Mathematical Researches
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نگاشتهای خطی حافظ ضربهای واحد
Linear maps preserving identity products
نظریۀ عملگر
Operator theorey
مقاله مستقل
Original Manuscript
<div>فرض کنید A و B دو C-جبر، A یکدار و <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>ϕ</m:r><m:r>:</m:r><m:r>A</m:r><m:r>⟶</m:r><m:r>B</m:r></span></span></span></i> یک نگاشت خطی پیوسته باشد. در این مقاله، تحت شرایط خاص ثابت میکنیم که <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>ϕ</m:r></span></span></span></i> یک <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i>-همریختی ژوردان است.<br>
همچنین، نشان میدهیم که اگر <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>X</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">یک </span></span></span><i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i><span dir="RTL" lang="FA" style="font-size:12.0pt"><span style="line-height:115%"><span b="" nazanin="" style="font-family:">-</span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>A</m:r></span></span></span></i>-مدول باناخ و <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>σ</m:r><m:r>:</m:r><m:r>A</m:r><m:r>⟶</m:r><m:r>X</m:r></span></span></span></i> یک نگاشت خطی پیوسته باشد که در رابطه اشتقاق (اشتقاق ژوردان) با شرط <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>x</m:r></span></span></span></i> <span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="bi"></m:sty></m:scr></m:rpr><b>1</b></m:r></span></span></span></i> <span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><i>∘</i></m:r></span></span></span> <span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="bi"></m:sty></m:scr></m:rpr><b>1</b></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="bi"></m:sty></m:scr></m:rpr><b>)</b></m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span>صدق کند،</div>
<div>آنگاه <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>σ</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">یک </span></span></span><i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i>-اشتقاق است. <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"> <span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span></div>
<span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"> Let </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>A</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">and </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>B</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">be two </span></span></span> <span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span> <i><span style="font-size:11.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>C</m:r></span></span></span></i> <i><span style="font-size:11.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:4.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">-algebras, </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>A</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">is unital and </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>ϕ</m:r><m:r>:</m:r><m:r>A</m:r><m:r>⟶</m:r><m:r>B</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">be a continuous linear map. In this paper, under certain condition, we prove that </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>ϕ</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">is a Jordan </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">-homomorphism. We also prove that if </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>X</m:r><m:r> </m:r></span></span></span></i><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">is a Banach </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span dir="RTL" lang="FA" style="font-size:12.0pt"><span style="line-height:115%"><span b="" nazanin="" style="font-family:">-</span></span></span><i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>A</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span dir="RTL" lang="FA" style="font-size:12.0pt"><span style="line-height:115%"><span b="" nazanin="" style="font-family:">-</span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">bimodule, and </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>σ</m:r><m:r>:</m:r><m:r>A</m:r><m:r>⟶</m:r><m:r>X</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">is a continuous linear map satisfying derivation (Jordan derivation) equation with </span></span></span> <span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="bi"></m:sty></m:scr></m:rpr><b>1</b></m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span> <span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>x</i></m:r><m:r><i>∘</i></m:r></span></span></span> <span style="font-size:12.0pt"><span cambria="" math="" style="font-family:"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>y</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>=</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="bi"></m:sty></m:scr></m:rpr><b>1</b></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="bi"></m:sty></m:scr></m:rpr><b>)</b></m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">, then </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>σ</m:r></span></span></span></i> <span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:5.5pt"> </span></span></span></span></span><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">is a </span></span></span> <i><span style="font-size:12.0pt"><span style="line-height:115%"><span cambria="" math="" style="font-family:"><m:r>*</m:r></span></span></span></i><span style="font-size:11.0pt"><span style="line-height:115%"><span calibri="" style="font-family:">-derivation</span></span></span>
همریختی, همریختی ژوردان, اشتقاق, اشتقاق ژوردان.
Homomorphism, Jordan homomorphism, derivation, Jordan derivation.
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http://mmr.khu.ac.ir/browse.php?a_code=A-10-558-3&slc_lang=fa&sid=1
Abbas
Zivari-Kazempour
عباس
زیوری کاظم پور
zivari@abru.ac.ir
10031947532846006827
10031947532846006827
Yes
Ayatollah Boroujerdi University
دانشگاه بروجرد