unitization of an essential ideal [closed]












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Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?






operator-theory operator-algebras c-star-algebras







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closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07


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    $begingroup$


    Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?






    operator-theory operator-algebras c-star-algebras







    share|cite|improve this question









    $endgroup$



    closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, metamorphy, Davide Giraudo, amWhy, Leucippus

    If this question can be reworded to fit the rules in the help center, please edit the question.



















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      -1








      -1





      $begingroup$


      Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?






      operator-theory operator-algebras c-star-algebras







      share|cite|improve this question









      $endgroup$




      Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?






      operator-theory operator-algebras c-star-algebras




      operator-theory operator-algebras c-star-algebras






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      share|cite|improve this question











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      asked Dec 19 '18 at 5:05









      mathrookiemathrookie

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      918512




      closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, metamorphy, Davide Giraudo, amWhy, Leucippus

      If this question can be reworded to fit the rules in the help center, please edit the question.







      closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, metamorphy, Davide Giraudo, amWhy, Leucippus

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          1 Answer
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          If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:



          enter image description here



          (see Theorem 3.1.8, [1])



          When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.




          Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.



          [1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.






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            1 Answer
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            1 Answer
            1






            active

            oldest

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            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$


            If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:



            enter image description here



            (see Theorem 3.1.8, [1])



            When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.




            Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.



            [1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.






            share|cite|improve this answer









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              1












              $begingroup$


              If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:



              enter image description here



              (see Theorem 3.1.8, [1])



              When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.




              Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.



              [1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$


                If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:



                enter image description here



                (see Theorem 3.1.8, [1])



                When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.




                Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.



                [1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.






                share|cite|improve this answer









                $endgroup$




                If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:



                enter image description here



                (see Theorem 3.1.8, [1])



                When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.




                Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.



                [1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 19 '18 at 12:39









                C.DingC.Ding

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