@article {
author = {Ghasemi Honary, Taher and Omidi, Mashaalah and Sanatpour, AmirHossein},
title = {Almost n-Multiplicative Maps between Frechet Algebras},
journal = {International Journal of Nonlinear Analysis and Applications},
volume = {8},
number = {1},
pages = {187-195},
year = {2017},
publisher = {Semnan University},
issn = {2008-6822},
eissn = {2008-6822},
doi = {10.22075/ijnaa.2017.2500},
abstract = {For 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$$q_k(Ta_1a_2\cdots a_n-Ta_1Ta_2\cdots Ta_n)\leq \varepsilon p_k(a_1) p_k(a_2)\cdots p_k(a_n),$$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$$q_k(Ta_1a_2\cdots a_n-Ta_1Ta_2\cdots Ta_n)\leq \varepsilon p_{n(k)}(a_1) p_{n(k)}(a_2)\cdots p_{n(k)}(a_n),$$for each $k \in \mathbb{N}$ and $a_1, a_2, \ldots, a_n\in A$.The linear map $T$ is called $n$-multiplicative if$$Ta_{1}a_{2} \cdots a_{n} = Ta_{1} Ta_{2} \cdots Ta_{n},$$for every $a_{1}, a_{2},\ldots, a_{n} \in A$.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 $n\geq 2$.},
keywords = {multiplicative maps (homomorphisms),Almost multiplicative maps,automatic continuity,Frechet algebras,Banach algebras},
url = {https://ijnaa.semnan.ac.ir/article_2500.html},
eprint = {https://ijnaa.semnan.ac.ir/article_2500_05334ad00015c7183c16bcaeeaa21ae5.pdf}
}