Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68228120170606Almost 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 $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that<br />$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$<br />for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. The linear map $T$ is called textit{weakly almost $n$-multiplicative}, if there exists $varepsilongeq 0$ such that for every $kin mathbb{N}$ there exists $n(k)in mathbb{N}$ with<br />$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_{n(k)}(a_1) p_{n(k)}(a_2)cdots p_{n(k)}(a_n),$$<br />for each $k in mathbb{N}$ and $a_1, a_2, ldots, a_nin A$.<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 $ngeq 2$. In particular, every continuous linear functional on $A$ is weakly almost $n$-multiplicative for each $ngeq 2$.https://ijnaa.semnan.ac.ir/article_2500_05334ad00015c7183c16bcaeeaa21ae5.pdf