Algebras of holomorphic functions and the Michael problem (Q259733)

One version of what is known as the Michael problem (1952) is the following: Let \(\mathcal A\) be a complete, commutative complex locally \(m\)-convex algebra. Is every homomorphism \(\varphi:\mathcal A \to \mathbb C\) automatically bounded? It has been shown by \textit{D. Clayton} [Rocky Mt. J. Math. 5, 337--344 (1975; Zbl 0325.46055)], \textit{M. Schottenloher} [Arch. Math. 37, 241--247 (1981; Zbl 0471.46036)] and \textit{J. Mujica} [Complex analysis in Banach spaces. Holomorphic functions and domains of holomorphy in finite and infinite dimensions. Amsterdam/New York/Oxford: North-Holland (1986; Zbl 0586.46040)] that, if the Michael problem holds for certain ``test algebras'' \(\mathcal A,\) then it holds in general. Here, the author provides a general result which yields, as special case, the results contained in the three aforementioned references. Among the fundamental ideas in this nice note, the author shows that for sequentially complete locally convex spaces \(E\) with compactly convergent Schauder basis, if \(\mathcal A\) is a sequentially complete complex commutative locally \(m\)-convex algebra which admits an unbounded complex homomorphism, then there is an unbounded complex homomorphism on the algebra \((\mathcal H(E),\tau_c)\) of holomorphic functions endowed with the compact-open topology. From this, it is evident that the same holds for the subalgebra \((\mathcal H_b(E), \tau_b)\) of entire functions that are bounded on all bounded subsets of \(E\) endowed with the topology of uniform convergence on bounded sets.