Mathematical Researches
پژوهش های ریاضی
mmr
Basic Sciences
http://mmr.khu.ac.ir
1
admin
2588-2546
2588-2554
10.61186/mmr
fa
jalali
1400
12
1
gregorian
2022
3
1
8
1
online
1
fulltext
fa
نگاشتهای خطی مشابه پادمشتقها در عناصر متعامد روی جبرهای فون-نویمان
Linear maps on von-Neumann algebras behaving like anti-derivations at orthogonal elements
نظریۀ عملگر
Operator theorey
علمی پژوهشی بنیادی
S
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin"">فرض کنید </span></span><m:omath><span lang="FA" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> جبری فون-نویمان و </span></span><m:omath><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r><m:r>: </m:r></span></span></i><span lang="FA" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>→</m:r></span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> نگاشت خطی پیوسته باشد. همچنین فرض کنید </span></span><m:omath><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i></m:omath><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> در یکی از شرایط زیر صدق کند:</span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omath><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r> </m:r><m:r>xy</m:r><m:r>=0</m:r><m:r>⟹</m:r><m:r>yδ</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>=0, (</m:r><m:r>x</m:r><m:r> , </m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>);</m:r></span></span></i></m:omath></m:omathpara><span dir="LTR" style="font-size:10.0pt"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omath><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr> </m:r><m:r>x</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0⟹</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>δ</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d></m:e><m:sup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>=0, </m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r> , </m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>;</m:r></span></span></i></m:omath></m:omathpara><span dir="LTR" style="font-size:10.0pt"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><m:omathpara><m:omath><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r><m:r>=0</m:r><m:r>⟹</m:r><m:r>yδ</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d></m:e><m:sup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0, (</m:r><m:r>x</m:r><m:r> , </m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>).</m:r></span></span></i></m:omath></m:omathpara><span dir="LTR" style="font-size:10.0pt"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin"">در این مقاله در هر یک از حالتهای ذکر شده ساختار </span></span><m:omath><i><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i></m:omath><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> را مشخصهسازی میکنیم.</span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="text-justify:kashida"><span style="text-kashida:0%"><span style="line-height:22.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><b><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""></span></span></b></span></span></span></span></span></span></span><br>
<span style="line-height:18.0pt"><b><span style="font-size:9.0pt">Introduction </span></b></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">Through this paper all algebras and linear spaces are on the complex field </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>C</m:r></span></span></i></m:omath><span style="font-size:9.0pt">. Let </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be an algebra and </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be an </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt">-bimodule. The linear mapping </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>d</m:r><m:r>:</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>→M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> is called an anti-derivation if </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>d</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>xy</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r><m:r>yd</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>d</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:omath> <m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>(</m:r><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt">. Also, </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>d</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> is called a derivation if </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>d</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>xy</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r><m:r>xd</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>d</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:omath> <m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>(</m:r><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt">. The linear mapping </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r><m:r>:</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>→M</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> is a Jordan derivation if </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>d</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>2</m:r></span></span></i></m:sup></m:ssup></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r><m:r>xd</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>d</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:omath> <m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>(</m:r><m:r>x</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt">. Any anti-derivation and derivation is a Jordan derivation, but the converse is not necessarily true. Jordan in [1] has shown that every continuous Jordan derivation on C*-algebra </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> into any Banach </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt">-bimodule is a derivation. Derivations and anti-derivations are important classes of mappings on algebras which have been used to study of structure of algebras. We refer to [2] and the references there in. </span></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">Bersar studied in [3] additive maps on prime ring contain a non-trivial idempotent satisfying </span></span><br>
<span style="line-height:18.0pt"><m:omathpara><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>, </m:r><m:r>xy</m:r><m:r>=0 </m:r><m:r>⟹</m:r><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r><m:r>+</m:r><m:r>xδ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0 .</m:r></span></span></i></m:omath></m:omathpara><span style="font-size:9.0pt"></span></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">Later, many studies have been done in this case and different results were obtained, for instance, see [4, 5, 6, 7, 8, 9] and the references therein. Recently [10, 11, 12, 13], the problem of characterizing continuous linear maps behaving like derivations or anti-derivations at orthogonal elements for several types of orthogonality conditions on <i>*</i>-algebras have been studied. In this paper we study the above problems on von Neumann algebra.</span></span><br>
