Automatic continuity of almost multiplicative linear functionals on Fréchet algebras (Q2812462)

Let \((A,\{p_n\})\) and \((B,\{q_n\})\) be Fréchet algebras. A linear map \(T:(A,\{p_n\})\to (B,\{q_n\})\) is called almost multiplicative if \(q_n(Tab-TaTb)\leqslant \varepsilon p_n(a)p_n(b)\) for all \(n\in\mathbb{N}\), for every \(a,b\in A\) and for some \(\varepsilon >0\), and \(T\) is called weakly almost multiplicative if for each \(k\in\mathbb{N}\) there is \(n(k)\in \mathbb{N}\) such that \(q_n(Tab-TaTb)\leqslant \varepsilon p_{n(k)}(a)p_{n(k)}(b)\) for all \(a,b\in A\) and for some \(\varepsilon >0\). It is shown that if \(T:(A,\{p_n\})\to \mathbb{C}\) is a weakly almost multiplicative linear functional, then \(T\) is multiplicative or continuous. Moreover, if \((A,\{p_n\})\) is a functionally continuous Fréchet algebra and \((B,\{q_n\})\) is a semisimple and commutative Fréchet algebra, then every weakly almost multiplicative map \(T:A\to B\) is automatically continuous.