fa مطالعه مرز فرستنبرگ برای زوج گروه ها آنالیز Anal علمی پژوهشی بنیادی S <span style="font-size:12pt"><span style="line-height:107%"><span style="direction:rtl"><span style="unicode-bidi:embed"><span microsoft="" sans="" serif="" style="font-family:">ﺩﺭ ﺍیﻦ ﻣﻘﺎﻟﻪ، ﺑﺎ ﻓﺮﺽ ﺁﻥ که<i> </i><i><span calibri="" dir="LTR" style="font-family:">G</span></i> ﮔﺮﻭﻩ ﺗﻮﭘﻮﻟﻮﮊیک ﮔﺴﺴﺘﻪ ﻭ<i> </i><i><span calibri="" dir="LTR" style="font-family:">H</span></i> یک ﺯیﺮﮔﺮﻭﻩ ﺁﻥ ﺍﺳﺖ، ﺑﻪ<b> </b>ﺑﯿﺎﻥ ﻣﻔﻬﻮﻡ <span calibri="" dir="LTR" style="font-family:">)</span><i>&nbsp;H</i><span calibri="" dir="LTR" style="font-family:">(<i>G,</i></span> ﺳﯿﺴﺘﻢ ﻋﻤلگرﯼ ﭘﺮﺩﺍﺧﺘﻪ ﻭ ﻣﺮﺯ ﻫﺎﻣﺎﻧﺎ ﺑﺮﺍﯼ ﺯﻭﺝ ﮔﺮﻭەﻫﺎﯼ <span calibri="" dir="LTR" style="font-family:">)</span><i>&nbsp; H</i><span calibri="" dir="LTR" style="font-family:">(<i>G,</i></span> ﺭﺍ ﺗﻌﺮیﻒ ﻧﻤﻮﺩەﺍیﻢ. ﺩﺭﻭﺍﻗﻊ، ﻧﺸﺎﻥ ﺩﺍﺩەﺍیﻢ ﮐﻪ ﻣﺮﺯ ﻓﺮﺳﺘﻨﺒﺮﮒ ﺑﺮﺍﯼ ﺯﻭﺝ ﮔﺮﻭەﻫﺎ ﺑﺎ ﻣﺮﺯ ﻫﺎﻣﺎﻧﺎ ﺑﺮﺍﯼ ﺯﻭﺝ ﮔﺮﻭەﻫﺎ،&nbsp;</span></span></span></span></span><span style="font-size:12pt"><span style="line-height:120%"><span style="direction:rtl"><span style="unicode-bidi:embed"><span microsoft="" sans="" serif="" style="font-family:">ﻫﻤﺮیﺨﺖ ﺍﺳﺖ. ﺑﻪ ﺑﯿﺎﻥ ﺩیگر، ﺗﻌﻤﯿﻢ طبیعی نظریه ﮐﻼﺳﯿک ﻣﺮﺯﻫﺎﯼ ﻓﺮﺳﺘﻨﺒﺮﮒ ﺑﺮﺍﯼ ﻣﺮﺯ ﻓﺮﺳﺘﻨﺒﺮﮒ ﺑﺮﺍﯼ ﺯﻭﺝ ﮔﺮﻭەﻫﺎﯼ <span calibri="" dir="LTR" style="font-family:">)</span><i>&nbsp;H</i><span calibri="" dir="LTR" style="font-family:">(<i>G,</i></span> ﻧﯿﺰ ﺑﺮﻗﺮﺍﺭ ﺍﺳﺖ.&nbsp;</span></span></span></span></span><br> &nbsp; <span style="font-size:11pt"><span style="line-height:102%"><span microsoft="" sans="" serif="" style="font-family:"><span style="font-size:10.5pt"><span style="line-height:102%"><span new="" roman="" style="font-family:" times="">In this paper, assuming that </span></span></span><i><span style="font-size:12.0pt"><span style="line-height:102%"><span calibri="" style="font-family:">G </span></span></span></i><span style="font-size:10.5pt"><span style="line-height:102%"><span new="" roman="" style="font-family:" times="">is a discrete topological group and </span></span></span><i><span style="font-size:12.0pt"><span style="line-height:102%"><span calibri="" style="font-family:">H </span></span></span></i><span style="font-size:10.5pt"><span style="line-height:102%"><span new="" roman="" style="font-family:" times="">is a subgroup of </span></span></span><i><span style="font-size:12.0pt"><span style="line-height:102%"><span calibri="" style="font-family:">G</span></span></span></i><span style="font-size:10.5pt"><span style="line-height:102%"><span new="" roman="" style="font-family:" times="">, we introduce the notion of a </span></span></span><span style="font-size:12.0pt"><span style="line-height:102%"><span calibri="" style="font-family:">(<i>G, H</i>)</span></span></span><span style="font-size:10.5pt"><span style="line-height:102%"><span new="" roman="" style="font-family:" times="">-operator system and define the Hamana boundary for pairs of groups </span></span></span><span style="font-size:12.0pt"><span style="line-height:102%"><span calibri="" style="font-family:">(<i>G, H</i>)</span></span></span><span style="font-size:10.5pt"><span style="line-height:102%"><span new="" roman="" style="font-family:" times="">. In fact, we show that the Furstenberg boundary for pairs of groups is isomorphic to the Hamana boundary for pairs of groups. In other words, the natural extension of the classical theory of Furstenberg boundaries remains valid for the Furstenberg boundary associated with pairs of groups </span></span></span><span style="font-size:12.0pt"><span style="line-height:102%"><span calibri="" style="font-family:">(<i>G, H</i>)</span></span></span><span style="font-size:10.5pt"><span style="line-height:102%"><span new="" roman="" style="font-family:" times="">.</span></span></span></span></span></span><br> &nbsp;&nbsp; ﻣﺮﺯ ﻓﺮﺳﺘﻨﺒﺮﮒ, ﻣﺮﺯ ﻫﺎﻣﺎﻧﺎ, ﭘﻮﺷﺶ ﺍﻧﮋﮐﺘﯿﻮ Furstenberg boundary, Hamana boundary, Injective envelope 1 7 http://mmr.khu.ac.ir/browse.php?a_code=A-12-183-1&slc_lang=fa&sid=1 Mohammad Bagher Asadi محمد باقر اسدی mb.asadi@ut.ac.ir 10031947532846006960 10031947532846006960 Yes University of Tehran دانشگاه تهران Zahra Hassanpour-Yakhdani زهرا حسن پور یخدانی zhassanpour@gmail.com 10031947532846006961 10031947532846006961 No University of Tehran دانشگاه تهران