\newtheorem{theo}{\siah\Large}[section] >\newtheorem{conseq}{\siah\Large}[section] >\newtheorem{prop}{\siah\Large𥐤}[section] >\newtheorem{defn}{\siah\Large}[section] >\newtheorem{lemm}{\siah\Large}[section] >\newtheorem{exam}{\siah\Large}[section] >\newtheorem{rmark}{\siah\Large}[section] < >\newcommand{\real}{I\!\!R} >\newcommand{\p}{I\!\!P} >\newcommand{\fel}{\longrightarrow} >\newcommand{\La}{\frak{L}} >\newcommand{\oo}{\circ} < >\def\t{\textstyle} >\newcommand{\ron}{\frac{\rond}{\rond x^i}} >\newcommand{\z}{\times} >\newcommand{\fe}{\varphi} >\newcommand{\zir}{\subseteq} >\newcommand{\rond}{\partial} >\newcommand{\baar}{\overline} >\newcommand{\ro}{\rho} >\newcommand{\ba}{(E,\pi,M)} >\newcommand{\jj}{j^1_ps} >\renewcommand{\j}{J^1E} >\newcommand{\fa}{꤭} >\newcounter{fnote}[section] >\newcommand{\fnote}[1]{\setcounter{footnote}{\value{fnote}}\footnote{#1}\addtocounter{fnote}{1}} >%\newcommand{\fnotemark}{\setcounter{footnote}{\value{fnote}}\footnotemark\addtocounter{fnote}{1}} >%\newcommand{\fnotetext}[1]{\setcounter{footnote}{\value{fnote}}\footnotetext{#1}} >%\setcounter{page}{1} >%\pagestyle{headings} >%----------------------------------------------------------------------------------------------- >\newcommand{\graphicx}[2]{ >\input{epsf} >\epsfxsize=#1 >\begin{figure}[htb] >\centerline{\epsffile{#2}} >\end{figure} >} >%--------------------------------------------------------------------------------------------- >%______________________________________________________________________________________________ >%______________________________ړꑤڕ_________________________ > > >%\include{fcover} > >\pagenumbering{adad} %T0 create page with roman numbe > >%\include{ackonowledge} > >%\include{fabstract} < < < >\thispagestyle{empty} >\pagenumbering{farsifoo} >%\setcounter{page}{1} >\renewcommand{\baselinestretch}{1.2} < \thispagestyle{empty} < < \pagestyle{headings} >%ړڢړڢړڬڐ\pagestyle{headings}ڢ >%ړڢڐtableofcontents >%ړڬڐڟڢڕڐmaketitleڢ < < < < {\large \ph 1) M. Adib, A. Riazi and J. Bracic; {\em Quasi-multipliers of the dual of a Banach algebra}, Banach j. Math. Anal, >{$\bf $} (2010).} < >{\large \ph 2) M. Adib, A. Riazi and L. A. Khan; {\em Quasi-multipliers on F-algebras}, Abstract and Applied Analysis, (2011).} >% {$\bf 3$} (2009), 121-132} < < >{\large \ph 3) A. Riazi, M. Adib; {\em $\varphi-$Multipliers on Banach algebras without order}, Int. Journal of Math. Analysis , >{$\bf 3$} (2009), 121-132} < >{\large \ph 4) M. Adib, A. Riazi; {\em Double $\varphi-$multipliers and some of their properties}, Int. Math. Forum, > {$\bf 3$} (2010), 2497-2504} < < >{\large \ph 5) M. Adib and A. Riazi ; {\em $\varphi_{m}-$Multipliers on topological modules }, submitted. >%{$\bf 3$} < >{\large \ph 6) M. Adib, A. Riazi and J. Bracic; {\em Quasi-multipliers on weak Arens regular Banach algebras }, International Conference on Mathematical Analysis, Thailand, >{$\bf $} (2010)} < >{\large \ph 7) M. Adib, A. Riazi; {\em Quasi-multipliers on locally bounde algebras }, 41st Annual Iranian Mathematics Conference, Urmia, >{$\bf $} (2010)} < < < < < %pagestyle{headings} >%ړڢړڢړڬڐ\pagestyle{headings}ڢ >%ړڢڐtableofcontents %{\large \ph Akbar-Zadeh, H.; {\em Generalized Einstein manifolds}, Journal of Geometry and Physics, Elsevier Science B.V., >%{$\bf 17$} (1995), 342-380} %\pagestyle{headings} %ړڬڐڟڢڕڐmaketitleڢ <ʕ 0$ڢړگ$$.|f_{\alpha}(x)|\leq C,\hspace{.3cm} (\alpha\in J, x\in B)$$ =,f\in A^{*}$ڗڪڍ𑪖fnotecanonical embeddingڐ$A$ړ$A^{**}$ = \hspace{1.7cm}< a\cdot f,b> =< f, ba>$$$\vspace{-1.2cm}$ <$$< G\cdot f,a>=< G,f\cdot a> \hspace{1cm}=< F,a\cdot f>$$$\vspace{-1.2cm}$ <$$=\hspace{.7cm}=$$ <$\circ', \circ$ړڮڐڢڍڪ <$A$ڍڐڮڐڢڍړړړڐ$F,G\in A^{**}$$$F\circ G=F\circ' G.$$ <ړڐ$a\in A$$F\in A^{**}$$$a\circ F=\pi(a)\circ 'F,\hspace{.3cm}F\circ a=F\circ '\pi(a).$$$\vspace*{-2cm}$ ==$ \graphicx{14cm}{nahid2.eps} \begin{eqnarray*} >T\circ S(xy)&=&T(\varphi(x)S(y))=\varphi(S(y))T(\varphi(x))=S(\varphi(y))\varphi(T(x))\\ >&=&S(\varphi(y)T(x))=S\circ T (yx)=S\circ T (xy)\\ >\end{eqnarray*} <ڢ$M_{\varphi}(A)$ڛڐ <ڢڐڢ$M_{\varphi}(A)$ꑢڐ꤭$T\in M_{\varphi}(A)$ړگړڐ <$S\in M_{\varphi}(A)$ڢړ$.T\circ S=0$ڝ𑪖$\varphi$$-\varphi$ڐ$T\circ \varphi=0$ڢ < <$$T(xy)=\varphi(x)T(y)=\varphi(\varphi(x))T(y)=T(\varphi(x))\varphi(y)=0.$$ڝ$AA=A$$.T=0$$\Box$ %\begin{eqnarray*} >%.T^{-1}(x)\varphi^{-1}(y)&=&(T^{-1} \circ T) [T^{-1}(x)\varphi^{-1}(y)]\\ >%&=&T^{-1}[T\circ T^{-1}(x)\hspace{.2cm}\varphi\circ \varphi^{-1}(y)]\\ >%&=&T^{-1}(xy)\\ >%\end{eqnarray*} <ڐڢ$.