Automatic continuity of almost multiplicative linear functionals on Fr'echet algebras

摘要

A linear functional $T$ on a Fr$cute{mathbf{ext{e}}}$chet algebra $(A, (p_n))$ is called extit{almost multiplicative} with respect to the sequence $(p_n)$, if there exists $arepsilongeq0$ such that $|Tab - Ta Tb|leq arepsilon p_n(a) p_n(b)$ for all $n in mathbb{N}$ and for every $a, b in A$. We show that an almost multiplicative linear functional on a Fr$cute{mathbf{ext{e}}}$chet algebra is either multiplicative or it is continuous, and hence every almost multiplicative linear functional on a functionally continuous Fr$cute{mathbf{ext{e}}}$chet algebra is continuous.