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On an affirmative solution to the non-commutative Michael's problem, with applications to automatic continuity theory
Functional Analysis · Submitted 19 Jun 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We settle this in the affirmative along with its various equivalent forms, even for the non-commutative case.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In 1952, Michael posed a question about the functional continuity of commutative Frechet algebras in his memoir, known as Michael problem in the literature. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | We settle this in the affirmative along with its various equivalent forms, even for the non-commutative case. | result 0.83 | 0.96 | 0.95 | asserted 1.00 |
| s2 | Indeed, we continue our recent works, and develop two approaches to directly attack these problems. | contribution 0.80 | 0.91 | 0.05 | asserted 0.99 |
| s3 | 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. | method 0.82 | 0.85 | 0.07 | asserted 0.88 |
| s4 | 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). | method 0.88 | 0.50 | 0.04 | asserted 0.53 |
| s5 | 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. | method 0.98 | 0.87 | 0.04 | asserted 1.00 |
| s6 | 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. | background 0.99 | 0.05 | 0.03 | asserted 1.00 |
| s7 | We summarize effects of our affirmative solutions on these attempts in addition to giving various new (important) applications in automatic continuity theory. | contribution 0.97 | 0.92 | 0.09 | asserted 0.98 |
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Original abstract
In 1952, Michael posed a question about functional continuity of commutative Fr\'{e}chet algebras in his memoir, known as Michael's problem in the literature. We settle this in the affirmative for the non-commutative Fr\'{e}chet algebras, along with its equivalent form. Indeed, we continue our previous works to directly attack these problems. We show that the existence of a discontinuous character on the non-commutative test case for this problem -- the Fr\'{e}chet algebra ${\cal U}_{nc}$ of all entire functions on $\ell^\infty$ -- allows us to generate another Fr\'{e}chet algebra topology on ${\cal U}_{nc}$, inequivalent to the usual Fr\'{e}chet algebra topology. An important tool is a topological version of the tensor algebra over a Banach space. The elementary, but crucial, idea is to express the test algebra ${\cal U}_{nc}$ as a {\it weighted Fr\'{e}chet tensor algebra} over the Banach space $\ell^1$. We give some important applications in automatic continuity theory. In particular, we provide affirmative answers to significant, long-standing problems from 1967 (extending Johnson's result for non-commutative, semi-simple Fr\'{e}chet algebras; Dales' survey, Question 9), 1976 (extending Esterle's result, giving another approach for Michael's problem for more general $(F)$-algebras), and 1978 (a weaker version of an equivalent form of the non-commutative Michael's problem; Dales' survey, p. 143).
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