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\centerline{\textbf{\Large{Abstract}}}
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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 $.

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 $B$.

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.

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.

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 $\phi\in A''$.

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
 $h:A\to B$ 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.

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.

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This MSc thesis has been prepared based on the following articles:
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\begin{enumerate}
\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.}
 \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.}
\item {A. Zivari Kazempour, A characterization of Jordan homomorphism on Banach algebras, \textit{Chinese J. Math.} (2014), 3 pages.} 
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\noindent\textbf{Keywords:}
Banach algebras, Frechet algebras, $n$-homomorphism, $n$-ring homomorphism, $n$-Jordan homomorphism, Arens product, generalized Jordan homomorphism, stability of functional equations.
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\noindent\textbf{Mathematics Subject Classifications (2010):}
 \lr{17C65}, \lr{39A15}, \lr{39B52}, \lr{39B82}, \lr{39B72}, \lr{46L05}, \lr{46H25}, \lr{47B45}, \lr{47B48}, \lr{47H15}. 
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