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 $n\in \mathbb{N}$, $n\ge 2$, a linear map $T:A\to B$ is called \textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $\epsilon \ge 0$ such that
$$q_k(T{a}_1{a}_2\cdots a_n-T{a}_{1}T{a}_2\cdots T{a}_n)\le \epsilon p_k(a_1)p_k(a_2)\cdots p_k(a_n),$$
for each $k\in \mathbb{N}$ and $a_1,a_2,\dots ,a_n\in A$. The linear map $T$ is called \textit{weakly almost $n$-multiplicative}, if there exists $\epsilon \ge 0$ such that for every $k\in \mathbb{N}$ there exists $n(k)\in \mathbb{N}$ with
$$q_k(T{a}_1{a}_2\cdots a_n-T{a}_{1}T{a}_2\cdots T{a}_n)\le \epsilon p_{n(k)}(a_1)p_{n(k)}(a_2)\cdots p_{n(k)}(a_n),$$
for each $k\in \mathbb{N}$ and $a_1,a_2,\dots ,a_n\in A$.
The linear map $T$ is called $n$-multiplicative if
$$T{a}_1{a}_2\cdots a_n=T{a}_{1}T{a}_2\cdots T{a}_n,$$
for every $a_1,a_2,\dots ,a_n\in A$.

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 $T:A\to \mathbb{C}$ 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 $T:A\to B$ is a continuous linear map, then under certain conditions $T$ is weakly almost $n$-multiplicative for each $n\ge 2$. In particular, every continuous linear functional on $A$ is weakly almost $n$-multiplicative for each $n\ge 2$.

Keywords

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