Can distinct elements of a $C^*$-algebra be separated by a maximal left ideal?












3














Let $A$ be a $C^*$-algebra, and let $fneq gin A$. Does there exist a maximal ideal $Jtrianglelefteq A$ with $f+Jneq g+J$?



I'm particularly interested in the case of $A=B(mathcal H)$, and why things aren't obvious. In this case, $fneq g$ implies that there is some $hinmathcal H$ with $f(h)neq g(h)$, so the left ideal of the annihilator of $h$ certainly separates them. However, $text{Ann}_{B(mathcal H)}(h)$ is not maximal, and I don't immediately see how to upgrade this to a maximal ideal.



An alternative approach is to attempt to find an irreducible representation. We have a good family of representations from GNS: for each positive linear functional $rho$ on $A$, we have a representation $pi_rho$ on a Hilbert space, with the image of $1$ being cyclic for the representation. If $fneq g$, then at least one of these representations has $fcdot [1]neq gcdot [1]$, because the GNS representation is faithful. I know that the annihilator of a simple module is the intersection of maximal left ideals, so if I believe that if I can find a simple $A$-module on which $f$ and $g$ act differently, there is such an ideal.






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    3














    Let $A$ be a $C^*$-algebra, and let $fneq gin A$. Does there exist a maximal ideal $Jtrianglelefteq A$ with $f+Jneq g+J$?



    I'm particularly interested in the case of $A=B(mathcal H)$, and why things aren't obvious. In this case, $fneq g$ implies that there is some $hinmathcal H$ with $f(h)neq g(h)$, so the left ideal of the annihilator of $h$ certainly separates them. However, $text{Ann}_{B(mathcal H)}(h)$ is not maximal, and I don't immediately see how to upgrade this to a maximal ideal.



    An alternative approach is to attempt to find an irreducible representation. We have a good family of representations from GNS: for each positive linear functional $rho$ on $A$, we have a representation $pi_rho$ on a Hilbert space, with the image of $1$ being cyclic for the representation. If $fneq g$, then at least one of these representations has $fcdot [1]neq gcdot [1]$, because the GNS representation is faithful. I know that the annihilator of a simple module is the intersection of maximal left ideals, so if I believe that if I can find a simple $A$-module on which $f$ and $g$ act differently, there is such an ideal.






    c-star-algebras







    share|cite|improve this question

























      3












      3








      3







      Let $A$ be a $C^*$-algebra, and let $fneq gin A$. Does there exist a maximal ideal $Jtrianglelefteq A$ with $f+Jneq g+J$?



      I'm particularly interested in the case of $A=B(mathcal H)$, and why things aren't obvious. In this case, $fneq g$ implies that there is some $hinmathcal H$ with $f(h)neq g(h)$, so the left ideal of the annihilator of $h$ certainly separates them. However, $text{Ann}_{B(mathcal H)}(h)$ is not maximal, and I don't immediately see how to upgrade this to a maximal ideal.



      An alternative approach is to attempt to find an irreducible representation. We have a good family of representations from GNS: for each positive linear functional $rho$ on $A$, we have a representation $pi_rho$ on a Hilbert space, with the image of $1$ being cyclic for the representation. If $fneq g$, then at least one of these representations has $fcdot [1]neq gcdot [1]$, because the GNS representation is faithful. I know that the annihilator of a simple module is the intersection of maximal left ideals, so if I believe that if I can find a simple $A$-module on which $f$ and $g$ act differently, there is such an ideal.






      c-star-algebras







      share|cite|improve this question













      Let $A$ be a $C^*$-algebra, and let $fneq gin A$. Does there exist a maximal ideal $Jtrianglelefteq A$ with $f+Jneq g+J$?



      I'm particularly interested in the case of $A=B(mathcal H)$, and why things aren't obvious. In this case, $fneq g$ implies that there is some $hinmathcal H$ with $f(h)neq g(h)$, so the left ideal of the annihilator of $h$ certainly separates them. However, $text{Ann}_{B(mathcal H)}(h)$ is not maximal, and I don't immediately see how to upgrade this to a maximal ideal.



