Mathematical Researches

fa یک توپولوژی موضعاً محدب روی جبرهای بورلینگ A locally Convex Topology on the Beurling Algebras جبر alg مقاله مستقل Original Manuscript فرض کنید  یک گروه موضعاً فشرده،  یک تابع وزن و   فضای توابع اندازهپذیر روی   باشد که اساساً کراندار و در بینهایت صفر میشوند. در این مقاله توپولوژی موضعاً محدب  را روی فضای وزندار  بررسی میکنیم. نشان می‌دهیم که دوگان   با این توپولوژی برابر فضای باناخ  است. علاوه بر این، برخی ویژگی‌های فضای    با توپولوژی مذکور را بررسی می‌کنیم.   Introduction Let G be a locally compact group with a fixed left Haar measure λ  and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="11" > be a weight function on G;  that is a Borel measurable function <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="91" > with  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="122" > for all <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="53" >.   We denote by <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" > the set of all measurable  functions <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="10" > such that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif" width="74" >; the group algebra of  G  as defined in [2]. Then  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" > with the convolution product “*” and the norm  <img alt="" height="20" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="38" > defined by <img alt="" height="20" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif" width="119" >  is a Banach algebra known as Beurling algebra. We denote by n(G,<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="11" >) the topology generated by the  norm <img alt="" height="20" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="38" >.    Also, let <img alt="" height="26" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="54" > denote the space of all measurable functions 𝑓  with <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="83" >, the Lebesgue space as defined in [2]. Then <img alt="" height="26" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="54" >  with   the product <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="12" > defined by <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif" width="83" >, the   norm <img alt="" height="20" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif" width="42" > defined by  <img alt="" height="20" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif" width="119" >, and the complex conjugation as involution is a commutative <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif" width="29" >algebra. Moreover, <img alt="" height="26" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="54" > is the dual of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >. In fact, the mapping   <img alt="" height="28" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif" width="382" >is an isometric isomorphism.  We denote by <img alt="" height="28" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gif" width="65" >the <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.gif" width="14" >-subalgebra of <img alt="" height="26" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="54" > consisting of all functions  𝘨 on G such that for each <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.gif" width="34" >, there is a compact subset K of G for which <img alt="" height="26" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.gif" width="104" >.  For a study of <img alt="" height="28" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.gif" width="62" >in the unweighted case see  [3,6].  We introduce and study a locally convex topology <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif" width="54" > on <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" > such that <img alt="" height="28" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif" width="58" > can be identified with the strong dual of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >. Our work generalizes  some interesting results of  [15] for group algebras to a more general setting of weighted group algebras. We also show that (<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >,<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif" width="58" >)  could be a normable or bornological space only if G is compact. Finally, we prove that <img alt="" height="28" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif" width="58" > is complemented in <img alt="" height="26" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="54" >  if and only if G is compact. For some similar recent studies see [4,7,8,10,12-14]. One may be interested to see the work [9] for an application of these results. Main results We denote by  𝒞  the set of increasing sequences of compact subsets of G and by ℛ the set of increasing sequences <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.gif" width="26" > of real numbers in <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.gif" width="38" > divergent to infinity. For any <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image056.gif" width="54" > and <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image058.gif" width="54" >, set <img alt="" height="22" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image060.gif" width="326" >and note that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif" width="80" > is a convex balanced absorbing set in the space <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >. It is easy to see that the family 𝒰 of all sets <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif" width="80" > is a base of neighbourhoods of zero for a locally convex topology on <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.gif" width="56" > see for example [16]. We denote this topology by <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif" width="54" >.  Here we use some ideas from  [15], where this topology has been introduced and studied for  group algebras. Proposition 2.1 Let G be a locally compact group, and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif" width="14" >be a weight function on G.   The norm topology n(G,<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="11" >) on <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" > coincides with the topology <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif" width="54" > if and only if G is compact. Proposition 2.2 Let G be a locally compact group, and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif" width="14" >be a weight function on G.  Then the dual of (<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >,<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif" width="58" >)  endowed with the strong topology can be identified with <img alt="" height="28" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif" width="58" >endowed with <img alt="" height="20" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif" width="42" >-topology. Proposition 2.3 Let G be a locally compact group, and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif" width="14" >be a weight function on G.  Then the following assertions are equivalent: a) (<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >,<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif" width="58" >)  is barrelled. b) (<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >,<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif" width="58" >)  is bornological. c) (<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="52" >,<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif" width="58" >)  is metrizable. d) G  is compact. Proposition 2.4 Let G be a locally compact group, and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif" width="14" >be a weight function on G.  Then  is not complemented in .<a href="./files/site1/files/52/10.pdf">./files/site1/files/52/10.pdf</a> گروه موضعاً فشرده, توپولوژی موضعاً محدب, فضای لبگ وزندار, دوگان. Locally compact group, Locally convex topology, Weighted Lebesgue space, Dual. 221 228 http://mmr.khu.ac.ir/browse.php?a_code=A-10-80-1&slc_lang=fa&sid=1 Saeid Maghsoudi سعید مقصودی s_maghsodi@znu.ac.is 10031947532846003074 10031947532846003074 Yes University of Zanjan دانشگاه زنجان، گروه ریاضی