# Automatic continuity of almost multiplicative maps between Frechet algebras

Document Type: Research Paper

Authors

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.

Keywords

Main Subjects

### History

• Receive Date: 07 February 2014
• Revise Date: 03 September 2014
• Accept Date: 19 September 2014