\varphi^{-1}(x)T^{-1}(y)=T^{-1}(xy)$$.T^{-1}\in M_{\varphi^{-1}}(A)$ < <ڐڢڐڐ$-\varphi$ڮڤڛړڐڍڢ$n=2$ڐڐ \begin{equation}\label{el1} >T((x+y)^{2})=\varphi(x+y)T(x+y)=\varphi(x)T(x)+\varphi(x)T(y)\vspace{-2cm} >\end{equation} < < $$+\varphi(y)T(x)+\varphi(y)T(y)$$ <گڢ >\begin{equation}\label{el2} >T((x+y)^{2})=T(x^{2}+2xy+y^{2})=\varphi(x)T(x)+2T(xy)+\varphi(y)T(y) >\end{equation} \begin{equation}\label{el3} >.2T(xy)=\varphi(x)T(y)+\varphi(y)T(x) >\end{equation} %\begin{equation} < <ڐڛړڛ$A$䓑ھrefel3׽ڊڪړڐڢ$\{z_{n}\}^{\nfty}_{n=1}^{\infty}\subset A$ >\begin{equation} > 2T(xyz_{n})&=& \varphi(y)T(xz_{n})+\varphi(xz_{n})T(y)\vspace{-2cm} >\end{equation} >% < >\begin{eqnarray*} >&=& \varphi(y)[\varphi(x)T(z_{n})+\varphi(z_{n})T(x)]/2\\ >&+& \varphi(xz_{n})T(y).\\ >\end{eqnarray*} <ڢڡڢ < >\begin{equation}\label{el5} > 2T(xyz_{n})&=& [\varphi(y)\varphi(x)T(z_{n})+\varphi(y)\varphi(z_{n})T(x)\vspace{-2cm} >\end{equation} < <$$+2\varphi(x)\varphi(z_{n})T(y)]/2$$ <ڐꑢڐھrefel3׽ڊڢ < >\begin{equation}\label{el6} > 2T(xyz_{n})&=&[\varphi(x)\varphi(y)T(z_{n})+\varphi(x)\varphi(z_{n})T(y)\vspace{-2cm} >\end{equation} < <$$+2\varphi(y)\varphi(z_{n})T(x)]/2.$$ <ڤھrefel5׽ھrefel6׽ڊړڐ$x,y,z_{n}\in A$ړڍ <$$.\lim_{n\rightarrow \infty}\varphi(x)\varphi(z_{n})T(y)=\lim _{n\rightarrow \infty}\varphi(y)\varphi(z_{n})T(x)$$ <ڐڐ$A$ڛڛꑢړ$\varphi$ڢ$A$ ڝڐڊ$\varphi(x)T(y)=T(x)\varphi(y)$$T$$-\varphi$ڐ$\Box$ \begin{eqnarray*} >.F'\circ G'&=&\mu^{**}(F)\circ \mu^{**}(G)=\mu^{**}(F\circ G)=\mu^{**}(F\circ 'G)\\ >&=&\mu^{**}(F)\circ ' \mu^{**}(G)=F'\circ 'G'\\ >\end{eqnarray*} <ڢ$M_{\varphi}(A)$ڍڐ <$(2)$ڛړڕڪڗڐprotectedciteRU꤭$T\in M_{\varphi}(A)$ <$\{e_{\alpha}:\alpha \in I\}$ڗڢ$A$$\mu$𑪖ڗڪڢ$(1)$ړڨڢ <$T=\lim_{\alpha} _{T(e_{\alpha})}\varphi$ڢ$.\overline{\mu(A)}=M_{\varphi}(A)$ړڗړ\ref{OO}ڊ <$M_{\varphi}(A)$ڕڐ \begin{eqnarray*} >(\varphi(a)-\lambda)\lim_{n}T(c_{n})&=&\lim_{n}( \varphi(a)-\lambda)T(c_{n})=\lim_{n}\varphi(a-\lambda)T(c_{n})\\ >&=&\lim_{n} T(a-\lambda)\varphi(c_{n})=(T(a)-\lambda)\lim_{n} \varphi(c_{n})\\ >&=&1\\ >\end{eqnarray*} <ڢ$\lim _{n}T(c_{n})(\varphi(a)-\lambda)$=1ڢ$.\lambda\notin \sigma(\varphi(a))$ڐڐ$\sigma(\varphi(a))=\sigma(a)$ڢ$.\lambda\notin \sigma(a)$ <꤭$.\lambda\notin \sigma(a)$$b\in A$ڢړگ$(a-\lambda)b=1$ړ$$.(T(a)-\lambda)\varphi(b)=T(a-\lambda)\varphi(b)=T((a-\lambda)b)=1$$ <$.\varphi(b)(T(a)-\lambda)=1$ړ$.\lambda\notin \sigma(T(a))$$.\sigma(T(a))=\sigma(a)$ < < \neq 0$꤭$\pi:A\rightarrow A^{**}$𑪖ړڢ < >\begin{eqnarray*} ><\varphi^{**}(\pi(x)),f>&=&<\pi(x),\varphi^{*}(f)>=<\varphi^{*}(f),x>\\ >&=&=\neq 0\\ >\end{eqnarray*} <ڢ$.\varphi^{**}(\pi(x))\neq 0$$\varphi^{**}$𑪖ڬڐ <꤭$.m,n\in A^{**}$ړڐ$f\in A^{*}$$x,y\in A$ <ڢ$$=<\varphi^{**}(m)\circ T^{**}(n),f>.$$ <ڐꑢڐڗڮڍڢ <$$==$$ < >\begin{eqnarray*} >&=&<\varphi^{**}(n)\cdot f,T(x)>=<\varphi^{**}(n),f\cdot T(x)>\\ >&=&\\ >\end{eqnarray*} < < >\begin{equation}\label{el7} ><\varphi^{*}(f\cdot T(x)),y>==. >\end{equation} <گڢ <$$<\varphi^{**}(m)\circ T^{**}(n),f>=<\varphi^{**}(m),T^{**}(n)\cdot f>=$$ < < >\begin{eqnarray*} ><\varphi^{*}(T^{**}(n)\cdot f),x>&=&=\\ >&=&\\ >\end{eqnarray*} < < >\begin{equation}\label{el8} >== >\end{equation} < <ڐڐ$T$$-\varphi$ڐړھrefel7׽ھrefel8׽ڡڢ$$=<\varphi^{**}(m)\circ T^{**}(n),f>.$$ <ڐڢڐ$\varphi$ڤ$A$ړڍ$\varphi^{**}$ڤ$A^{**}$ڐڢ$T^{**}$$-\varphi^{**}$ڤ$A^{**}$ڐ \begin{equation}\label{el9} >.\lim_{n\rightarrow \infty}||T^{**}(T^{**}(u_{n}))-T^{**}(\varphi^{**}(e))||=0 >\end{equation} < <ڝ$\varphi^{**}$ڡ$T^{**}$$-\varphi^{**}$ڤ$A^{**}$ڐړ$m\in A^{**}$ڢ <$(10)$ <$$\varphi^{**}(m)\circ (T^{**}(T^{**}(u_{n}))-T^{**}(\varphi^{**}(e)))=\varphi^{**}(m)\circ T^{**}(T^{**}(u_{n}))-\varphi^{**}(m)\circ T^{**}(\varphi^{**}(e))\vspace{-1cm}$$ < >\begin{eqnarray*} >&=& T^{**}(m)\circ \varphi^{**}(T^{**}(u_{n}))-T^{**}(m)\circ \varphi^{**}(\varphi^{**}(e))\\ >&=&T^{**}(m)\circ \varphi^{**}(T^{**}(u_{n}))-T^{**}(m)\circ \varphi^{**}(e)\\ >\end{eqnarray*} <$S_{n}$ڤړڮڥڗ <$$S_{n}:T^{**}(A^{**})\rightarrow T^{**}(A^{**})\vspace{-1cm}$$ <$$S_{n}(T^{**}(m))=T^{**}(m)\circ \varphi^{**}(T^{**}(u_{n})),\hspace*{.5cm}m\in A^{**}$$ <$S_{n}$ڡڐڟڐڍ$e$ڤڢ$A^{**}$ڐ$T^{**}(m)\circ \varphi^{**}(e)=T^{**}(m\circ e)=T^{**}(m)$ <ڢړڐꑢڐھrefel9׽$(10)$ڡڢ < >\begin{eqnarray*} >||S_{n}(T^{**}(m))-T^{**}(m)||&=&||T^{**}(m)\circ \varphi^{**}(T^{**}(u_{n}))-T^{**}(m)\circ \varphi^{**}(e)||\\ >&=&||\varphi^{**}(m)\circ (T^{**}(T^{**}(u_{n}))-T^{**}(\varphi^{**}(e)))||\\ >&\leq & ||\varphi^{**}(m)||\hspace{.1cm}||T^{**}(T^{**}(u_{n}))-T^{**}(\varphi^{**}(e))||\rightarrow 0\\ >\end{eqnarray*} <ڐ$n\rightarrow \infty$ڍ$$.