论文标题
在肯定地解决迈克尔在Fréchet代数理论中备受赞誉的问题，并应用于自动连续性理论
On affirmative solution to Michael's acclaimed problem in the theory of Fréchet algebras, with applications to automatic continuity theory
论文作者
论文摘要
1952年，迈克尔提出了一个问题，上面关于他的回忆录中的交换弗雷切特代数的功能连续性，即文学中被称为迈克尔问题。我们以肯定的及其各种同等形式解决了这一点，即使是针对非共同案件的。确实，我们继续最近的作品，并开发了两种直接攻击这些问题的方法。第一种方法是证明该问题的测试案例，即所有有界复杂序列的Banach空间上所有功能的特定代数，实际上，如果存在不连续的字符，则一项不确定的所有复杂形式功率序列的特征代数。 In the second approach, the existence of a discontinuous character would allow us to generate other Frechet algebra topology, inequivalent to the usual Frechet algebra topology, by applying the method of Read (he used this method to show that the famous Singer-Wermer conjecture (1955) fails in the Frechet case).在这两种方法中，一个重要的工具都是在Banach空间上（对称）张量代数的拓扑版本。基础但至关重要的想法是将测试代数表达为在所有绝对可总结的复杂序列的Banach空间上的加权特里切特对称代数。 Several mathematicians have worked on two problems of Michael since 1952, giving affirmative solutions for special classes of Frechet algebras under various conditions, or discussing various test cases, or discussing various approaches, or discussing various other equivalent forms, or deriving other important automatic continuity results such as the (non-)uniqueness of the Frechet algebra topology for certain commutative Frechet algebras by alternate, difficult or lengthy methods.除了在自动连续性理论中提供各种新的（重要）应用外，我们总结了肯定解决方案对这些尝试的影响。
In 1952, Michael posed a question about the functional continuity of commutative Frechet algebras in his memoir, known as Michael problem in the literature. We settle this in the affirmative along with its various equivalent forms, even for the non-commutative case. Indeed, we continue our recent works, and develop two approaches to directly attack these problems. The first approach is to show that the test case for this problem, the Frechet algebra of all entire functions on the Banach space of all bounded complex sequences, is, in fact, a Frechet algebra of all complex formal power series in one indeterminate, if there exists a discontinuous character. In the second approach, the existence of a discontinuous character would allow us to generate other Frechet algebra topology, inequivalent to the usual Frechet algebra topology, by applying the method of Read (he used this method to show that the famous Singer-Wermer conjecture (1955) fails in the Frechet case). In both the approaches, an important tool is a topological version of the (symmetric) tensor algebra over a Banach space; the elementary, but crucial, idea is to express the test algebra as a weighted Frechet symmetric algebra over the Banach space of all absolutely summable complex sequences. Several mathematicians have worked on two problems of Michael since 1952, giving affirmative solutions for special classes of Frechet algebras under various conditions, or discussing various test cases, or discussing various approaches, or discussing various other equivalent forms, or deriving other important automatic continuity results such as the (non-)uniqueness of the Frechet algebra topology for certain commutative Frechet algebras by alternate, difficult or lengthy methods. We summarize effects of our affirmative solutions on these attempts in addition to giving various new (important) applications in automatic continuity theory.
