Bulletin of the
Korean Mathematical Society BKMS

A linear functional $T$ on a Fr$\acute{\mathbf{\text{e}}}$chet algebra $(A, (p_n))$ is called \textit{almost multiplicative} with respect to the sequence $(p_n)$, if there exists $\varepsilon\geq0$ such that $|Tab - Ta Tb|\leq \varepsilon p_n(a) p_n(b)$ for all $n \in \mathbb{N}$ and for every $a, b \in A$. We show that an almost multiplicative linear functional on a Fr$\acute{\mathbf{\text{e}}}$chet algebra is either multiplicative or it is continuous, and hence every almost multiplicative linear functional on a functionally continuous Fr$\acute{\mathbf{\text{e}}}$chet algebra is continuous.

Keywords: multiplicative maps (homomorphisms), almost multiplicative maps, almost multiplicative linear functionals, automatic continuity, Fr$\acute{\text{e}}$chet algebras, $Q$-algebras

MSC numbers: Primary 46H40, 47A10; Secondary 46H05, 46J05, 47B33