On the uniqueness sets of pointwise continuity in topological algebras

Problem's Description

Let $(A, \tau)$ be a complex topological algebra(e.g. a Fréchet algebra) . Let $S(A)$ denote the set of all multiplicative functionals $\psi\colon A\to \mathbb{C}$.

A set $E\subset S(A)$ is called a uniqueness set for $A$ if $$ \psi(f)=0,\, \forall \psi\in E \implies \psi (f)=0,\, \forall \psi \in S(A). $$ The problem is to find more or less general assumptions on $(A,\tau)$ under which the continuity (in $\tau$) of the multiplicative functionals in a uniqueness set $E$ implies the continuity of all functionals in $S(A)$.

An interesting particular case of this problem is when $A\subset C(X)$ where $X$ is a given Hausdorff topological space. In this case we can consider the analog of the above problem by considering only Dirac functionals $\psi_x\colon A\to \mathbb{C}:$ $$ \psi_x(f):=f(x),\, \forall f\in A. $$

This problem is motivated by the Michaels's conjecture on the continuity of multiplicative functionals in Fréchet algebras. It also aims to generalize analogous properties of the Fréchet algebra of holomorphic functions in a complex domain.

1. Book Functional analysis

   • Walter Rudin

2. Book Topological algebras

   • E. Beckenstein,
   • L. Narici,
   • C. Suffel

3. Book Multiplicative functionals on topological algebras

   • T. Husain