\sup _{||m||\leq 1}||S_{n}(T^{**}(m))-T^{**}(m)||\rightarrow 0$$ <ڐꑢڐ\ref{lemm2}ړڐڛڗ㢐$n\in \Bbb{N}$$S_{n}$ڕڢڕ <ړڐ$m\in A^{**}$$m'\in A^{**}$ڐړ < < >\begin{eqnarray*} >T^{**}(m)&=&S_{n}(T^{**}(m'))=T^{**}(m')\circ \varphi^{**}(T^{**}(u_{n}))\\ >&=&\varphi^{**}(m')\circ T^{**}(T^{**}(u_{n}))=T^{**}(\varphi^{**}(m')\circ T^{**}(u_{n}))\\ >&=&T^{**2}(m'\circ u_{n})\\ >\end{eqnarray*} <ڢ$.T^{**}(A^{**})\subseteq T^{**2}(A^{**})$ړ$.T^{**2}(A^{**})\subseteq T^{**}(A^{**})$$.T^{**}(A^{**})=T^{**2}(A^{**})$ <ڐڐړ$T$ړڐړڐꑢڐڗ$1.10$ڐprotectedciteCO׊$T^{**2}(A^{**})$ړڐڟڝ <$T^{**2}(A^{**})=(T^{2})^{**}(A^{**})$ړڐꑢڐ$1.10$protectedciteCO׊$T^{2}(A)$ڢ$A$ړڐړڐڝړ <$T^{2}(A)$ڢ$T(A)$ڡڢ$.T^{2}(A)=T(A)$ <$(2)\Rightarrow (3)$꤭$.T^{2}(A)=T(A)$ړڗ$\{v_{\alpha}\}_{\alpha \in I }$ڢ$A$ڐړگ <$.T(e_{\alpha})=T^{2}(v_{\alpha})$ڐڗ <$\{t_{\alpha}\}_{\alpha}=\{\varphi(T(v_{\alpha}))\}_{\alpha}$ڗڢ$T(A)$ڐ <ڐڐ$T^{2}(A)$ړڐړڐꑢڐ𑪖ړڗ$\{\varphi(T(v_{\alpha}))\}_{\alpha}$ڐ <ړ <$a\in A$ڢ$$T(ae_{\alpha})=\varphi(a)T(e_{\alpha})=\varphi(a)T^{2}(v_{\alpha})=T(a)\varphi(T(v_{\alpha}))$$ <ڢ$$.T(a)=\lim _{\alpha}T(ae_{\alpha})=\lim _{\alpha}T(a)\varphi(T(v_{\alpha}))=\lim _{\alpha} T(a) t_{\alpha}$$ <$.T(a)=\lim _{\alpha}t_{\alpha}T(a)$ڟڐڐ$\varphi$ړ$T$ڛڪ$$.\{t_{\alpha}\}_{\alpha}=\{\varphi(T(v_{\alpha}))\}_{\alpha}=\{T(\varphi(v_{\alpha}))\}_{\alpha}\subseteq T(A)$$ <$\{t_{\alpha}\}$ڗڢ$T(A)$ڐ <$(3)\Rightarrow (1)$꤭$T(A)$ڗڢړڢڐڬړڗړڗprotectedciteHR׊$AA=A$$.T(A)T(A)=T(A)$ <ڟڐڕ$\varphi$ڢ < < >\begin{eqnarray*} >T^{2}(A)&=&T(T(AA))=T(T(A)\varphi(A))=\varphi(T(A))T(\varphi(A))\\ >&=&\varphi(T(A))\varphi(T(A))=\varphi(T(A))\\ >&=&T(\varphi(A))=T(A).\\ >\end{eqnarray*} < < < %\begin{eqnarray*} >%S(a)&=&S(\lim e_{\alpha}a)=\lim S(e_{i})\varphi(a)=\lim S(e_{\alpha})\varphi(\varphi(a))\\ >%&=&\lim S(e_{\alpha}\varphi(a))=S(\lim e_{\alpha}\varphi(a))=S(\varphi(a))\\ >%&=&(S\circ \varphi)(a)=0\\ >%\end{eqnarray*} <$||.||_{\varphi}$ڤ$M_{\varphi}^{l}(A)$ڐړ$M_{\varphi}^{l}(A)$ڛڐڟڢ$M_{\varphi}^{l}(A)$ڐ <꤭$\{S_{n}\}_{n}$ڢڢ$M_{\varphi}^{l}(A)$ړڍړ$x\in A$ڢ$\{(S_{n}\circ \varphi)(x)\}_{n}$ڢڢ$A$ڐ <𑪖$S:A\rightarrow A$ڤړڮ$S(x)=\lim (S_{n}\circ \varphi)(x)$ڗڢڐڬ$||S_{n}-S||_{\varphi}\rightarrow 0$ڢ < >%\begin{eqnarray*} >%S(ab)&=&\lim (S_{n}\circ \varphi)(ab)=\lim S_{n}(\varphi(a)\varphi(b))\\ >%&=&\lim S_{n}(\varphi(a))\varphi(\varphi(b))=S(a)\varphi(b).\\ >%\end{eqnarray*} <$.S\in M_{\varphi}^{l}(A)$ < < =$$ <ڐꑢڐprotectedciteDS$4$$(8.5)$ړڐ$x^{*}\in (L_{1}(G))^{*}$$K\in L_{\infty}(G)$ڢړگ$$.=\int _{G} f(t)K(t^{-1})d\lambda \hspace{.5cm}(f\in L_{1}(G))$$ < >\begin{eqnarray*} >&=&\int _{G} (T(f)*\varphi(g))(t)\hspace{.1cm}K(t^{-1})d\lambda=\widehat{(T(f)*\varphi(g))}(K)\hspace{2cm}(*)\\ >&=&[H_{T}\cdot\widehat{\varphi(f)}\widehat{\varphi(g)}](K)=[\widehat{\varphi(f)}\widehat{T(g)}](K)\\ >\end{eqnarray*} <گڢ < >\begin{eqnarray*} >&=&\int_{G}(\varphi(f)*T(g))(t)\hspace{.1cm}K(t^{-1})d\lambda =\widehat{(\varphi(f)*T(g))}(K)\hspace{2cm}(**)\\ >&=&[\widehat{\varphi(f)}\widehat{T(g)}] (K)\\ >\end{eqnarray*} < <$(**),(*)$ڡڢ$.\varphi(f)*T(g)=T(f)*\varphi(g)$ 0$ڗ$f\in L_{1}(G)$ <ڢړگړڐ$1\leq i\leq n$$\widehat{f}(\gamma_{i})=1$$.||\widehat{f}||<1+\epsilon$ڐڝړړ$\varphi$ <ڢ$\{f_{m}\}_{m}\subseteq L_{1}(G)$ڢړگ$.\lim_{m}\varphi(f_{m})=f$ړ$(\varphi(\lim_{m}f_{m}))^{\hat{}}(\gamma_{i})=1$ <$.\lim_{m}||\hat{\varphi(f_{m})}||<1+\epsilon$ >\begin{eqnarray*} >|\Sigma_{i=1}^{n}c_{i}\hat{\mu_{T}}(\gamma_{i})|&=&|\Sigma_{i=1}^{n}c_{i}(\varphi(\lim _{m}f_{m}))^{\hat{}}(\gamma_{i})\hat{\mu_{T}}(\gamma_{i})|=|\Sigma_{i=1}^{n}c_{i}(\varphi(\lim_{m} f_{m})*\mu_{T})^{\hat{}}(\gamma_{i})|\\ >&=&|\Sigma_{i=1}^{n}c_{i}(T(\lim f_{m}))^{\hat{}}(\gamma_{i})|=\lim_{m}|\Sigma_{i=1}^{n} c_{i}(T(f_{m}))^{\hat{}}(\gamma_{i})|\\ >&=&\lim _{m}|\int\Sigma_{i=1}^{n} c_{i}\gamma_{i}^{-1}(t)Tf_{m}(t)d\lambda|\leq \lim_{m}||Tf_{m}||_{1}||\Sigma_{1}^{n}c_{i}\gamma_{i}^{-1}||_{\infty}\\ >&\leq & \lim _{m}||T||\hspace{.1cm}||\varphi(f_{m})||_{1}||\Sigma_{i=1}^{n}c_{i}\gamma_{i}^{-1}||_{\infty}\\ >&=&\lim_{m}||T||\hspace{.1cm}||(\varphi(f_{m}))^{\hat{}}||\hspace{.1cm}||\Sigma_{i=1}^{n}c_{i}\gamma_{i}^{-1}||_{\infty}\\ >&\leq &||T||(1+\epsilon)||\hspace{.1cm}||\Sigma_{i=1}^{n}c_{i}\gamma_{i}^{-1}||_{\infty}\\ >\end{eqnarray*} <$|\Sigma_{i=1}^{n}c_{i}\hat{\mu_{T}}(\gamma_{i})|\leq |\Sigma_{i=1}^{n}c_{i}\gamma_{i}^{-1}||_{\infty}$ڐ$4.2$protectedciteGA׊ <$.||\mu_{T}||\leq ||T||$ڟړڗړڐ$||T||\leq ||\mu_{T}||$ڟړ줐ڐ < \begin{eqnarray*} >.