Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68228120170701Almost n-Multiplicative Maps between Frechet Algebras187195250010.22075/ijnaa.2017.2500ENTaherGhasemi HonaryDepartment of Mathematics, Kharazmi University, Tehran, IranMashaalahOmidiDepartment of Basic Sciences, Kermanshah University of Technology, Kermanshah, IranAmirHosseinSanatpourDepartment of Mathematics, Kharazmi University, Tehran, IranJournal Article20160522For the Fr\'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n \in \mathbb{N}$, $n\geq 2$, a linear map $T:A \rightarrow B$ is called \textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $\varepsilon\geq 0$ such that<br />$$q_k(Ta_1a_2\cdots a_n-Ta_1Ta_2\cdots Ta_n)\leq \varepsilon p_k(a_1) p_k(a_2)\cdots p_k(a_n),$$<br />for each $k\in \mathbb{N}$ and $a_1, a_2, \ldots, a_n\in A$. The linear map $T$ is called \textit{weakly almost $n$-multiplicative}, if there exists $\varepsilon\geq 0$ such that for every $k\in \mathbb{N}$ there exists $n(k)\in \mathbb{N}$ with<br />$$q_k(Ta_1a_2\cdots a_n-Ta_1Ta_2\cdots Ta_n)\leq \varepsilon p_{n(k)}(a_1) p_{n(k)}(a_2)\cdots p_{n(k)}(a_n),$$<br />for each $k \in \mathbb{N}$ and $a_1, a_2, \ldots, a_n\in A$.<br />The linear map $T$ is called $n$-multiplicative if<br />$$Ta_{1}a_{2} \cdots a_{n} = Ta_{1} Ta_{2} \cdots Ta_{n},$$<br />for every $a_{1}, a_{2},\ldots, a_{n} \in A$.<br /><br />In this paper, we investigate automatic continuity of (weakly) almost $n$-multiplicative maps between certain classes of Fr\'{e}chet algebras, including Banach algebras. We show that if $(A, (p_k))$ is a Fr\'{e}chet algebra and $T: A \rightarrow \mathbb{C}$ is a weakly almost $n$-multiplicative linear functional, then either $T$ is $n$-multiplicative, or it is continuous. Moreover, if $(A, (p_k))$ and $(B, (q_k))$ are Fr\'{e}chet algebras and $T:A \rightarrow B$ is a continuous linear map, then under certain conditions $T$ is weakly almost $n$-multiplicative for each $n\geq 2$. In particular, every continuous linear functional on $A$ is weakly almost $n$-multiplicative for each $n\geq 2$.https://ijnaa.semnan.ac.ir/article_2500_05334ad00015c7183c16bcaeeaa21ae5.pdf