      An alternative approach is to attempt to find an irreducible representation. We have a good family of representations from GNS: for each positive linear functional $rho$ on $A$, we have a representation $pi_rho$ on a Hilbert space, with the image of $1$ being cyclic for the representation. If $fneq g$, then at least one of these representations has $fcdot [1]neq gcdot [1]$, because the GNS representation is faithful. I know that the annihilator of a simple module is the intersection of maximal left ideals, so if I believe that if I can find a simple $A$-module on which $f$ and $g$ act differently, there is such an ideal.






      c-star-algebras




      c-star-algebras






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      asked Dec 10 '18 at 0:59









      Ashwin Trisal

      1,2141516




      1,2141516






















          2 Answers
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          $forall aneq 0$, there is a pure state $tau$ on $A$ such that $tau(a^*a)=|a^*a|neq 0$(Theorem 5.1.11, [1]). Let $N_tau={a|tau(a^*a)=0},$ then $N_tau$ is a maximal left ideal(Theorem 5.3.5, [1]), and $anotin N_tau$.



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






          share|cite|improve this answer





























            1














            Your ideal is maximal when $A=B(H)$. Let $J={T: Th=0}$. Let $Sin B(H)setminus J$. Then $Shne0$. Choose $k$ with $langle Sh,krangle=1$, and let $Wx=langle x,krangle,h$. Put $R=I-WS$. Then
            $$ Rh=h-WSh=h-h=0, $$ so $Rin J$. Then $$ I=R+WSin J+mathbb C,WS,$$ and $J$ is maximal (its codimension is 1).






            share|cite|improve this answer





















            • But $J$ is not an ideal.
              – C.Ding
              Dec 10 '18 at 3:59












            • I might be missing something. In what sense is $J $ not a left ideal?
              – Martin Argerami
              Dec 10 '18 at 4:14










            • The OP needs an ideal, not just an left ideal.
              – C.Ding
              Dec 10 '18 at 4:19










            • Separating points by maximal ideals in $B (H) $? And, did you read the title of the question?
              – Martin Argerami
              Dec 10 '18 at 4:21











            Your Answer





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            2 Answers
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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1














            $forall aneq 0$, there is a pure state $tau$ on $A$ such that $tau(a^*a)=|a^*a|neq 0$(Theorem 5.1.11, [1]). Let $N_tau={a|tau(a^*a)=0},$ then $N_tau$ is a maximal left ideal(Theorem 5.3.5, [1]), and $anotin N_tau$.



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






            share|cite|improve this answer


























              1














              $forall aneq 0$, there is a pure state $tau$ on $A$ such that $tau(a^*a)=|a^*a|neq 0$(Theorem 5.1.11, [1]). Let $N_tau={a|tau(a^*a)=0},$ then $N_tau$ is a maximal left ideal(Theorem 5.3.5, [1]), and $anotin N_tau$.



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






              share|cite|improve this answer
























                1












                1








                1






                $forall aneq 0$, there is a pure state $tau$ on $A$ such that $tau(a^*a)=|a^*a|neq 0$(Theorem 5.1.11, [1]). Let $N_tau={a|tau(a^*a)=0},$ then $N_tau$ is a maximal left ideal(Theorem 5.3.5, [1]), and $anotin N_tau$.



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






                share|cite|improve this answer












                $forall aneq 0$, there is a pure state $tau$ on $A$ such that $tau(a^*a)=|a^*a|neq 0$(Theorem 5.1.11, [1]). Let $N_tau={a|tau(a^*a)=0},$ then $N_tau$ is a maximal left ideal(Theorem 5.3.5, [1]), and $anotin N_tau$.



                [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 10 '18 at 4:37









                C.Ding

                1,3931321




                1,3931321























                    1














                    Your ideal is maximal when $A=B(H)$. Let $J={T: Th=0}$. Let $Sin B(H)setminus J$. Then $Shne0$. Choose $k$ with $langle Sh,krangle=1$, and let $Wx=langle x,krangle,h$. Put $R=I-WS$. Then
                    $$ Rh=h-WSh=h-h=0, $$ so $Rin J$. Then $$ I=R+WSin J+mathbb C,WS,$$ and $J$ is maximal (its codimension is 1).






                    share|cite|improve this answer





















                    • But $J$ is not an ideal.
                      – C.Ding
                      Dec 10 '18 at 3:59












                    • I might be missing something. In what sense is $J $ not a left ideal?
                      – Martin Argerami
                      Dec 10 '18 at 4:14










                    • The OP needs an ideal, not just an left ideal.
                      – C.Ding
                      Dec 10 '18 at 4:19