(a*T)(b\cdot x)&=&T(b\cdot x\cdot a)=b\cdot T(\varphi(x\cdot a))=b\cdot T(\varphi(x)\cdot a)\\ >&=&b\cdot (a*T)(\varphi(x))\\ >\end{eqnarray*} <ڝ$X$$-A$ڢگ$T$ڡڐ$M_{\varphi_{m}}^{l}(X,Y)$$-A$ڝڐ$\Box$ %\begin{eqnarray*} >%(T_{1}\circ T_{2})(a\cdot x)&=&T_{1}(T_{2}(a\cdot x))=T_{1}(a\cdot T_{2}(\varphi(x)))\\ >%&=&a\cdot T_{1}(\varphi(T_{2}(\varphi(x)))=a\cdot (T_{1}\circ T_{2})(\varphi(\varphi(x)))\\ >%&=&a\cdot (T_{1}\circ T_{2})(\varphi(x)). >%\end{eqnarray*} <$T_{1}\circ T_{2}\in M_{\varphi_{m}}^{l}(X,X)$$M_{\varphi_{m}}^{l}(X,X)$ڛڐ \begin{eqnarray*} >(a*T_{\alpha}-a*T)(\varphi(b))&=&T_{\alpha}(\varphi(b)\cdot a)-T(\varphi(b)\cdot a)\in(T_{\alpha}-T)(\varphi(D)\cdot a)\\ >&=& (T_{\alpha}-T)(D\cdot a)\in V\\ >\end{eqnarray*} <ڢ$.a*T_{\alpha}\rightarrow ^{u_{\varphi}}a*T$$(M_{\varphi_{m}}^{l}(A,X),u_{\varphi})$ڗڝڐړڐ$(M_{\varphi_{m}}^{l}(A,X),s_{\varphi})$ڗڝڐ \begin{eqnarray*} >_{\{T(\varphi(e_{\lambda}))\}_{\lambda}} \varphi(\varphi(a))-T(\varphi(a))&=&T(\varphi(e_{\lambda}))\cdot \varphi(\varphi(a))-T(\varphi(a))\\ >&=&T(e_{\lambda}\varphi(a))-T(\varphi(a))\in V\\ >\end{eqnarray*} <$._{\{T(\varphi(e_{\lambda}))\}_{\lambda} }\varphi\rightarrow^{s_{\varphi}} T$ \begin{eqnarray*} ><(F\cdot f)\cdot G,a>&=&<\pi(a),(F\cdot f)\cdot G>=\\ >&=&<(G\circ' \pi(a))\circ F,f>=\\ >&=&<\pi(a)\circ F,f\cdot G>=\\ >\end{eqnarray*} <ڢ$A$ڍڮڐ \begin{eqnarray*} ><(F\cdot f)\cdot G,a>&=&=\\ >&=&=\\ >&=&<\pi(a)\circ F,f\cdot G>=<\pi(a),F\cdot (f\cdot G)>\\ >&=&\\ >\end{eqnarray*} <$A$ڍڮڐ \begin{eqnarray*} >&=&==<\pi(e),(G\cdot f)\cdot F>\\ >&=&<\pi(e),G\cdot (f\cdot F)>=<\pi(e)\circ G,f\cdot F>=\\ >&=&\\ >\end{eqnarray*} <$A$ڍڐ \begin{equation}\label{els} >m(F\cdot f,G)=F\cdot m(f,G)\hspace{1cm}, \hspace{1cm}m(f,G\circ F)=m(f,G)\cdot F >\end{equation} <𑪖ڢڡ$m':A^{**}\times A^{*}\rightarrow A^{*}$ڪڮڝڐ$A^{*}$ڪړڐ$f\in A^{*}$$F,G\in A^{**}$ڤڥړ줐ړ <$$m'(F\circ G,f)=F\cdot m'(G,f)\hspace{1cm}, \hspace{1cm}m'(G,f\cdot F)=m'(G,f)\cdot F$$ \begin{equation}\label{ele} >.m(F\cdot f,G\circ H)=F\cdot m(f,G)\cdot H >\end{equation} \begin{eqnarray*} >(H* m)(F\cdot f,G)=m((F\cdot f)\cdot H,G)=m(F\cdot (f\cdot H),G)\\ >=F\cdot m(f\cdot H,G)=F\cdot (H* m)(f,G) >\end{eqnarray*} < < >\begin{eqnarray*} >(H* m)(f,G\circ F)=m(f\cdot H,G\circ F)=(H* m)(f,G)\cdot F >\end{eqnarray*} <ڝ$m$ڡڐ$H* m$𑪗ڡڐڢ$.H* m\in QM_{r}(A^{*})$ڐ <ڢ$.m* H\in QM_{r}(A^{*})$ ===$$ <ڢ$.T=R_{T^{*}(E)}$ 0$ڢړ$f\in A^{*}$ڢړ <$||f||\leq 1$$.||T||-\epsilon<||Tf||$ڢڐڬ$$||\rho_{T}||\geq ||\rho_{T}(f,E)||=||Tf||>||T||-\epsilon$$ <ڢ$\rho$گڐ <ڐڢڐ$A^{**}$ړڍ$\rho$ڕ꤭$E$ڐ$A^{**}$ <$m\in QM_{r}(A^{*})$ڢړ𑪖$T:A^{*}\rightarrow A^{*}$ړڮ$Tf=m(f,E )$ڮڤڐ$A^{*}$ڤڗ <ڍړڐ$f\in A^{*}$$F\in A^{**}$ڗ$$\rho_{T}(f,F)=(Tf)\cdot F=m(f,E)\cdot F=m(f,E\circ F)=m(f,F)$$ <줐ڐ𑪖$\rho$ڕڐ \begin{eqnarray*} >.\xi(m_{1}\circ_{\rho}m_{2})(F,f)&=&(\lambda(L_{{(T_{2}T_{1})}^{*}(E)}))(F,f)=F\cdot ((T_{2}T_{1})^{*}(E)\cdot f)\\ >&=&F\cdot(T^{*}_{1}(T_{2}^{*}(E))\cdot f)\\ >\end{eqnarray*} <$T_{1}$ڮڤڐ$A^{*}$ڐ$T_{1}^{*}$ڮڝڤ$A^{**}$ڡړڢ >\begin{eqnarray*} >.(\xi(m_{1})\circ _{\lambda} \xi(m_{2}))(F,f)&=&(\lambda(L_{T_{1}^{*}(E)})\circ_{\lambda }\lambda(L_{T_{2}^{*}(E)}))(F,f)=\lambda(L_{T_{1}^{*}(E)}L_{T_{2}^{*}(E)})(F,f)\\ >&=&F\cdot ((T_{1}^{*}(E)\circ T_{2}^{*}(E))\cdot f)=F\cdot (T_{1}^{*}(E\circ T_{2}^{*}(E))\cdot f)\\ >&=&F\cdot (T_{1}^{*}(T_{2}^{*}(E))\cdot f)\\ >\end{eqnarray*} <$\xi$ڐ <ڢ$\xi$ړڐ꤭$m\in QM_{r}(A^{*})$ړ$\xi(m)=0$گڗ$T\in M_{r}(A^{*})$ڢړگ$.\lambda(L_{T^{*}(E)})=0$ <ڐړړ$\lambda$ڪړڐ$f\in A^{*}$$.T^{*}(E)\cdot f=0$ڝ$A^{*}$ڐڝڗڕڐ$.T^{*}(E)=0$ړ <ڐ$F\in A^{**}$$T^{*}(F)=T^{*}(E\circ F)=T^{*}(E)\circ F=0$$.m=\rho(T)=0$ <ڐڢ$\xi $ڕڐ꤭$.m'\in QM_{l}(A^{*})$ڐڕړ$\lambda$$S\in M_{l}(A^{*})$ڢړگ$m'=\lambda(S)$ڗ <$T=R_{S^{*}(E)}\in M_{r}(A^{*})$ړڢڪ$S=L_{T^{*}(E)}$$.m'=\lambda(S)=\lambda(L_{T^{*}(E)})$ \begin{eqnarray*} >(\psi(H_{1})\circ _{\rho}\psi(H_{2}))(f,F)&=&\rho_{T_{2}T_{1}}(f,F)=T_{2}(T_{1}(f))\circ F=T_{1}f\cdot (H_{2}\circ F)\\ >&=&f\cdot (H_{1}\circ H_{2}\circ F)=\psi(H_{1}\circ H_{2})(f,F)\\ >\end{eqnarray*} <$\psi$ڐڟ꤭ړ$H\in A^{**}$$.\psi(H)=0$ڐړړ𑪖$\rho$$.R_{H}=0$ړڐ$f\in A^{*}$$.f\cdot H=0$ڝ$A^{*}$ <گڤڗڕڐ$H=0$ڢ$\psi$ړڐړڐڕړ꤭$.m\in QM_{r}(A^{*})$$T\in M_{r}(A^{*})$ڢ <ړ$$m=\rho_{T}=\rho_{R_{T^{*}(E)}}=\psi(T^{*}(E))$$ړ$\pdi$ڕڐ \begin{eqnarray*} >||m_{\alpha}(f,F)-m(f,F)||&=&||m_{\alpha}(f,G\circ H)-m(f,G\circ H)||\\ >&=&||(m_{\alpha}*G)(f,H)-(m*G)(f,H)||\rightarrow 0\\ >\end{eqnarray*} <$\{m_{\alpha}\}$ڢڗ$\gamma$ړ$m$ڐړ$.\gamma\subseteq \beta$ \begin{equation}\label{el88} >.||T_{F}^{\alpha}-T_{F}||\rightarrow 0 >\end{equation} <ڝ$\gamma\subseteq \beta$ڗ$\{m_{\alpha}\}$ <ڗ$\gamma$ڐړڗړ$(1)$ڊ$(QM_{r}(A^{*}),\gamma)$ڐڢ$m\in QM_{r}(A^{*})$ڢړگ >%\begin{equation} >% >%\end{equation} <$$$$\lim _{\alpha}m_{\alpha}(f,F)=m(f,F) ,\hspace{.2cm} f\in A^{*},F\in A^{**} < <ڐ$G\in A^{**}$ڢ >\begin{eqnarray*}\label{el77} >\rho_{T_{F}}(f,G)&=&\lim _{\alpha}\rho_{T_{F}^{\alpha}}(f,G)=\lim_{\alpha}(m_{\alpha}* F)(f,G)=\lim _{\alpha}m_{\alpha}(f,F\circ G)\\ >&=&m(f,F\circ G)=(m* F)(f,G)\\ >\end{eqnarray} < <ڢړڐꑢڐ(\ref{el88})(\ref{el77})$$.||m_{\alpha}* F-m*F||=||\rho_{T_{F}^{\alpha}}-\rho_{T_{F}}||=||T_{F}^{\alpha}-T_{F}||\rightarrow 0$$ <$m_{\alpha}\rightarrow ^{\beta}m$ړ$(QM_{r}(A^{*}),\beta)$ڐ 0$ڢړگړ$m\in H$ >\begin{equation}\label{els1} >. ||m(f,F)||\leq r >\end{equation} >% <$f\in A^{*}$$m\in H$𑪖$M_{f}:A^{**}\rightarrow A^{*}$ڤړڮڥڗ$$M_{f}(F):=m(f,F), \hspace{.5cm}F\in A^{**}$$ <줐ڢ$.