Automatic continuity of almost multiplicative linear functionals on Fr\'echet algebras
Bull. Korean Math. Soc. 2016 Vol. 53, No. 3, 641-649
https://doi.org/10.4134/BKMS.b140116
Published online May 31, 2016
Taher Ghasemi Honary, Mashaallah Omidi, and Amir Hossein Sanatpour
Kharazmi University, Kharazmi University, Kharazmi University
Abstract : 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
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