                    • Separating points by maximal ideals in $B (H) $? And, did you read the title of the question?
                      – Martin Argerami
                      Dec 10 '18 at 4:21
















                    1














                    Your ideal is maximal when $A=B(H)$. Let $J={T: Th=0}$. Let $Sin B(H)setminus J$. Then $Shne0$. Choose $k$ with $langle Sh,krangle=1$, and let $Wx=langle x,krangle,h$. Put $R=I-WS$. Then
                    $$ Rh=h-WSh=h-h=0, $$ so $Rin J$. Then $$ I=R+WSin J+mathbb C,WS,$$ and $J$ is maximal (its codimension is 1).






                    share|cite|improve this answer





















                    • But $J$ is not an ideal.
                      – C.Ding
                      Dec 10 '18 at 3:59












                    • I might be missing something. In what sense is $J $ not a left ideal?
                      – Martin Argerami
                      Dec 10 '18 at 4:14










                    • The OP needs an ideal, not just an left ideal.
                      – C.Ding
                      Dec 10 '18 at 4:19










                    • Separating points by maximal ideals in $B (H) $? And, did you read the title of the question?
                      – Martin Argerami
                      Dec 10 '18 at 4:21














                    1












                    1








                    1






                    Your ideal is maximal when $A=B(H)$. Let $J={T: Th=0}$. Let $Sin B(H)setminus J$. Then $Shne0$. Choose $k$ with $langle Sh,krangle=1$, and let $Wx=langle x,krangle,h$. Put $R=I-WS$. Then
                    $$ Rh=h-WSh=h-h=0, $$ so $Rin J$. Then $$ I=R+WSin J+mathbb C,WS,$$ and $J$ is maximal (its codimension is 1).






                    share|cite|improve this answer












                    Your ideal is maximal when $A=B(H)$. Let $J={T: Th=0}$. Let $Sin B(H)setminus J$. Then $Shne0$. Choose $k$ with $langle Sh,krangle=1$, and let $Wx=langle x,krangle,h$. Put $R=I-WS$. Then
                    $$ Rh=h-WSh=h-h=0, $$ so $Rin J$. Then $$ I=R+WSin J+mathbb C,WS,$$ and $J$ is maximal (its codimension is 1).







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Dec 10 '18 at 3:40









                    Martin Argerami

                    124k1176174




                    124k1176174












                    • But $J$ is not an ideal.
                      – C.Ding
                      Dec 10 '18 at 3:59












                    • I might be missing something. In what sense is $J $ not a left ideal?
                      – Martin Argerami
                      Dec 10 '18 at 4:14










                    • The OP needs an ideal, not just an left ideal.
                      – C.Ding
                      Dec 10 '18 at 4:19










                    • Separating points by maximal ideals in $B (H) $? And, did you read the title of the question?
                      – Martin Argerami
                      Dec 10 '18 at 4:21


















                    • But $J$ is not an ideal.
                      – C.Ding
                      Dec 10 '18 at 3:59












                    • I might be missing something. In what sense is $J $ not a left ideal?
                      – Martin Argerami
                      Dec 10 '18 at 4:14










                    • The OP needs an ideal, not just an left ideal.
                      – C.Ding
                      Dec 10 '18 at 4:19










                    • Separating points by maximal ideals in $B (H) $? And, did you read the title of the question?
                      – Martin Argerami
                      Dec 10 '18 at 4:21
















                    But $J$ is not an ideal.
                    – C.Ding
                    Dec 10 '18 at 3:59






                    But $J$ is not an ideal.
                    – C.Ding
                    Dec 10 '18 at 3:59














                    I might be missing something. In what sense is $J $ not a left ideal?
                    – Martin Argerami
                    Dec 10 '18 at 4:14




                    I might be missing something. In what sense is $J $ not a left ideal?
                    – Martin Argerami
                    Dec 10 '18 at 4:14












                    The OP needs an ideal, not just an left ideal.
                    – C.Ding
                    Dec 10 '18 at 4:19




                    The OP needs an ideal, not just an left ideal.
                    – C.Ding
                    Dec 10 '18 at 4:19












                    Separating points by maximal ideals in $B (H) $? And, did you read the title of the question?
                    – Martin Argerami
                    Dec 10 '18 at 4:21




                    Separating points by maximal ideals in $B (H) $? And, did you read the title of the question?
                    – Martin Argerami
                    Dec 10 '18 at 4:21


















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