<span style="line-height:18.0pt"><b><span style="font-size:9.0pt">Material and methods</span></b></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt"> In this article, the subsequent conditions on a continuous linear map </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r><m:r>:</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>→</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> where </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><i> </i><span style="font-size:9.0pt">is a <i>*</i>-algebra has been considered:</span></span><br>
<span style="line-height:18.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><m:omathpara><m:omath><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr> </m:r><m:r>x</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0</m:r><m:r>⟹</m:r><m:r>xδ</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0, (</m:r><m:r>x</m:r><m:r> , </m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>);</m:r></span></span></i></m:omath></m:omathpara><span lang="FA" style="font-size:9.0pt"><span style="font-family:"Arial",sans-serif"></span></span></span></span></span><br>
<span style="line-height:18.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><m:omathpara><m:omath><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr> </m:r><m:r>x</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0⟹</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>xδ</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d></m:e><m:sup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0, </m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r> , </m:r><m:r>y</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>.</m:r></span></span></i></m:omath></m:omathpara><span lang="FA" style="font-size:9.0pt"><span style="font-family:"Arial",sans-serif"></span></span></span></span></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">We consider following conditions on continuous linear map on von Neumann algebras:</span></span><br>
<span style="line-height:18.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><m:omathpara><m:omath><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r><m:r><i>xy</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0</m:r><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>yδ</i></m:r></span></span><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>δ</i></m:r></span></span><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0</m:r><m:r><i>, (</i></m:r><m:r><i>x</i></m:r><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr><i>)</i></m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>;</m:r></span></span></m:omath></m:omathpara><span dir="LTR" style="font-size:9.0pt"></span></span></span></span><br>
<span style="line-height:18.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><m:omathpara><m:omath><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r><m:r><i>x</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0⟹</m:r></span></span><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>δ</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d></m:e><m:sup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0, (</m:r><m:r><i>x</i></m:r><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>);</m:r></span></span></m:omath></m:omathpara><span dir="LTR" style="font-size:9.0pt"></span></span></span></span><br>
<span style="line-height:18.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><m:omathpara><m:omath><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0</m:r><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>yδ</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d></m:e><m:sup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>δ</i></m:r></span></span><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><span dir="LTR" style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0, (</m:r><m:r><i>x</i></m:r><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>).</m:r></span></span></m:omath></m:omathpara><span lang="FA" style="font-size:9.0pt"><span style="font-family:"Arial",sans-serif"></span></span></span></span></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">Over methods are based on structure of von Neumann algebras and the fact that every derivation on von Neumann algebras is inner. </span></span><br>
<span style="line-height:18.0pt"><b><span style="font-size:9.0pt">Main Results</span></b></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">The followings are the main results of our paper.</span></span><br>
<span style="line-height:18.0pt"><b><span style="font-size:9.0pt">Theorem. </span></b><span style="font-size:9.0pt">Let </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be a von Neumann algebra and </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r><m:r>:</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>→</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> is a </span><span style="font-size:9.0pt">continuous linear map. Then </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> satisfies </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r><m:r> </m:r><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>=0</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> for all </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span style="font-size:9.0pt"> with </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>xy</m:r><m:r>=0</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> if only if there are elements </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>,</m:r><m:r>ν</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> such that </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r><m:r>x</m:r><m:r> </m:r><m:r>μ</m:r><m:r>-</m:r><m:r>νx</m:r></span></span></i></m:omath><span style="font-size:9.0pt">, where </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>-</m:r><m:r>ν</m:r><m:r>∈</m:r><m:r>Z</m:r><m:r> (</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> and </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"><m:lit></m:lit></m:sty></m:scr></m:rpr>[</m:r></span></span></i><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>μ</m:r><m:r>]+2</m:r></span></span></i><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>-</m:r><m:r>ν</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> for all </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span style="font-size:9.0pt">.</span></span><br>