\cal{H}$$=\{M_{f}: m\in H\}\subseteq CL(A^{**},A^{*})$ړڐꑢڐ(\ref{els1})ړ$G\in A^{**}$$$||M_{f}(G)||=||m(f,G)||\leq r(f,G),\hspace{.5cm}m\in H$$ <$\cal{H}$ړڐڐ$c=c(F)>0$ڢړ >% >\begin{equation}\label{els2} >||M_{f}||\leq c,\hspace{.5cm}m\in H >\end{equation} <꤭$P=\{p_{m}:m\in H\}$ڡڐڐڪ$p_{m}$ڤ$A^{*}$ړړگ <$$.p_{m}(f)=||M_{f}||=\sup _{||F||\leq 1}||M_{f}(F)||=\sup _{||F||\leq 1}||m(f,F)||,\hspace{.2cm}f\in A^{*}$$ <ڐڙړ$m$$p_{m}$ڤ$A^{*}$ڕڐ꤭$\{f_{n}\}\subseteq A^{*}$ړگ$f_{n}\rightarrow f_{0}\in A^{*}$ < < < >\begin{eqnarray*} >|p_{m}(f_{n})-p_{m}(f_{0})|\leq p_{m}(f_{n}-f_{0})&=&\sup _{||F||\leq 1}||M_{f_{n}-f_{0}}(F)||\\ >&=&\sup _{||F||\leq 1}||m(f_{n}-f_{0},F)||\rightarrow 0\\ >\end{eqnarray*} <ڢ$p_{m}$ڕڐ <ڐڤ(\ref{els2})ڡ$P$ړڐړڐꑢڐ\ref{1}ڙ$K_{0}\geq 0$ <$ball B=B(f_{0},r)=\{f\in A^{*}:||f-f_{0}||\leq r\}$ڢړگړ$f\in ball B$ <$.p_{m}(f)\leq K_{0}$ڟ꤭$f\in A^{*}$ړ$.||f||\leq 1$ڢڐڬ <$$p_{m}(f)=\frac{p_{m}(rf+f_{0}-f_{0})}{r}\leq \frac{1}{r}(p_{m}(rf+f_{0})+p_{m}(f_{0}))\leq\frac { 2K_{0}}{r}$$ < <$$||m||=\sup _{||f||\leq 1,||F||\leq 1}||m(f,F)||=\sup _{||f||\leq 1}p_{m}(f)\leq \frac{2K_{0}}{r}$$ڢ$H$$-\tau$ڐ <$(2)$ړڐꑢڐ$(1)$ڤ$\gamma\subseteq \beta\subseteq \tau$ڟړ줐ڐ < < \begin{eqnarray*} >(\theta_{\mu})(G\circ f,F)&=&(G\circ f)\circ (\mu\circ F)=G\circ(f\circ(\mu\circ F))\\ >&=&G\circ [(\theta_{\mu})(f,F)]\\ >\end{eqnarray*} <گڢڝ$L_{\infty}(G)$$-M(G)$ڐ$.f\circ\mu\in L_{\infty}(G)$ڟړڐꑢڐڡڡڢ <$$\theta_{\mu}(f,F\circ G)=(f\circ\mu)\circ (F\circ G)=((f\circ \mu)\circ F)\circ G=[\theta_{\mu}(f,F)]\circ G$$ <ڢ$.\theta_{\mu}\in QM_{r}(L_{1}(G)^{*})$ <ڙ$\theta$ړڐ꤭$.\theta_{\mu}=0$ړڐ$f\in L_{\infty}(G)$$F\in (L_{\infty}(G))^{*}$ڢ$.(f\circ \mu)\circ F=0$ <ڝ$L_{1}(G)$ڪڗڐڊړ$f\in L_{\infty}(G)$$$f\circ \mu=0$$ڢڟڡړ$f\in C_{0}(G)$ڊ$$.f\circ \mu=0$$ڟڐڐ$M(G)$ڢ$C_{0}(G)$ <$C_{0}(G)$ڪڗڐ$.\mu=0$ \begin{eqnarray*} >m(ab,c)&=&m_{(S,T)}(ab,c)=(ab)S(c)=a[bS(c)]\\ >&=&a[m_{(S,T)}(b,c)]=am(b,c).\\ >\end{eqnarray*} <$.m(a,bc)=m(a,b)c$$m$ڪڮڐڐړڐڐꑢڛ$A$ڐ𑪖ړ \begin{equation}\label{elF1} >.m(a^{2},b)=am(a,b),\hspace{1cm} m(a,b^{2})=m(a,b)b >\end{equation} \begin{eqnarray*}\label{elF2} >.m((a+b)^{2},c)&=&(a+b)m(a+b,c)\\ >&=&am(a,c)+am(b,c)+bm(a,c)+bm(b,c)\\ >\end{eqnarray} <گڢ < < < >\begin{eqnarray*}\label{elF3} >.m((a+b)^{2},c)&=&m(a^{2}+b^{2}+2ab,c)\\ >&=&m(a^{2},c)+m(b^{2},c)+2m(ab,c)\\ >&=&am(a,c)+bm(b,c)+2m(ab,c)\\ >\end{eqnarray} <(\ref{elF2})(\ref{elF3})ڢ < >\begin{equation}\label{elF4} >2m(ab,c)=am(b,c)+bm(a,c) >\end{equation} <ڢڐꑢڐ(\ref{elF4})ڊړ$a,b,c,d\in A$ڡڢ < >\begin{eqnarray*}\label{elF5} >2m(abd,c)&=&abm(d,c)+dm(ab,c)\\ >&=&abm(d,c)+\frac{1}{2} d[2m(ab,c)]\\ >&=&abm(d,c)+\frac{1}{2} dam(b,c)+\frac {1}{2} dbm(a,c)\\ >\end{eqnarray} <ڐڡڛ$A$ڢړڐꑢڐ(\ref{elF4})ړڍ < < < >\begin{eqnarray*}\label{elF6} >2m(abd,c)&=&2m(adb,c)=adm(b,c)+bm(ad,c)\\ >&=&adm(b,c)+\frac{1}{2}[bam(d,c)+bdm(a,c)]\\ >&=&adm(b,c)+\frac {1}{2} abm(d,c)+\frac{1}{2} dbm(a,c)]\\ >\end{eqnarray} <(\ref{elF5})(\ref{elF6})ڢ$$.abm(d,c)=ad m(b,c)$$ڝ$A$ꑢڐ$.bm(d,c)=dm(b,c)$ړړڐ$d\in A$$$dm(ab,c)=abm(d,c)=a[bm(d,c)]=adm(b,c)=dam(b,c)$$ <ڢ$.m(ab,c)=am(b,c)$ڐڢ$.m(a,bc)=m(a,b)c$ \begin{equation}\label{elF7} >m(ab,cd)=am(b,c)d >\end{equation} < \begin{eqnarray*} >m(a+b,c)&=&m(xw+yw,c)=(x+y)m(w,c)=xm(w,c)+ym(w,c)\\ >&=&m(xw,c)+m(yw,c)=m(a,c)+m(b,c)\\ >\end{eqnarray*} <$.m(c,a+b)=m(c,a)+m(c,b)$ <$$m(\alpha a,c)=m(\alpha xw,c)=(\alpha x)m(w,c)=\alpha m(xw,c)=\alpha m(a,c)$$ڢ$m$ڢڐ <$(3)$꤭$m$ڪڮړڐڢ$m$ړڐڕڐ꤭$a\in A$$\{x_{n}\}\subseteq A$ڢڐړ$x$ړڍ$\{x_{n}-x\}$ڢڐ <ړڬڐڝ$A$ڗڕڐڢ$\{z_{n}\}$$z$ڢ$A$ړ$z_{n}\rightarrow 0$$.x_{n}-x=zz_{n}$ <$$.m(a,x_{n})-m(a,x)=m(a,x_{n}-x)=m(a,zz_{n})=m(a,z)z_{n}\rightarrow 0$$ <ڐڢ$m$ړڗڕڐ꤭$\{x_{n }\}\in A$$\{y_{n}\}\in A$ڢړ$x$$y$ړ꤭$W$ڐڬړ <$U$ڐڬڤڝڐ$.U+U\subseteq W$ڐڢړ$m$ڢ < < < >\begin{eqnarray*}\label{elF8} >.m(x_{n},y_{n})-m(x,y)&=&m(x_{n}-x,y_{n})+m(x,y_{n})-m(x,y)\\ >&=&m(x_{n}-x,y_{n})+m(x,y_{n}-y)\\ >\end{eqnarray} <$n$𑪖$T_{n}:A\rightarrow A$ڤړڮ$$T_{n}(a)=m(a,y_{n}),\hspace{1cm}a\in A$$ڗڝ$m$ړڐڕڐ𑪖$T_{n}$ <ڗڕڐڢ$\{T_{n}(x)\}=\{m(x,y_{n})\}$ڢڐڐ\ref{2}ڢ$\{T_{n}\}$ <ڡړ$V$ڐڬڢړگړ$n\geq 1$$$.T_{n}(V)\subseteq U,\hspace{1cm} $$ <$\{x_{n}\}$ړ$x$䢢گ$n_{1}$ڢړگړ$n\geq n_{1}$$.x_{n}-x\in V$ړڐ$n\geq n_{1}$$$.m(x_{n}-x,y_{n})=T_{n}(x_{n}-x)\in U$$ <ڐڐ$m$ړڐڕڐ$n_{2}\in N$ڢ <ړړ$n\geq n_{2}$$.m(x,y_{n}-y)\in U$ړڐꑢڐ(\ref{elF8})ړڐ$n\geq max\{n_{1},n_{2}\}$ <$$.m(x_{n},y_{n})-m(x,y)\in U+U\subseteq W$$ \begin{equation}\label{elF9} >||T||_{q}=\sup \{\frac{q(T(x))}{ q(x)}:x\in A,x\neq 0\} >\end{equation} <ڐڬڢ \begin{eqnarray*} >||T||_{q}&=&\sup \{q(T(x)):x\in A, q(x)=1\}\\ >&=&\sup\{q(T(x)):x\in A,q(x)\leq 1\}\\ >\end{eqnarray*} \begin{eqnarray*} >xS(y)&=&x\lim_{\lambda}m(e_{\lambda},y)=\lim _{\lambda} m(xe_{\lambda},y)=m(x,y)\\ >&=&\lim_{\lambda}m(x,e_{\lambda}y)=\lim_{\lambda}m(x,e_{\lambda})y=T(x)y\\ >\end{eqnarray*} <ړ$.