Author/Authors :
Ghasemi Honary، Taher نويسنده Kharazmi University , , Omidi، Mashaallah نويسنده Department of Mathematics, Kharazmi University , , Sanatpour، Amir Hossein نويسنده Kharazmi University ,
Abstract :
For Fr$\acute{\mathbf{\text{e}}}$chet algebras $(A, (p_n))$
and $(B, (q_n))$, a linear map $T:A\rightarrow B$ is
\textit{almost multiplicative} with respect to $(p_n)$ and
$(q_n)$, if there exists $\varepsilon\geq 0$ 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 $\varepsilon\geq 0$ 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.