<span style="line-height:18.0pt"><b><span style="font-size:9.0pt">Theorem. </span></b><span style="font-size:9.0pt">Let </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be a von Neumann algebra and </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r><m:r>:</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>→</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> is a </span><span style="font-size:9.0pt">continuous linear map. Then </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> satisfies </span><m:omath><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>δ</i></m:r></span></span><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e></m:d></m:e><m:sup><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0</m:r></span></span></m:omath><span style="font-size:9.0pt"> for all </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span style="font-size:9.0pt"> with </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>y</m:r></span></span></i></m:e><m:sup><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0</m:r></span></span></m:omath><span style="font-size:9.0pt"> if only if there are elements </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>,</m:r><m:r>ν</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> such that </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r><m:r>νx</m:r><m:r>-</m:r><m:r>μx</m:r></span></span></i></m:omath><span style="font-size:9.0pt">, where Re</span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>∈</m:r><m:r>Z</m:r><m:r> (</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> and </span></span><br>
<span style="line-height:18.0pt"><m:omath><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>μ</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>ν</m:r><m:r>-</m:r><m:r>μ</m:r></span></span></i></m:e></m:d></m:e><m:sup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r></span></span></i><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>ν</m:r><m:r>-</m:r><m:r>μ</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0</m:r></span></span></i></m:omath><span style="font-size:9.0pt">,</span></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">for all </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span style="font-size:9.0pt">.</span></span><br>
<span style="line-height:18.0pt"><b><span style="font-size:9.0pt">Theorem. </span></b><span style="font-size:9.0pt">Let </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be a von Neumann algebra and </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r><m:r>:</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>→</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> is a </span><span style="font-size:9.0pt">continuous linear map. Then </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> satisfies </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>y</m:r></span></span></m:e></m:d><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r><m:r>yδ</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d></m:e><m:sup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0</m:r></span></span></m:omath><span style="font-size:9.0pt"> for all </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span style="font-size:9.0pt"> with </span><m:omath><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:ssuppr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e><m:sup><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></span></span></m:sup></m:ssup><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>y=0</m:r></span></span></m:omath><span style="font-size:9.0pt"> if only if there are elements </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>,</m:r><m:r>ν</m:r><m:r>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> such that </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r><m:r>xμ</m:r><m:r>-</m:r><m:r>νx</m:r></span></span></i></m:omath><span style="font-size:9.0pt">, where Re</span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>∈</m:r><m:r>Z</m:r><m:r> (</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> and </span></span><br>
<span style="line-height:18.0pt"><m:omath><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>μ</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r></span></span></i><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><m:ssup><m:ssuppr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>-</m:r><m:r>ν</m:r></span></span></i></m:e></m:d></m:e><m:sup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>*</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+</m:r></span></span></i><m:d><m:dpr><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>μ</m:r><m:r>-</m:r><m:r>ν</m:r></span></span></i></m:e></m:d><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r><m:r>,</m:r><m:r>y</m:r></span></span></i></m:e></m:d><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0</m:r></span></span></i></m:omath><span style="font-size:9.0pt">,</span></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">for all </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>x</m:r></span></span></i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><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>y</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>A</m:r></span></span></m:omath><span style="font-size:9.0pt">.</span></span><br>
<span style="line-height:18.0pt"><b><span style="font-size:9.0pt">Conclusion</span></b></span><br>
<span style="line-height:18.0pt"><span style="font-size:9.0pt">Let </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be a von Neumann algebra and </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r><m:r>:</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>→</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be a </span><span style="font-size:9.0pt">continuous linear map. Let </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> be anti-derivation at orthogonal elements. We characterized the structure of </span><m:omath><i><span style="font-size:9.0pt"><span style="font-family:"Cambria Math",serif"><m:r>δ</m:r></span></span></i></m:omath><span style="font-size:9.0pt"> according to the </span><span dir="RTL" lang="FA" style="font-size:9.0pt"><span style="font-family:"Arial",sans-serif">)</span></span><span style="font-size:9.0pt">generalized) </span><span style="font-size:9.0pt">inner derivation. </span></span><br>
<span style="font-size:9.0pt"><span style="line-height:107%"><span style="font-family:"Calibri",sans-serif">We guess that the results obtained can also be proved on standard operator algebras.</span></span></span>
پادمشتق, جبرهای فون-نویمان, عناصر متعامد.
anti-derivations, orthogonal elements, von-Neumann algebras.
224
234
http://mmr.khu.ac.ir/browse.php?a_code=A-13-120-2&slc_lang=fa&sid=1
Hoger
Ghahranmani
هوگر
قهرمانی
h.ghahramani@uok.ac.ir
10031947532846005774
10031947532846005774
Yes
University of Kurdistan
دانشگاه کردستان
Behrooz
Fadaee
بهروز
فدایی
b.fadaee@sci.uok.ac.ir
10031947532846005775
10031947532846005775
No
University of Kurdistan
دانشگاه کردستان
Kamal
Fallahi
کمال
فلاحی
fallahi1361@gmail.com
10031947532846005776
10031947532846005776
No
Department of Mathematics, Payam Noor University of Technology
دانشگاه پیام نور تهران