Mathematical Researches

fa آرنز منظم نگاشت‌های دو خطی کران‌دار On the Properties of the Arens Regularity of Bounded Bilinear Mappings جبر alg مقاله استخراج شده از پایان نامه Research Paper در این مقاله به خواص آرنز منظم نگاشت دو خطی کراندار میپردازیم و نشان میدهیم که نگاشت دو خطی کراندار <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" >  آرنز منظم است اگر و تنها اگر نگاشت خطی <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.png" >  با ضابطۀ <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image003.png" >   ضعیف فشرده باشد. سپس قضیه‌ای را اثبات میکنیم که ویژگی ضعیف فشردگی نگاشت دو خطی کراندار و آرنز منظم را به یک‌دیگر مرتبط میسازد. هم‌چنین به بررسی آرنز منظم و خاصیت ضعیف فشردگی نگاشتهای خطی کراندار میپردازیم و نتایجی مشابه نتایج دیلز، اولگر و آریکان را بیان میکنیم. در ادامه ارتباط بین آرنز منظم جبرهای باناخ و انعکاسی بودن را بررسی می‌کنیم.     Introduction       Let <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="10" >, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="9" > and<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="12" > be Banach spaces and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="80" > be a bilinear mapping. In 1951 Arens found two extension for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" > as <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="25" > and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="37" > from <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="58" > into <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="20" >.  The mapping <img alt="" height="26" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="31" > is the unique extension of <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="17" > such that <img alt="" height="26" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="116" > from <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="31" > into <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="26" > is <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image030.gif" width="100" > continuous for every <img alt="" height="26" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image032.gif" width="57" >, but the mapping <img alt="" height="26" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image034.gif" width="120" > is not in general <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image030.gif" width="100" > continuous from <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image037.gif" width="26" > into <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image039.gif" width="26" > unless <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image041.gif" width="50" >.  Thus for all <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image043.gif" width="57" > the mapping <img alt="" height="26" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image034.gif" width="120" >is <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image046.gif" width="100" > continuous if and only if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" > is Arens regular. Regarding <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="17" > as a Banach <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image050.gif" width="89" >, the operation <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image052.gif" width="92" > extends to <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image054.gif" width="31" > and <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image056.gif" width="33" > defined on <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="59" >. These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image060.gif" width="26" > becomes a Banach algebra. Material and methods    The constructions of the two Arens multiplications in <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image062.gif" width="24" > lead us to definition of topological centers for <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image062.gif" width="24" > with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in some manuscripts. It is known that the multiplication map of every non-reflexive,  <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image065.gif" width="24" >-algebra is Arens regular.  In this paper, we extend some problems from Banach algebras to the general criterion on module actions and bilinear mapping with some applications in group algebras. Results and discussion We will investigate on the Arens regularity of bounded bilinear mappings and we show that a bounded bilinear mapping <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="80" > is Arens regular if and only if the linear map <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image067.gif" width="62" > with <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image069.gif" width="112" > is weakly compact, so we prove a theorem that establish the relationships between Arens regularity and weakly compactness properties for any bounded bilinear mappings. We also study on the Arens regularity and weakly compact property of bounded bilinear mapping and we have analogous results to that of Dalse, <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image071.gif" width="10" >lger and Arikan. For Banach algebras, we establish the relationships between Arens regularity and reflexivity.   Conclusion The following conclusions were drawn from this research. <ul> <li><img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image073.gif" width="119" >if and only if the bilinear mapping  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="80" > is Arens regular.</li> <li>A bounded bilinear mapping <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image075.gif" width="73" > is Arens regular if and only if the linear map <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image077.gif" width="61" > with <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image079.gif" width="104" > is weakly compact.</li> <li><img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image081.gif" width="129" > if and only if the bilinear mapping  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="80" > is Arens regular.</li> <li>Assume that <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image083.gif" width="82" > has approximate identity. Then  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image085.gif" width="15" > is Arens regular if and only if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="10" > is reflexive.<a href="./files/site1/files/62/9Abstract.pdf">./files/site1/files/62/9Abstract.pdf</a></li> </ul>     آرنز منظم, جبر باناخ, دوگان دوم, ضربهای آرنز, ضعیف فشردگی, نگاشت دو خطی. Arens product, Arens regularity, Banach algebra, bilinear map, second dual, weakly compact. 235 242 http://mmr.khu.ac.ir/browse.php?a_code=A-10-83-7&slc_lang=fa&sid=1 Abotalb Shikh Ali ابوطالب شیخعلی Ebadian.ali@gmail.com 10031947532846004037 10031947532846004037 Yes Departement of Mathematics دانشگاه پیام نور، دانشکدۀ علوم پایه، گروه ریاضی Kazem Haghnejad azar کاظم حق نژاد آذر haghnejad@uma.ac.ir 10031947532846004038 10031947532846004038 No دانشگاه محقق اردبیلی، دانشکدۀ علوم، گروه ریاضیات و کاربردها Ali Abadian علی عبادیان a.ebadian@urmia.ac.ir 10031947532846004039 10031947532846004039 No Department of mathematics, Faculty of Science, Urmia University, Urmia, Iran. دانشگاه ارومیه. دانشکدۀ علوم. گروه ریاضی