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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\centerline{\textbf{\Large{Abstract}}}‎</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\noindent‎ </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎A linear map $h:A\to B$ between algebras $A$ and $B$ is called $n$-homomorphism‎, ‎if $h(a_1\cdots a_n)=h(a_1)\cdots h(a_n)$‎, ‎for every $a_1,\cdots,a_n\in A $‎, ‎and it is called $n$-anti-homomorphism if $h(a_1\cdots a_n)=h(a_n)\cdots h(a_1)$ for all $a_1,\cdots,a_n\in A $‎.</p>
<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"></p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎In 2010‎, ‎T‎. ‎G‎. ‎Honary and H‎. ‎Shayanpour showed that if $A$ and $B$ are Banach algebras‎, ‎$B$ is semisimple and factorizable‎, ‎and $h:A\to B$ is a surjective $n$-homomorphism‎, ‎then $h$ is automatically continuous‎. ‎This result‎, ‎which is an extension of the famous theorem of B‎. ‎Johnson‎, ‎has also been proved by Eshaghi Gordji et al‎. ‎in 2015 without the factorizability condition on $B$‎.</p>
<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"></p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎In this Msc thesis we will investigate conditions under which surjective $n$-homomorphisms and  $n$-anti-homomorphisms between Banach algebras (not necessarily commutative) are continuous‎.</p>
<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"></p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎An additive map $h:A\to B$ between algebras (rings) $A$ and $B$ is called $n$-ring homomorphism if $h(a_1\cdots a_n)=h(a_1)\cdots h(a_n)$ for all $a_1,\cdots,a_n\in A $‎. ‎If $h(a^n)=h(a)^n$ for all $a\in A$‎, ‎then the additive map $h$ is called $n$-Jordan homomorphism‎. ‎We will study $n$-Jordan homomorphism $h$ between Banach algebras $A$ and $B$ and impose conditions on $A$‎, ‎$B$‎, ‎or the Jordan homomorphism $h$ so that $h$ turns out into a homomorphism‎.</p>
<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"></p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎Let $A$ and $B$ be Banach algebras and $h:A\to B$ be a homomorphism‎. ‎Let $A',B',A'',B''$ be the first and second duals of $A$ and $B$‎, ‎respectively‎. ‎Then $A''$ and $B''$ are Banach algebras under the first and second Arens products‎. ‎We will show that $h'':A''\to B''$ is also a homomorphism‎, ‎where $h''(\phi)=\phi\circ h'$‎, ‎$h'(f)=f\circ h$‎, ‎for $f\in B'$ and $\phi\in A''$‎.</p>
<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"></p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎Let‎ $A$ and $B$ be $C^{\ast}$-algebras such that the ‎closed unit ball of ‎$‎‎A$ ‎is ‎compact ‎with ‎respect ‎to ‎the ‎weak ‎topoloy ‎of ‎‎$‎‎A$‎. Let</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> $h:A\to B$ be a Jordan homomorphism which preserves the involution‎, ‎and moreover‎, let ‎$h$ be weakly-norm continuous‎, whenever $A$ is considered with the weak topology‎. ‎Then $h$ is bounded with the norm topologies of $A$ and $B$‎, ‎and moreover $\|h\|\leqslant 1$‎. ‎We will also prove that when $A$ and $B$ are Banach algebras and $h:A\to B$ is a 3-homomorphism‎, ‎then $h'':A''\to B''$ is also a 3-homomorphism‎. ‎However‎, ‎this result is also proved for $n$-homomorphisms‎, ‎whenever $h$ is continuous‎.</p>
<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"></p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎Let $h:A\to B$ be a linear mapping between algebras $A$ and $B$‎. ‎Then $h$ is called a generalized Jordan homomorphism if there exists a Jordan homomorphism $h':A\to B$ such that\\ $h(a^2)=h(a)h'(a)$ for all $a\in A$‎. ‎Similarly‎, ‎$h$ is called a generalized homomorphism if there exists a  homomorphism $h':A\to B$ such that‎ $h(ab)=h(a)h'(b)$ for all $a,b\in A$‎. ‎In 2014‎, ‎Eshaghi Gordji et al‎. ‎studied generalized Jordan homomorphisms between Banach algebras as well as Frechet algebras‎. ‎We will investigate these kinds of homomorphisms‎, ‎in relation with the stability of functional equations‎, ‎following the Hyers-Ulam stability method‎.</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\vspace*{1.5cm}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎This MSc thesis has been prepared based on the following articles‎:</p>
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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{enumerate}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\item  {M‎. ‎Eshaghi Gordji‎, ‎A‎. ‎Jabbari and E‎. ‎Karapinar‎, ‎Automatic continuity of surjective $n$-homomorphisms on Banach algebra‎,  ‎\textit{Bull‎. ‎Iran‎. ‎Math‎. ‎Soc.} {\bf{41}} (2015)‎, ‎no‎. ‎5‎, ‎1207--1211.}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎\item {M‎. ‎Eshaghi Gordji‎, ‎M‎. ‎Bavand Savadkouhi‎, ‎C‎. ‎H‎. ‎Park and J‎. ‎Lee‎, ‎Generalized Jordan homomorphism in Frechet algebras‎, ‎\textit{J‎. ‎Comput‎. ‎Anal‎. ‎Appl.} {\bf 16}  (2014)‎, ‎no‎. ‎2‎, ‎391--401.}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\item {A‎. ‎Zivari Kazempour‎, ‎A characterization of Jordan homomorphism on Banach algebras‎, ‎\textit{Chinese J‎. ‎Math.} (2014)‎, ‎3 pages.}‎ </p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{enumerate}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\noindent\textbf{Keywords:}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎Banach algebras‎, ‎Frechet algebras‎, ‎$n$-homomorphism‎, ‎$n$-ring homomorphism‎, ‎$n$-Jordan homomorphism‎, ‎Arens product‎, ‎generalized Jordan homomorphism‎, ‎stability of functional equations‎.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\vskip 0.4cm‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\noindent\textbf{Mathematics Subject Classifications (2010):}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> ‎\lr{17C65}‎, ‎\lr{39A15}‎, ‎\lr{39B52}‎, ‎\lr{39B82}‎, ‎\lr{39B72}‎, ‎\lr{46L05}‎, ‎\lr{46H25}‎, ‎\lr{47B45}‎, ‎\lr{47B48}‎, ‎\lr{47H15}‎. </p>
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