Do all *-isomorphisms between von Neumann algebras preserve strong operator topology?












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Do all $*$-isomorphisms between von Neumann algebras preserve the strong operator topology?



Seems clearly true for $*$-isomorphisms with a unitary implementation, but I don't see the answer for other cases ... perhaps there is an easy argument from the fact that von Neumann algebras are closed in this topology, but I've spent a while looking for one and don't see it.










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    2














    Do all $*$-isomorphisms between von Neumann algebras preserve the strong operator topology?



    Seems clearly true for $*$-isomorphisms with a unitary implementation, but I don't see the answer for other cases ... perhaps there is an easy argument from the fact that von Neumann algebras are closed in this topology, but I've spent a while looking for one and don't see it.










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      2












      2








      2







      Do all $*$-isomorphisms between von Neumann algebras preserve the strong operator topology?



      Seems clearly true for $*$-isomorphisms with a unitary implementation, but I don't see the answer for other cases ... perhaps there is an easy argument from the fact that von Neumann algebras are closed in this topology, but I've spent a while looking for one and don't see it.










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      Do all $*$-isomorphisms between von Neumann algebras preserve the strong operator topology?



      Seems clearly true for $*$-isomorphisms with a unitary implementation, but I don't see the answer for other cases ... perhaps there is an easy argument from the fact that von Neumann algebras are closed in this topology, but I've spent a while looking for one and don't see it.







      operator-algebras von-neumann-algebras






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      asked Nov 28 '18 at 18:02









      Doug McLellanDoug McLellan

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          No. Take $M$ to be any II$_1$-factor. Let $pi:Mto B(H)$ be an irreducible representation (it exists because you can do GNS of a pure state). As $M$ is simple (as a C$^*$-algebra!), $pi$ is injective. And $pi(M)$ is dense in $B(H)$, but it cannot be everything.



          So $pi:Mtopi(M)$ is a $*$-isomorphism that does not preserve the sot/wot/ultrasot/ultrawot topologies.



          As mentioned in the comments, if $Msubset B(H)$ and $Nsubset B(K)$ are von Neumann algebras (in the usual "double commutant" sense) then a $*$-isomorphism $pi:Mto N$ is sot-continuous by passing through normality.






          share|cite|improve this answer























          • But in this case $pi$ is not a $ast$-isomorphism since it is not surjective.
            – Adrián González-Pérez
            Nov 29 '18 at 12:17










          • I am also not convinced since $ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $pi$. Perhaps I am confused about that.
            – Adrián González-Pérez
            Nov 29 '18 at 12:18






          • 1




            @Adrián: of course, I can take $pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra.
            – Martin Argerami
            Nov 29 '18 at 15:09












          • Thanks! I didn't realize that.
            – Adrián González-Pérez
            Nov 29 '18 at 15:25






          • 1




            No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me).
            – Martin Argerami
            Nov 29 '18 at 15:33











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          No. Take $M$ to be any II$_1$-factor. Let $pi:Mto B(H)$ be an irreducible representation (it exists because you can do GNS of a pure state). As $M$ is simple (as a C$^*$-algebra!), $pi$ is injective. And $pi(M)$ is dense in $B(H)$, but it cannot be everything.



          So $pi:Mtopi(M)$ is a $*$-isomorphism that does not preserve the sot/wot/ultrasot/ultrawot topologies.



          As mentioned in the comments, if $Msubset B(H)$ and $Nsubset B(K)$ are von Neumann algebras (in the usual "double commutant" sense) then a $*$-isomorphism $pi:Mto N$ is sot-continuous by passing through normality.






          share|cite|improve this answer























          • But in this case $pi$ is not a $ast$-isomorphism since it is not surjective.
            – Adrián González-Pérez
            Nov 29 '18 at 12:17










          • I am also not convinced since $ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $pi$. Perhaps I am confused about that.
            – Adrián González-Pérez
            Nov 29 '18 at 12:18






          • 1




            @Adrián: of course, I can take $pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra.
            – Martin Argerami
            Nov 29 '18 at 15:09












          • Thanks! I didn't realize that.
            – Adrián González-Pérez
            Nov 29 '18 at 15:25






          • 1




            No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me).
            – Martin Argerami
            Nov 29 '18 at 15:33
















          3














          No. Take $M$ to be any II$_1$-factor. Let $pi:Mto B(H)$ be an irreducible representation (it exists because you can do GNS of a pure state). As $M$ is simple (as a C$^*$-algebra!), $pi$ is injective. And $pi(M)$ is dense in $B(H)$, but it cannot be everything.



          So $pi:Mtopi(M)$ is a $*$-isomorphism that does not preserve the sot/wot/ultrasot/ultrawot topologies.



