On an affirmative solution to Michael's problem, with applications to automatic continuity theory
Abstract
In 1952, Michael posed a question about functional continuity of commutative Fréchet algebras in his memoir, known as Michael's problem in the literature. We settle this in the affirmative along with its equivalent forms. Indeed, we continue our previous works to directly attack these problems. We show that the test case for this problem -- the Fréchet algebra cal U of all entire functions on ℓ∞ -- is, in fact, a Fréchet algebra cc[[X]], if there exists a discontinuous character on cal U. An important tool is a topological version of the tensor (symmetric) algebra over a Banach space. The elementary, but crucial, idea is to express the test algebra cal U as a {\it weighted Fréchet symmetric algebra} over the Banach space ℓ1. We summarize the effects of our affirmative solution in terms of important applications in automatic continuity theory, including affirmative answers to two significant, long-standing problems from 1966 and 1978.
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BibTeX
@misc{patel2026affirmative,
title = {{On an affirmative solution to Michael's problem, with applications to automatic continuity theory}},
author = {S. R. Patel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39260},
eprint = {2610.39260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39260},
}
RIS
TY - UNPB
TI - On an affirmative solution to Michael's problem, with applications to automatic continuity theory
AU - S. R. Patel
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39260
UR - https://arxiv.org/abs/2610.39260
AB - In 1952, Michael posed a question about functional continuity of commutative Fréchet algebras in his memoir, known as Michael's problem in the literature. We settle this in the affirmative along with its equivalent forms. Indeed, we continue our previous works to directly attack these problems. We show that the test case for this problem -- the Fréchet algebra cal U of all entire functions on ℓ^∞ -- is, in fact, a Fréchet algebra cc[[X]], if there exists a discontinuous character on cal U. An important tool is a topological version of the tensor (symmetric) algebra over a Banach space. The elementary, but crucial, idea is to express the test algebra cal U as a {\it weighted Fréchet symmetric algebra} over the Banach space ℓ^1. We summarize the effects of our affirmative solution in terms of important applications in automatic continuity theory, including affirmative answers to two significant, long-standing problems from 1966 and 1978.
ER -