Document Type : Research Paper

Authors

• Taher Ghasemi Honary ¹
• Mashaalah Omidi ²
• AmirHossein Sanatpour ¹

¹ Department of Mathematics, Kharazmi University, Tehran, Iran

² Department of Basic Sciences, Kermanshah University of Technology, Kermanshah, Iran

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