          As mentioned in the comments, if $Msubset B(H)$ and $Nsubset B(K)$ are von Neumann algebras (in the usual "double commutant" sense) then a $*$-isomorphism $pi:Mto N$ is sot-continuous by passing through normality.






          share|cite|improve this answer























          • But in this case $pi$ is not a $ast$-isomorphism since it is not surjective.
            – Adrián González-Pérez
            Nov 29 '18 at 12:17










          • I am also not convinced since $ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $pi$. Perhaps I am confused about that.
            – Adrián González-Pérez
            Nov 29 '18 at 12:18






          • 1




            @Adrián: of course, I can take $pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra.
            – Martin Argerami
            Nov 29 '18 at 15:09












          • Thanks! I didn't realize that.
            – Adrián González-Pérez
            Nov 29 '18 at 15:25






          • 1




            No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me).
            – Martin Argerami
            Nov 29 '18 at 15:33














          3












          3








          3






          No. Take $M$ to be any II$_1$-factor. Let $pi:Mto B(H)$ be an irreducible representation (it exists because you can do GNS of a pure state). As $M$ is simple (as a C$^*$-algebra!), $pi$ is injective. And $pi(M)$ is dense in $B(H)$, but it cannot be everything.



          So $pi:Mtopi(M)$ is a $*$-isomorphism that does not preserve the sot/wot/ultrasot/ultrawot topologies.



          As mentioned in the comments, if $Msubset B(H)$ and $Nsubset B(K)$ are von Neumann algebras (in the usual "double commutant" sense) then a $*$-isomorphism $pi:Mto N$ is sot-continuous by passing through normality.






          share|cite|improve this answer














          No. Take $M$ to be any II$_1$-factor. Let $pi:Mto B(H)$ be an irreducible representation (it exists because you can do GNS of a pure state). As $M$ is simple (as a C$^*$-algebra!), $pi$ is injective. And $pi(M)$ is dense in $B(H)$, but it cannot be everything.



          So $pi:Mtopi(M)$ is a $*$-isomorphism that does not preserve the sot/wot/ultrasot/ultrawot topologies.



          As mentioned in the comments, if $Msubset B(H)$ and $Nsubset B(K)$ are von Neumann algebras (in the usual "double commutant" sense) then a $*$-isomorphism $pi:Mto N$ is sot-continuous by passing through normality.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 29 '18 at 21:16

























          answered Nov 28 '18 at 18:32









          Martin ArgeramiMartin Argerami

          124k1176175




          124k1176175












          • But in this case $pi$ is not a $ast$-isomorphism since it is not surjective.
            – Adrián González-Pérez
            Nov 29 '18 at 12:17










          • I am also not convinced since $ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $pi$. Perhaps I am confused about that.
            – Adrián González-Pérez
            Nov 29 '18 at 12:18






          • 1




            @Adrián: of course, I can take $pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra.
            – Martin Argerami
            Nov 29 '18 at 15:09












          • Thanks! I didn't realize that.
            – Adrián González-Pérez
            Nov 29 '18 at 15:25






          • 1




            No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me).
            – Martin Argerami
            Nov 29 '18 at 15:33


















          • But in this case $pi$ is not a $ast$-isomorphism since it is not surjective.
            – Adrián González-Pérez
            Nov 29 '18 at 12:17










          • I am also not convinced since $ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $pi$. Perhaps I am confused about that.
            – Adrián González-Pérez
            Nov 29 '18 at 12:18






          • 1




            @Adrián: of course, I can take $pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra.
            – Martin Argerami
            Nov 29 '18 at 15:09












          • Thanks! I didn't realize that.
            – Adrián González-Pérez
            Nov 29 '18 at 15:25






          • 1




            No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me).
            – Martin Argerami
            Nov 29 '18 at 15:33
















          But in this case $pi$ is not a $ast$-isomorphism since it is not surjective.
          – Adrián González-Pérez
          Nov 29 '18 at 12:17




          But in this case $pi$ is not a $ast$-isomorphism since it is not surjective.
          – Adrián González-Pérez
          Nov 29 '18 at 12:17












          I am also not convinced since $ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $pi$. Perhaps I am confused about that.
          – Adrián González-Pérez
          Nov 29 '18 at 12:18




          I am also not convinced since $ast$-homomorphism must preserve the suprema of ascending families of projections and that will imply normality for $pi$. Perhaps I am confused about that.
          – Adrián González-Pérez
          Nov 29 '18 at 12:18




          1




          1




          @Adrián: of course, I can take $pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra.
          – Martin Argerami
          Nov 29 '18 at 15:09






          @Adrián: of course, I can take $pi(M)$ to be the codomain. And you seem to have a misunderstanding of the equivalence normal $iff$ sot-continuous (and you need selfadjoints, not just projections). The equivalence is true, if you are doing it in a von Neumann algebra. Unless you claim that every monotone-complete C$^*$-algebra is a von Neumann algebra.
          – Martin Argerami
          Nov 29 '18 at 15:09














          Thanks! I didn't realize that.
          – Adrián González-Pérez
          Nov 29 '18 at 15:25




          Thanks! I didn't realize that.
          – Adrián González-Pérez
          Nov 29 '18 at 15:25




          1




          1




          No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me).
          – Martin Argerami
          Nov 29 '18 at 15:33




          No problem. I have mixed feelings about the fact that von Neumann algebras are always considered represented. On the one hand, it makes a lot of sense because it is how you would usually use them. But, on the other hand, because this is usually not really considered in textbooks, together with AW$^*$-algebras becoming unfashionable a few decades ago, there is little knowledge about all this (including me).
          – Martin Argerami
          Nov 29 '18 at 15:33


















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