We provide a study of Köthe sequence algebras. These are Fréchet sequence algebras which can be viewed as abstract analogoues of algebras of smooth or holomorphic functions. Of particular treatment are the following properties: unitality, m‐convexity, Q‐property and variants of amenability. These properties are then checked against the topological $(\mathrm{DN})$‐(Ω) type conditions of Vogt–Zaharjuta. Description of characters and closed ideals in Köthe echelon algebras is provided. We show that these algebras are functionally continuous and we characterize completely which of them factor. Moreover, we prove that closed subalgebras of Montel m‐convex Köthe echelon algebras are Köthe echelon algebras as well and we give a version of Stone–Weiestrass theorem for these algebras. We also emphasize connections with the algebra of smooth operators.

中文翻译：

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Köthe梯形代数的一般方法

我们提供了Köthe序列代数的研究。这些是Fréchet序列代数，可以看作是光滑或全纯函数的代数的抽象类比。以下属性是特殊的处理方法：单位性，m凸性，Q属性和舒适性变量。然后根据拓扑检查这些属性$（\mathrm{DN}）$Vogt–Zaharjuta的（Ω）型条件。提供了科特梯形代数中的字符和封闭理想的描述。我们证明了这些代数在功能上是连续的，并且我们可以完全表征它们中的哪一个。此外，我们证明了的蒙特尔封闭代数米-凸Köthe梯队代数是Köthe梯队代数，以及我们给这些代数版本石Weiestrass定理。我们也强调与光滑算子的代数的联系。