(S,T)\in M_{d}(A)$ <ړڐ$(a,b)\in A\times A$ڢ <$$[\varphi_{d}(S,T)](a,b)=aS(b)=a\lim_{\lambda} m(e_{\lambda},b)=\lim_{\lambda} m(ae_{\lambda},b)=m(a,b).$$ <$\phi_{d}(S,T)=m$ڢ$\varphi_{d}$ڕڐ <ڢ$\varphi_{d}$ړڐ꤭$(S_{1},T_{1}),(S_{2},T_{2})\in M_{d}(A)$$.\phi_{d}(S_{1},T_{1})=\phi_{d}(S_{2},T_{2})$ <ړ$a,b\in A$ <$$\phi_{d}(S_{1},T_{1})(a,b)=\phi_{d}(S_{2},T_{2})(a,b)$$ڢ$.aS_{1}(b)=aS_{2}(b)$ <$A$ꑢڐړڐ$b\in A$$S_{1}(b)=S_{2}(b)$ڢ$.S_{1}=S_{2}$ڟړڐꑢڐ\ref{3.3}$(3)$$.T_{1}=T_{2}$ړ$.(S_{1},T_{1})=(S_{2},T_{2})$ <$(2)$꤭$T\in M(A)$$.x,y\in A$ړڐꑢڐ\ref{3.2}$(2)$$T\in M_{l}(A)\cap M_{r}(A)$ڢ$(T,T)\in M_{d}(A)$ <$$\phi_{d}(T)(x,y)=\phi_{d}(T,T)(x,y)=xT(y)=T(xy).$$ <$$\phi_{l}(T)(x,y)=xT(y)=T(xy); \hspace{.3cm}\phi_{r}(T)(x,y)=T(x)y=T(xy).$$ <$(3)$ړڐ$a\in A$$(x,y)\in A\times A$ڢ$$\phi_{d}(\mu_{d}(a))(x,y)=\phi_{d}(L_{a},R_{a})(x,y)=xL_{a}(y)=xay=\phi_{A}(a)(x,y)$$ <ڢ$.\phi_{d}|\mu_{d}(A)=\phi_{A}$ \begin{eqnarray*} >\lim _{\lambda}q(e_{\lambda}ae_{\lambda}-a)&=&\lim_{\lambda}q[(e_{\lambda}xye_{\lambda}-e_{\lambda}xy)+(e_{\lambda}xy-xy)]\\ >&\leq & \lim _{\lambda} q(e_{\lambda}x).q(ye_{\lambda}-y)+\lim_{\lambda} q(e_{\lambda}xy-xy)\\ >&=&q(x).q(0)+q(0)=0.\\ >\end{eqnarray*} <$(2)$ڛ$A$$\{e_{\lambda}:\lambda\in I\}$ꤐڗڢ$A$ڐړ$x\in A$ڢ$y\in A$ړگ <$\lim_{\lambda} m(e_{\lambda},x)=y$ڟڐڗړڗ$q(e_{\lambda})\leq 1$ڢ < < < >\begin{eqnarray*} >q[y-e_{\lambda}m(e_{\lambda},x)]&\leq & q(y-e_{\lambda }y)+q[e_{\lambda} y-e_{\lambda}m(e_{\lambda},x)]\\ >&\leq & q(y-e_{\lambda}y)+q(e_{\lambda})q(y-m(e_{\lambda},x))\\ >&\leq & q(y-e_{\lambda}y)+q(y-m(e_{\lambda},x))\rightarrow 0.\\ >\end{eqnarray*} \begin{eqnarray*} >||m||_{q}&=&0\Leftrightarrow \sup _{x\neq 0,y\neq 0} \frac{q[m(x,y)]}{q(x)q(y)}=0\\ >&\Leftrightarrow & q[m(x,y)]=0, \hspace{.1cm}x,y\in A,x,y\neq 0\\ >&\Leftrightarrow & q[m(x,y)]=0 ,\hspace{.1cm}x,y\in A\\ >&\Leftrightarrow & m=0\\ >\end{eqnarray*} < <ړڐ$\alpha\in K$$q(\alpha x)=|\alpha|^{k}q(x)$ڢ$.||\alpha m||_{q}=|\alpha|^{k}||m||_{q}$ <꤭$m,u\in QM(A)$$\epsilon>0$ڢړ$x,y\in A$ڢړگ$$||m+u||_{q}\leq \frac{q[(m+u)(x,y)]}{q(x)q(y)} +\epsilon$$ <ڐڬ >\begin{eqnarray*} >||m+u||_{q}&\leq & \frac{q[m(x,y)+u(x,y)]}{q(x)q(y)}+\epsilon\\ >&\leq & \frac{q[m(x,y)]}{q(x)q(y)}+\frac{q[u(x,y)]}{q(x)q(y)}+\epsilon\\ >&\leq &||m||_{q}+||u||_{q}+\epsilon.\\ >\end{eqnarray*} <\end{eqnarray*} <$||m+u||_{q}\leq ||m||_{q}+||u||_{q}$ړ$||.||_{q}$$-k$ڤ$QM(A)$ڐ <$(2)$꤭$\{m_{i}:i\in N\}$ڢ$-||.||_{q}$ڢ$QM(A)$ړڍړڐ$x,y\in A$ >\begin{eqnarray*} >\lim _{i,j}q[m_{i}(x,y)-m_{j}(x,y)]&=&\lim_{i,j} q[(m_{i}-m_{j})(x,y)]\\ >&\leq & \lim _{i,j} ||m_{i}-m_{j}||_{q}.q(x)q(y)=0\\ >\end{eqnarray*} <ړڐ$x,y\in A$ڢ$\{m_{i}(x,y)\}$ڢڢ$A$ڐړڐꑢڐڡړ$A$𑪖$m$ړڮ$$m(x,y)=\lim_{i}m_{i}(x,y), \hspace{.2cm}x,y\in A$$ڡڗڐ <$m$ڢړڐꑢڐ\ref{3}$(2)$ڕڐ <ړړ$a,b,x,y\in A$$$m(ax,yb)=\lim_{i}m_{i}(ax,yb)=\lim_{i}[am_{i}(x,y)b]=a[\lim_{i}m_{i}(x,y)]b=am(x,y)b$$ <ڢ$.m\in QM(A)$ڢ$.||m_{i}-m||_{q}\rightarrow 0$꤭$.\epsilon>0$ <ڐ$\{m_{i}:i\in N\}$ڢ$-||.||_{q}$ڐ䢢$N\geq 1$ <ڢړگړ$i,j\geq N$ڢ$$.||m_{i}-m_{j}||_{q}<\frac {\epsilon}{2}$$ <$$\frac{q[(m_{i}-m_{j})(x,y)]}{q(x)q(y)}<\frac{\epsilon}{2},\hspace{.5cm}i,j\geq N, x,y\in A, x,y\neq 0\hspace{2cm}(*)$$ <꤭$i_{0}\geq N$ڙ$.j\rightarrow \infty$ڝ$m_{i}(x,y)\rightarrow m(x,y)$ړڐ$x,y\in A$$x,y\neq 0$ڢ < $$\frac{q[m_{i_{0}}(x,y)-m(x,y)]}{q(x)q(y)}<\frac {\epsilon}{2}\hspace{7cm}(**)$$ <ړڐꑢڐ$(*)$$(**)$ړڐ$i\geq N$ڢ < < >\begin{eqnarray*} >\frac{q[m_{i}(x,y)-m(x,y)]}{q(x)q(y)&}&\leq \frac {q[m_{i}(x,y)-m_{i_{0}}(x,y)]}{q(x)q(y)}+ \frac {q[m_{i_{0}}(x,y)-m(x,y)]}{q(x)q(y)}\\ >&< &\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon\\ >\end{eqnarray*} <ڢ$.m_{i}\rightarrow^{||.||_{q}} m$ < \begin{eqnarray*} >||\phi_{d}(S,T)||_{q}&=&\sup _{x\neq 0,y\neq 0} \frac{q[\phi_{d}(S,T)(x,y)]}{q(x)q(y)}=\sup _{x\neq 0,y\neq 0} \frac{q[xS(y)]}{q(x)q(y)}\\ >&\leq &\sup _{x\neq 0,y\neq 0} \frac{q(x)q(S(y))}{q(x)q(y)}=\sup _{y\neq 0} \frac{q(S(y))}{q(y)}=||S||_{q}\\ >\end{eqnarray*} <꤭$\epsilon>0$ڢړ$(y\neq 0)\in A$ڢړگ$.||S||_{q}<\frac{q(S(y))}{q(y)}+\epsilon$ڐڡڥڮړ$q$ <$.0||S||_{q}-\epsilon$$ <ڢ$.||\phi_{d}(S,T)||_{q}=||S||_{q}=||(S,T)||_{q}$$\phi_{d}$𑪖گڐ <$(2)$ڐگړ𑪖$\phi_{A}$ڡڢ < >\begin{eqnarray*} >||a\circ m\circ b||_{q}&=&\sup _{x\neq 0,y\neq 0}\frac{q[(a\circ m\circ b)(x,y)]}{q(x)q(y)}=\sup _{x\neq 0,y\neq 0} \frac{q[m(xa,by)]}{q(x)q(y)}\\ >&=&\sup _{x\neq 0,y\neq 0} \frac{q[x.m(a,b)y]}{q(x)q(y)}=\sup _{x\neq 0,y\neq 0} \frac{q[\phi_{A}(m(a,b))(x,y)]}{q(x)q(y)}\\ >&=&||\phi_{A}(m(a,b))||_{q}=q(m(a,b))\\ >\end{eqnarray*} \begin{eqnarray*} >m\circ_{\phi_{d}}\phi_{A}(a)&=&\phi_{d}[(S,T)(L_{a},R_{a})]=\phi_{d}(SL_{a},R_{a}T)\\ >&=&\phi_{d}(L_{S(a)},R_{S(a)})=\phi_{A}(S(a))\in \phi_{A}(A).