On an affirmative solution to Michael's problem, with applications to automatic continuity theory

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 $\ell^\infty$ -- 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 $\ell^1$. 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.