论文标题
矢量值Lipschitz代数及其第二个二元组的共同体学特性
Cohomological properties of vector-valued Lipschitz algebras and their second duals
论文作者
论文摘要
令$ \ frak {f}（x，a）$为Banach代数$ \ hbox {Lip}（x，a）$或$ \ hbox {lip}（x，a）$。在本文中，我们表明$ \ frak {f}（x，a）$在且仅当$ x $均匀离散并且$ a $是可正常的情况下。我们还证明，结果适用于$ \ hbox {lip}^\ circ（x，a）$而不是$ \ frak {f}（x，a）$。在$ a^*$是可分离的情况下，我们确定$ \ frak {f}（x，a）^{**} $在且仅当$ x $均匀离散并且$ a^{**} $时才可以amen amen，但是，可正态，但是，$ \ hbox {lbox} {lbox}^i is的不合适性与公平（ $ a^{**} $和$ x $的有限性。我们证明，如果$ \ hbox {lip}（x，a）$是（分别为弱），那么 $ x $是统一离散的，$ a $是（分别（弱））。尤其是，$ \ hbox {lip} x $在且仅当$ x $是离散的情况下是弱的。然后，我们研究了矢量值Banach代数$ C_0（x，a）$和$ l^1（g，a）$的共同体属性。最后，我们证明了$ a^{**} $的双原位性（分别为周期性弱的舒适性）表示$ a $的双向物品（分别为循环弱的不合适性）。当$ a $交换时，此结果适用于弱的舒适性和循环舒适性。
Let $\frak{F}(X, A)$ be one of the Banach algebras $\hbox{Lip}(X, A)$ or $\hbox{lip}(X, A)$. In this paper, we show that $\frak{F}(X, A)$ is amenable if and only if $X$ is uniformly discrete and $A$ is amenable. We also prove that the result holds for $\hbox{lip}^\circ(X, A)$ instead of $\frak{F}(X, A)$. In the case where $A^*$ is separable, we establish that $\frak{F}(X, A)^{**}$ is amenable if and only if $X$ is uniformly discrete and $A^{**}$ is amenable, however, amenability of $\hbox{lip}^\circ(X, A)^{**}$ is equivalent to amenability of $A^{**}$ and finiteness of $X$. We prove that if $\hbox{Lip}(X, A)$ is point (respectively, weakly) amenable, then $X$ is uniformly discrete and $A$ is point (respectively, weakly) amenable. In particular, $\hbox{Lip}X$ is weakly amenable if and only if $X$ is discrete. We then investigate cohomological properties for vector-valued Banach algebras $C_0(X, A)$ and $L^1(G, A)$. Finally, we prove that biprojectivity (respectively, cyclically weak amenability) of $A^{**}$ implies biprojectivity (respectively, cyclically weak amenability) of $A$. This result holds for weak amenability and cyclic amenability when $A$ is commutative.
