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\title{Fixed point property in Hilbert C$^*$-modules \\[0.3cm]}
\author{ Mehrdad Golabi$^1$\thanks{Corresponding author, e-mail: m.golabi@email.kntu.ac.ir}, \,\,\,{ Kourosh Nourouzi$^{2}$
}\\[0.4cm]
{\em $^{1,2}$ Faculty of Mathematics, K. N. Toosi University of}\\
{\em Technology,}\\
{\em P.O. Box 16765-3381, Tehran, Iran.}\\}



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\section{Preliminaries}
Assume $A$ is a C$^*$-algebra and $M$ is a right $A$-module such that $z(ma)=m(za)=(zm)a$ for all 
$z\in \mathbb{C}, a\in A, m\in M$. Besides assume there is a map $\langle .,.\rangle: M\times M \longrightarrow A$ with the following properties:
\begin{enumerate}
	\item 
	$\langle m_1,zm_2+m_3\rangle =z\langle m_1,m_2\rangle +\langle m_1,m_3\rangle$; 
	\item 
	$\langle m_1,m_1\rangle \geq 0$;
	\item 
	$\langle m_1,m_1\rangle =0 \Leftrightarrow m_1=0$;
	\item 
	$\langle m_1,m_2\rangle ^*=\langle m_2,m_1\rangle $;
	\item 
	$\langle m_1,m_2a\rangle =\langle m_1,m_2\rangle a$
\end{enumerate}
for all $m_1,m_2,m_3\in M, \ z\in \mathbb{C}, \ a\in A$. Then we say $M$ is a pre-Hilbert $A$-module. If the norm on $M$ defined by $\lVert m\rVert :=\lVert \langle m,m\rangle \rVert ^
\frac{1}{2} $  is complete, $M_A$ is called Hilbert $A$-module or a Hilbert C$^*$-module over $A$.
For example when $J$ is a closed right ideal of a C$^*$-algebra $A$ then $J$ is a Hilbert $A$-module that the module's product is defined naturally and the $A$-valued innerproduct is defined by 
$\langle a_1,a_2\rangle = a^{*}_{1}a_2$. For another example suppose $\{ M_i \}^{\infty }_{i=1} $ is a countably collection of Hilbert $A$-modules then Hilbert $A$-module 
$\oplus _{i=1}^{\infty }M_i$ is defined by the following:
\[
\oplus_{i=1}^{\infty }M_i=\left \lbrace (m_1,m_2,m_3,\ldots ){\Bigg |}  \forall i\in \mathbb{N}\ \ m_i\in M_i\ ,\ \  \sum_{i=1}^{\infty } \langle m_i,m_i\rangle \text{ is convergent } \right \rbrace
\]
such that $\langle (m_i)^{\infty}_{i=1},(m^{'}_{i})^{\infty }_{i=1} \rangle =\sum_{i=1}^{\infty }\langle 
m_i, m^{'}_{i}\rangle $ and $(m_i)^{\infty }_{i=1}.a=(m_ia)^{\infty }_{i=1}$. We will denote by $H_A$ the Hilbert $A$-module $\oplus_{i=1}^{\infty }M_i $ when $M_i=A$ for all 
$i\in \mathbb{N}$ and call it the standard Hilbert C$^*$-module over $A$.


For a Banach space $X$ we say $X$ has fixed point property (F$\cdot$P$\cdot$P) if for every closed convex bounded $C\subseteq X$ and any nonexpansive mapping
\mbox{ $ T:C\to C$} ($\lVert T(x)-T(y)\rVert \leq \lVert x-y\rVert$) there is an $x\in C$ such that $T(x)=x$.


In \cite{1},\cite{2} there are researches about F$\cdot$P$\cdot$P in C$^*$-algebras and Banach algebras. Recently some research about variational inequality theory in Hilbert C$^*$-modules has been done that is in connection with the fixed point theory and F$\cdot$P$\cdot$P(\cite{3}).



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