally, we show that if $A$ is a dual left strongly irregular Banach algebra such that its second dual is amenable, then $A$ is reflexive.
