Almost n-Multiplicative Maps‎ between‎ ‎Frechet Algebras

For the Fr\'{e}chet algebras (A, (p_k)) and (B, (q_k)) and n \in \mathbb{N}, n\geq ۲, 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 ۰ such thatq_k(Ta_۱a_۲\cdots a_n-Ta_۱Ta_۲\cdots Ta_n)\leq \varepsilon p_k(a_۱) p_k(a_۲)\cdots p_k(a_n),for each k\in \mathbb{N} and a_۱, a_۲, \ldots, a_n\in A. The linear map T is called \textit{weakly almost n-multiplicative}, if there exists \varepsilon\geq ۰ such that for every k\in \mathbb{N} there exists n(k)\in \mathbb{N} withq_k(Ta_۱a_۲\cdots a_n-Ta_۱Ta_۲\cdots Ta_n)\leq \varepsilon p_{n(k)}(a_۱) p_{n(k)}(a_۲)\cdots p_{n(k)}(a_n),for each k \in \mathbb{N} and a_۱, a_۲, \ldots, a_n\in A.The linear map T is called n-multiplicative ifTa_{۱}a_{۲} \cdots a_{n} = Ta_{۱} Ta_{۲} \cdots Ta_{n},for every a_{۱}, a_{۲},\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 ۲. In particular, every continuous linear functional on A is weakly almost n-multiplicative for each n\geq ۲.

Keywords:

multiplicative maps (homomorphisms) , Almost multiplicative maps , automatic continuity , Frechet algebras , Banach algebras