Department of Mathematics, Kharazmi University, 1561836314, Tehran, Iran
Abstract
For Fr$acute{mathbf{text{e}}}$chet algebras $(A, (p_n))$ and $(B, (q_n))$, a linear map $T:Arightarrow B$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(Tab - Ta Tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{N}$, $a, b in A$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that for every $k in mathbb{N}$, there exists $n(k) in mathbb{N}$, satisfying the inequality $q_k(Tab - Ta Tb)leq varepsilon p_{n(k)}(a) p_{n(k)}(b),$ for all $a, b in A$.
We investigate the automatic continuity of (weakly) almost multiplicative maps between certain classes of Fr$acute{mathbf{text{e}}}$chet algebras, such as Banach algebras and Fr$acute{mathbf{text{e}}}$chet $Q$-algebras. We also obtain some results on the automatic continuity of dense range (weakly) almost multiplicative maps between Fr$acute{mathbf{text{e}}}$chet algebras.
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