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• W4285307145 abstract "Abstract Let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} be a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -algebra, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:math> {mathcal{ {mathcal M} }} be a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast - <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} -bimodule, and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> </m:math> delta be a linear mapping from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} into <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:math> {mathcal{ {mathcal M} }} . <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> </m:math> delta is called a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast - derivation if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mi>B</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>A</m:mi> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>B</m:mi> </m:math> delta left(AB)=Adelta left(B)+delta left(A)B and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>δ</m:mi> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> delta left({A}^{ast })=delta {left(A)}^{ast } for each <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> </m:math> A,B in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} . Let <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>G</m:mi> </m:math> G be an element in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> </m:math> delta is called a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast - antiderivable mapping at <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>G</m:mi> </m:math> G if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>A</m:mi> <m:msup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mi>G</m:mi> <m:mo>⇒</m:mo> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>δ</m:mi> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> <m:mi>A</m:mi> </m:math> A{B}^{ast }=GRightarrow delta left(G)={B}^{ast }delta left(A)+delta {left(B)}^{ast }A for each <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> </m:math> A,B in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} . We prove that if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} is a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mo>∗</m:mo> </m:mrow> </m:msup> </m:math> {C}^{ast } -algebra, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:math> {mathcal{ {mathcal M} }} is a Banach <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast - <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} -bimodule and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>G</m:mi> </m:math> G in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} is a separating point of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:math> {mathcal{ {mathcal M} }} with <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>A</m:mi> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mi>G</m:mi> <m:mi>A</m:mi> </m:math> AG=GA for every <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>A</m:mi> </m:math> A in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} , then every <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -antiderivable mapping at <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>G</m:mi> </m:math> G from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} into <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:math> {mathcal{ {mathcal M} }} is a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -derivation. We also prove that if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} is a zero product determined Banach <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -algebra with a bounded approximate identity, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:math> {mathcal{ {mathcal M} }} is an essential Banach <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast - <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} -bimodule and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> </m:math> delta is a continuous <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -antiderivable mapping at the point zero from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} into <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:math> {mathcal{ {mathcal M} }} , then there exists a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -Jordan derivation <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Δ</m:mi> </m:math> Delta from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} into <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:mrow> <m:mrow> <m:mi>♯</m:mi> <m:mi>♯</m:mi> </m:mrow> </m:msup> </m:math> {{mathcal{ {mathcal M} }}}^{sharp sharp } and an element <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>ξ</m:mi> </m:math> xi in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi class=MJX-tex-caligraphic mathvariant=script>ℳ</m:mi> </m:mrow> <m:mrow> <m:mi>♯</m:mi> <m:mi>♯</m:mi> </m:mrow> </m:msup> </m:math> {{mathcal{ {mathcal M} }}}^{sharp sharp } such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi mathvariant=normal>Δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>A</m:mi> <m:mi>ξ</m:mi> </m:math> delta left(A)=Delta left(A)+Axi for every <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>A</m:mi> </m:math> A in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} . Finally, we show that if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} is a von Neumann algebra and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> </m:math> delta is a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -antiderivable mapping (not necessary continuous) at the point zero from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} into itself, then there exists a <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∗</m:mo> </m:math> ast -derivation <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=normal>Δ</m:mi> </m:math> Delta from <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} into itself such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi mathvariant=normal>Δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>A</m:mi> <m:mi>δ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>I</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> delta left(A)=Delta left(A)+Adelta left(I) for every <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>A</m:mi> </m:math> A in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi class=MJX-tex-caligraphic mathvariant=script>A</m:mi> </m:math> {mathcal{A}} ." @default.
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• W4285307145 title "Characterizations of *-antiderivable mappings on operator algebras" @default.
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