Automatic continuity of almost multiplicative maps between Frechet algebras

Abstract

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