\\ >\end{eqnarray*} < < < < \begin{eqnarray*} >(m_{1} \odot m_{2})(x,y)&=&[\phi_{d}(S_{1}S_{2},T_{2}T_{1})](x,y)=x(S_{1}S_{2})(y)\\ >&=&x[S_{1}(S_{2}(y))]=x[S_{1}(\lim_{\lambda}e_{\lambda} S_{2}(y))]\\ >&=&m_{1}(x,\lim_{\lambda}e_{\lambda}S_{2}(y))=m_{1}(x,\lim_{\lambda}m_{2}(e_{\lambda},y))\\ >\end{eqnarray*} <ڢڪ$||m_{1}\odot m_{2}||_{q}\leq ||m_{1}||_{q}||m_{2}||_{q}$ڢ$(QM(A)),||.||_{q})$ڛ$-F$ڐ <ړڐꑢڐ\ref{zz}$(QM(A),||.||_{q})$ڐ <$(2)$꤭$m\in QM(A)$$.a\in A$ړڐꑢڐ$(1)$ړ$(x,y)\in A\times A$ >\begin{eqnarray*} >[m\odot \phi_{A}(a)](x,y)&=&m(x,\lim_{\lambda} \phi_{A}(a)(e_{\lambda},y))=m(x,\lim_{\lambda}e_{\lambda}ay)\\ >&=&m(x,ay)=(m\circ a)(x,y)\\ >\end{eqnarray*} <$.[\phi_{A}(a)\odot m](x,y)=(a\circ m)(x,y)$ <$(3)$꤭$m\in QM(A)$$.a\in A$ڐڕړ$\varphi_{d}$$(S,T)\in M_{d}(A)$ڢړگ$.\phi_{d}(S,T)=m$ <ڗړ$(2)$ >\begin{eqnarray*} >[m\odot \phi_{A}(a)](x,y)&=&(m\circ a)(x,y)=m(x,ay)=\phi_{d}(S,T)(x,ay)\\ >&=&xS(ay)=xS(a)y=\phi_{A}(S(a))(x,y)\\ >\end{eqnarray*} <ڢ$.m\odot \phi_{A}(a)=\phi_{A}(S(a))\in \phi_{A}(A)$ړ$\phi_{A}(a)\odot m=\phi_{A}(T(a))\in \phi_{A}(A)$ <$(4)$꤭$A$ڗڕڐړڐꑢڐ\ref{10}ڐ𑪖$\phi_{d},\phi_{l},\phi_{r},\phi_{A}$ڡڕگ <꤭$.(S_{1},T_{1}),(S_{2},T_{2})\in M_{d}(A)$ڍړڐꑢڐڗ <$$\phi_{d}(S_{1},T_{1})\circ _{\phi_{d}} \phi_{d}(S_{2},T_{2})=\phi_{d}[(S_{1},T_{1})(S_{2},T_{2})]$$ <$S_{1},S_{2}\in M_{l}(A)$ڍڐڗ$$\phi_{l}(S_{1})\circ_{\phi_{l}}\phi_{l}(S_{2})=\phi_{l}(S_{1}S_{2})$$ <$T_{1},T_{2}\in M_{r}(A)$ڍڐڗ$$\phi_{r}(T_{1})\circ _{\phi_{r}}\phi_{r}(T_{2})=\phi_{r}(T_{2}T_{1})$$ <$\phi_{d}:M_{d}(A)\rightarrow QM(A)$$\phi_{l}:M_{l}(A)\rightarrow QM(A)$$\phi_{r}:M_{r}(A)\rightarrow QM(A)$ڕڐ < < \begin{eqnarray*} >||a\circ m||_{q}&=&\sup _{x\neq 0,y\neq 0} \frac{q[(a\circ m)(x,y)]}{q(x)q(y)}=\sup_{x\neq 0,y\neq 0} \frac{q[xm(a,y)]}{q(x)q(y)}\\ >&\leq & \sup_{x\neq 0,y\neq 0} \frac{q(x)q(m(a,y))}{q(x)q(y)}=\sup_{x\neq 0,y\neq 0} \frac{q(m(a,y))}{q(y)}\\ >&=&\sup_{x\neq 0,y\neq 0} \frac{||m||_{q}q(a)q(y)}{q(y)}=||m||_{q}q(a)\\ >\end{eqnarray*} <ړ$.||m\circ a||_{q}\leq ||m||_{q}q(a)$ <$(2)$ړڐꑢڐ\ref{10}$.q(m(a,b))=||a\circ m\circ b||_{q}$ړړڐꑢڐ$(1)$$$.||a\circ m\circ b||_{q}\leq ||a\circ m||_{q}q(b)\leq ||m||_{q}q(a)q(b)$$ \begin{eqnarray*} >q[m_{i}(x,y)-m_{j}(x,y)]&=&q[(m_{i}-m_{j})(x,y)]\\ >&=&||x\circ (m_{i}-m_{j})\circ y||_{q}=\xi_{x,y}(m_{i}-m_{j})\\ >\end{eqnarray*} <ڢ$\{m_{i}(x,y)\}$ڢ$A$ڐ𑪖$m:A\times A\rightarrow A$ڤړڮ$m(x,y)=\lim_{i}m_{i}(x,y)$ڗړ$m$ <ړڗړ\ref{3}$(2)$ڕڗڐړړڐ$a,b,x,y\in A$$$m(ax,yb)=\lim_{i}m_{i}(ax,yb)=a[\lim_{i}m_{i}(x,y)]b=am(x,y)b$$ <$.m\in QM(A)$ړڐ$a,b\in A$$$\xi_{a,b}(m_{i}-m)=||a\circ m_{i}\circ b-a\circ m\circ b||_{q}=q[(m_{i}-m)(a,b)]\rightarrow 0$$ <$m_{i}\rightarrow^{\gamma} m$ڢ$(QM(A),\gamma)$ړڢڐڐ <ڢ$(QM(A),\beta)$ړڢڐڐ꤭$.m\in QM(A)$ڢڐڬړڐ$c\in A$𑪖 <$S_{c},T_{c}:A\rightarrow A$ړڮ <$$S_{c}(x)=m(c,x),\hspace{.2cm}T_{c}(x)=m(x,c),\hspace{.1cm}x\in A$$𑪗ڢ$M_{l}(A),M_{r}(A)$ڗ <ړ$$.\phi_{l}(S_{c})=c\circ m, \hspace{.2cm}\phi_{r}(T_{c})=m\circ c$$ <꤭$c\in A$$\{m_{i}:i\in N\}$ڢ$-\beta$ڢ$QM(A)$ړڐڗ$-\beta$ <ڢ$\{\phi_{l}(S_{c})_{i}\}$$\{\phi_{r}(T_{c})_{i}\}$ڢ$-\tau$ڢ$QM(A)$ڢڍ$$.(T_{c})_{i}(x)=m_{i}(x,c),\hspace{.2cm}(S_{c})_{i}(x)=m_{i}(c,x)$$ڝ𑪖$\phi_{l},\phi_{r}$ <گڊڪ <$\{(S_{c})_{i}\}$$\{(T_{c})_{i}\}$$-||.||_{q}$ڢ$M_{l}(A)$$M_{r}(A)$ڟړڐꑢڐ\ref{3.9}$M_{l}(A)$$M_{r}(A)$ <$S^{(c)}$ڢ$M_{l}(A)$$T^{(c)}$ڢ$M_{r}(A)$ړگ$$||(S_{c})_{i}-S^{(c)}||_{q}\rightarrow 0, \hspace{.2cm} ||(T_{c})_{i}-T^{(c)}||_{q}\rightarrow 0\hspace{1cm}(*)$$ < <ڐ$\gamma\subseteq \beta$ڗ$\{m_{i}\}$ڢڗ$\gamma$ڐڢ$(1)$ڢ$QM(A)$ڢڗ$\gamma$ڐڢ$m_{0}\in QM(A)$ڢ <گ$$\lim_{i}m_{i}(x,y)=m_{0}(x,y)$$ <ړڐ$a,b\in A$ <$$[\phi_{l}(S^{(c)})](a,b)=\lim_{i}[\phi_{l}(S_{c})_{i}](a,b)=\lim_{i}am_{i}(c,b)=(c\circ m_{0})(a,b)$$ <ڢ$.\phi_{l}(S^{(c)})=c\circ m_{0}$ <ڙڪ$.\phi_{r}(T^{(c)})=m_{0}\circ c$ <ڐꑢڐڤ$(*)$ <$$||c\circ m_{i}-c\circ m_{0}||_{q}=||\phi_{l}(S_{c})_{i}-\phi_{l}(S^{(c)})||_{q}= ||(S_{c})_{i}-S^{(c)}||_{q}\rightarrow 0$$ < <$$|| m_{i}\circ c-m_{0}\circ c||_{q}=||\phi_{r}(T_{c})_{i}-\phi_{r}(T^{(c)})||_{q}= ||(T_{c})_{i}-T^{(c)}||_{q}\rightarrow 0.$$ <$m_{i}\rightarrow^{\beta} m_{0}$ڢ$QM(A)$$-\beta$ڐ <$(2)$꤭$\{m_{\alpha}:\alpha\in J\}$ڗ$-\gamma$ڢ$QM(A)$ړړڛڢ$\{m_{i}:i\in N\}$ <ړڗ$\{m_{\alpha}:\alpha\in J\}$ڢ$(1)$𑪖$m:A\times A\rightarrow A$ړڮ$$m(x,y)=\lim _{\alpha}m_{\alpha}(x,y)$$ڡڐړړڐ$a,b,x,y\in A$ <$$m(ax,yb)=\lim_{\alpha}m_{\alpha}(ax,yb)=a[\lim_{\alpha} m_{\alpha}(x,y)]b=am(x,y)b.$$ <ڐڐ$A$ڗڕڐړڐꑢڐ\ref{k}$(3)$$m$ڕڗڢ$.m\in QM(A)$ <ڐڢ$(1)$$m_{\alpha}\rightarrow^{\gamma} m$ڢ$(QM(A),\gamma)$ڐ <$(QM(A),\beta)$ڐ 0$ڢړگړ$m\in H$$$||a\circ m\circ b||_{q}\leq r$$ړڐꑢڐ\ref{10}$(2)$ <$$q[m(a,b)]\leq r.\hspace{4cm}(1)$$ <$a\in A$$m\in H$𑪖$M_{a}:A\rightarrow A$ڤړڮڥڗ$$M_{a}(x)=m(a,x),\hspace{.2cm}x\in A.