论文标题
Banach代数及其第二个二元组的同源和同源特性
Homological and cohomological properties of Banach algebras and their second duals
论文作者
论文摘要
在本文中，我们研究了Banach代数的同源特性。我们表明，缩回Banach代数可保留双向物t，订约性和双足。我们还证明，BANACH代数的第二双二双的合同意味着Banach代数的签约性。对于带有$δ（a）\ neq \ emptySet $的Banach代数$ a $，令$ \ frak {f}（x，x，a）为Banach代数$ c_b（x，a）$，$ c_0（x，x，a）$，a）$，$ \ hbox {lip} lip {lip {lip}_α（lip） a）$。在下文中，我们研究Banach代数$ \ frak {f}（x，a）$的同源性能，尤其是其合同。我们证明，$ \ frak {f}（x，a）$的合同等同于$ x $的有限和$ a $的合同。如果$ a $是可交换的，则我们表明$ \ frak {f}（x，a）$在且仅当$ a $ as a $ as a $ c^* - $代数以及$ x $和$ x $和$δ（a）$都是有限的时。特别是，$ \ hbox {lip}_α^0（x，a）$在且仅当$ x $是有限时才可以签约。我们还调查$ l^1（g，a）$的合同性，并在$ g $有限的情况下，$ l^1（g，a）$是合同的，并且$ a $是合同的。最后，我们表明，beurling代数$ l^1（g，ω）$的双向物质相当于$ g $的紧凑性，但是，Banach代数$ l^1（g，ω）^{**} $的双原位性相当于$ g $。此结果适用于Banach代数$ M（g，ω）^{**} $，而不是$ l^1（g，ω）^{**} $。
In this paper, we investigate homological properties of Banach algebras. We show that retractions Banach algebras preserve biprojectivity, contractibility and biflatness. We also prove that contractibility of second dual of a Banach algebra implies contractibility of the Banach algebra. For a Banach algebra $A$ with $Δ(A)\neq\emptyset$, let $\frak{F}(X, A)$ be one of the Banach algebras $C_b(X, A)$, $C_0(X, A)$, $\hbox{Lip}_α(X, A)$ or $\hbox{lip}_α(X, A)$. In the following, we study homological properties of Banach algebra $\frak{F}(X, A)$, especially contractibility of it. We prove that contractibility of $\frak{F}(X, A)$ is equivalent to finiteness of $X$ and contractibility of $A$. In the case where, $A$ is commutative, we show that $\frak{F}(X, A)$ is contractible if and only if $A$ is a $C^*-$algebra and both $X$ and $Δ(A)$ are finite. In particular, $\hbox{lip}_α^0(X, A)$ is contractible if and only if $X$ is finite. We also investigate contractibility of $L^1(G, A)$ and establish $L^1(G, A)$ is contractible if and only if $G$ finite and $A$ is contractible. Finally, we show that biprojectivity of the Beurling algebra $L^1(G, ω)$ is equivalent to compactness of $G$, however, biprojectivity of the Banach algebras $L^1(G, ω)^{**}$ is equivalent to finiteness of $G$. This result holds for the Banach algebra $M(G, ω)^{**}$ instead of $L^1(G, ω)^{**}$.
