论文标题
在Gleason-Kahane-这些代数定理中
On the Gleason-Kahane-Żelazko theorem for associative algebras
论文作者
论文摘要
古典的Gleason-Kahane-Zelazko定理指出，线性在复杂的Banach代数上发挥作用，而不是在单位上消失，并且$λ（\ Mathbf 1）= 1 $，是乘法的，即，$λ（ab）=λ（ab）=λ（a（a）λ（a）λ（a）λ（a）λ（b）$ a $ a，b \ in a $ in $ a $。我们研究了联合Unital代数的GK组特性，尤其是用于功能代数的gk该属性。在Gk组代数$ a $中，至少在$ 3 $元素的字段上，并且拥有Codimension $ 1 $的理想，每个元素都是有限的单位总和。一个真实或复杂的代数，其距离左（右）理想的最大最大（右）是GK组代数。如果$ a $是可交换的代数，则本地化$ a_ {p} $是每个主要理想$ p $ $ a $的gk组algebra。因此，GKK的财产不是本地全球财产。 GK组代数类别在同态图像下关闭。如果一个函数代数$ a \ subseteq \ mathbb f^{x} $上的子字段$ \ mathbb f $ of $ \ mathbb c $，包含$ \ mathbb f^{x} $中的所有有界函数，则每个元素的每个元素of $ a $ a $ a $ a $ a $ a $ a $均为两个umits。如果$ a $还包含一个离散函数，则$ a $是gk电子代数。我们证明了定期分布的代数，以及在$（0，\ infty）$中支持的分布代数的单位化满足了Gk组属性，而紧凑型分布的代数则不满意。
The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $Λ(\mathbf 1)=1$, is multiplicative, that is, $Λ(ab)=Λ(a)Λ(b)$ for all $a,b\in A$. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra $A$ over a field of at least $3$ elements, and having an ideal of codimension $1$, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If $A$ is a commutative algebra, then the localisation $A_{P}$ is a GKŻ-algebra for every prime ideal $P$ of $A$. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra $A\subseteq \mathbb F^{X}$ over a subfield $\mathbb F$ of $\mathbb C$, contains all the bounded functions in $\mathbb F^{X}$, then each element of $A$ is a sum of two units. If $A$ contains also a discrete function, then $A$ is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in $(0,\infty)$ satisfy the GKŻ property, while the algebra of compactly supported distributions does not.