$$ <줐ڢ$.\cal{F}$$=\{M_{a}:m\in H\}\subseteq CL(A)$ړڐꑢڐ$(1)$ړڐ$x\in A$$m\in H$ <$$q[M_{a}(x)]=q[m(a,x)]\leq r(a,x)$$ <ڢ$\cal{F}$ړڐڐ\ref{2}$c=c(a)>0$ڢړگ$$||M_{a}||_{q}\leq c.\hspace{4cm}(2)$$ <꤭$P=\{p_{m}:m\in H\}$ڡڐڐ$-k$$p_{m}$ڤ$A$ړړ <$$p_{m}(a)=||M_{a}||_{q}=\sup_{b\neq 0}\frac {q[M_{a}(b)]}{q(b)}=\sup _{b\neq 0}\frac{q[m(a,b)]}{q(b)},\hspace{.5cm}a\in A.$$ <ڐڢړڐ$m\in H$$p_{m}$ڤ$A$ڕڐ꤭$\{a_{n}\}\subseteq A$$.a_{n}\rightarrow a_{0}\in A$ڍ < >\begin{eqnarray*} >|p_{m}(a_{n})-p_{m}(a_{0})|&\leq & p_{m}(a_{n}-a_{0})=\sup _{b\neq 0} \frac{q[M_{a_{n}-a_{0}}(b)]}{q(b)}\\ >&=&\sup_{b\neq 0}\frac{q[m(a_{n}-a_{0},b)]}{q(b)}\leq \sup_{b\neq 0}\frac{||m||_{q}q(a_{n}-a_{0})q(b)}{q(b)}\\ >&=&||m||_{q}q(a_{n}-a_{0})\rightarrow 0\\ >\end{eqnarray*} <ڢ$p_{m}$ڕڐڟڐڤ$(2)$ڡ$P$ړڐړڐꑢڐ\ref{1}ڙ$C>0$ <$ball B=B(x_{0},r)=\{x\in A:q(x-x_{0})\leq r\}$ڢړگړڐ$m\in H$$x\in B(x_{0},r)$$$p_{m}(x)\leq C.\hspace{4cm}(3)$$ <꤭$.a\in A$ڐ$$p_{m}(a)\leq \frac{2C.q(a)}{r}.\hspace{3cm}(4)$$ <$a=0$ړړ줐ڐ꤭$.a\neq 0$ړڤ줐ڢ$.t=(\frac{r}{q(a)})^{\frac{1}{k}}$ <$q$$-k$ڐ$ta+x_{0},x_{0}\in B(x_{0},r)$ <$$q(ta+x_{0}-x_{0})=q(ta)=t^{k}.q(a)\leq \frac{r}{q(a)}q(a)=r,\vspace{-1cm}$$ <$$q(x_{0}-x_{0})=q(0)=0\begin{eqnarray*} >p_{m}(a)&=&p_{m}(\frac{1}{t}ta)=(\frac{1}{t})^{k}p_{m}(ta)\leq \frac{q(a)}{r}p_{m}[ta+x_{0}-x_{0}]\\ >&\leq & \frac{q(a)}{r}[p_{m}(ta+x_{0})+p_{m}(x_{0})]\\ >&\leq & \frac{q(a)}{r}[C+C]=\frac{2C.q(a)}{r}\\ >\end{eqnarray*} <ڐڐڪڟړڐꑢڐ$(4)$ړڐ$m\in H$ < >\begin{eqnarray*} >||m||_{q}&=&\sup_{a,b\neq 0}\frac{q[m(a,b)]}{q(a)q(b)}=\sup_{a,b\neq 0} \frac {1}{q(a)}\sup_{a,b\neq 0}\frac{q[m(a,b)]}{q(b)}\\ >&=&\sup_{a\neq 0} \frac{1}{q(a)}p_{m}(a)\leq \sup_{a\neq 0}\frac {1}{q(a)}.\frac{2C.q(a)}{r}\leq \frac{2C}{r}\\ >\end{eqnarray*} <ڢ$H$$-\tau$ڐ <$(2)$ړڐꑢڐ$(1)$ڤ$\gamma\subseteq \beta\subseteq \tau$ڟړ줐ڐ < < < <ڂ < < < >\appendix%ڷ >%{\large \ph 1) M. 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First we extend the >concept of multipliers on Banach algebras to $\varphi-$multipliers on faithful Banach algebras and $\varphi_{m}-$multipliers on topological algebra-modules >and investigate their properties. We prove some results concerning Arens regularity, amenability and faithfulness of the algebra $M_{\varphi}(A).$ < >$\hspace{.3cm}$ In the next chapter we introduced quasi-multipliers on the dual of a Banach algebra and quasi-multipliers on $F-$algebras. Which is > a generalization of the concept of multipliers on Banach algebras. In the first part we consider algebra A whose dual has a mixed identity. > Among others we prove that for an Arens regular Banach algebra which has a bounded approximate identity the space $QM_{r}(A^{*})$ of all bilinear and separately continuous right >quasi-multipliers of $A^{*}$ is isometrically isomorphic to $A^{**}.$ > >$\hspace{.3cm}$Also we consider the notion of quasi-multipliers on an $F-$algebra A. we study their bilinearity and joint continuity under some suitable conditions. > we also define the strict and quasi-strict topologies on the algebra $QM(A)$ and investigate their properties.\\ < > >$\hspace{.3cm}$\vsp{\bf Keywords: } Quasi-multiplier, Multiplier, Faithful Banach algebra, Arens regular Banach algebra, F-algebra, K-algebra, Strict topology. >\end{quote} >\end{abstract} >\pagestyle{empty} %\MakeEnglishTitle{Pure Mathematics (Geometry)}{Generalized Einstein manifolds}{Ataabak Baagherzadeh Hushmandi} >%{Dr. Behrooz Bidabad}{Dr. Naser Boroujerdian}{Winter 2005} >%---------------------------------------------------------------------------------------------------- >%______________________________ړڐڕ_________________________ >\begin{center} >%----------------------------------------------------------------------------------------------- >\vspace*{-20mm} >{\Large{\cal Amirkabir University of Technology}} >\vspace*{10mm}\\ >%{\Large\bf Amirkabir}\\{\large\bf University of Technology} >\vspace*{1mm}\\ >{\large (Tehran Polytechnic)} >\vspace*{10mm}\\ >Department of Mathematics and Computer Science >%\vspace*{-3mm}\\ >% and >%\vspace*{-3mm}\\ >% Computer Sciences >\vspace*{10mm}\\ >{\Large Ph.D Thesis in Pure Mathematics(Analysis)}\\ >%{\large Pure Mathematics (Geometry) } >\vspace*{1.2cm}\\ >Subject: >\vspace*{4mm}\\ >{\huge $\varphi-$multipliers on without order Banach algebras and $\varphi_{m}-$multipliers on topological algebra-modules } >\vspace*{2cm}\\ >{\large Student:}\\ >{\Large Marjan Adib } >\vspace*{1cm}\\ >{\large Supervisor:}\\ >{\Large Dr. Abdolhamid Riazi} >\vspace*{1cm}\\ >{\large Advisor:}\\ >{\Large Dr. Alireza Bagheri salec } >\vfill > February 2011 >\end{center} >\newpage < < < < < < %\english >%\begin{enumerate} >%