About “names” of von Neuman algebra morphisms












2












$begingroup$


I have actually a basic quastion about maps between von Neumann algebras.



If I have a map $f:N to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal and unital?



I suppose that $f$ is unital if $f(1)=1$, and suppose that $f$ is completely positive if $f$ maps any operator with positive spectrum to other with positive spectrum too. But I really don't know.



Many thanks in advance. And apologise for this basic question.






operator-theory c-star-algebras von-neumann-algebras







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$endgroup$

















    2












    $begingroup$


    I have actually a basic quastion about maps between von Neumann algebras.



    If I have a map $f:N to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal and unital?



    I suppose that $f$ is unital if $f(1)=1$, and suppose that $f$ is completely positive if $f$ maps any operator with positive spectrum to other with positive spectrum too. But I really don't know.



    Many thanks in advance. And apologise for this basic question.






    operator-theory c-star-algebras von-neumann-algebras







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      I have actually a basic quastion about maps between von Neumann algebras.



      If I have a map $f:N to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal and unital?



      I suppose that $f$ is unital if $f(1)=1$, and suppose that $f$ is completely positive if $f$ maps any operator with positive spectrum to other with positive spectrum too. But I really don't know.



      Many thanks in advance. And apologise for this basic question.






      operator-theory c-star-algebras von-neumann-algebras







      share|cite|improve this question









      $endgroup$




      I have actually a basic quastion about maps between von Neumann algebras.



      If I have a map $f:N to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal and unital?



      I suppose that $f$ is unital if $f(1)=1$, and suppose that $f$ is completely positive if $f$ maps any operator with positive spectrum to other with positive spectrum too. But I really don't know.



      Many thanks in advance. And apologise for this basic question.






      operator-theory c-star-algebras von-neumann-algebras




      operator-theory c-star-algebras von-neumann-algebras






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 28 '18 at 19:04









      Gabriel PalauGabriel Palau

      1106




      1106






















          1 Answer
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          1












          $begingroup$

          You are right about unital.



          Normal means that $f$ respect suprema of bounded nets of selfadjoints. That is, if ${a_j}subset M$ and $a=sup a_j$, then $f(a)=sup f(a_j)$. This is the same as saying that $f$ is sot-sot continuous.



          Positive means that $f(x)geq0$ if $xgeq0$. Note that $xgeq0$ not only means that $sigma(x)subset[0,infty)$ but also that $x$ is selfadjoint.



          Completely positive means that $f^{(n)}=fotimes I_n:Motimes M_n(mathbb C)to Notimes M_n(mathbb C)$ is positive for all $ninmathbb N$. That is if $Xin M_n(M)$ is positive, then $[f(X_{kj})]_{k,j}in M_n(N)$ is positive.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators.
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 3:52










          • $begingroup$
            Supremum="least upper bound". If you have an order, you have notion of supremum.
            $endgroup$
            – Martin Argerami
            Dec 29 '18 at 3:53










          • $begingroup$
            yes, you are right. Thanks again
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 4:01











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          1 Answer
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          active

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          1












          $begingroup$

          You are right about unital.



          Normal means that $f$ respect suprema of bounded nets of selfadjoints. That is, if ${a_j}subset M$ and $a=sup a_j$, then $f(a)=sup f(a_j)$. This is the same as saying that $f$ is sot-sot continuous.



          Positive means that $f(x)geq0$ if $xgeq0$. Note that $xgeq0$ not only means that $sigma(x)subset[0,infty)$ but also that $x$ is selfadjoint.



          Completely positive means that $f^{(n)}=fotimes I_n:Motimes M_n(mathbb C)to Notimes M_n(mathbb C)$ is positive for all $ninmathbb N$. That is if $Xin M_n(M)$ is positive, then $[f(X_{kj})]_{k,j}in M_n(N)$ is positive.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators.
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 3:52










          • $begingroup$
            Supremum="least upper bound". If you have an order, you have notion of supremum.
            $endgroup$
            – Martin Argerami
            Dec 29 '18 at 3:53










          • $begingroup$
            yes, you are right. Thanks again
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 4:01
















          1












          $begingroup$

          You are right about unital.



          Normal means that $f$ respect suprema of bounded nets of selfadjoints. That is, if ${a_j}subset M$ and $a=sup a_j$, then $f(a)=sup f(a_j)$. This is the same as saying that $f$ is sot-sot continuous.



          Positive means that $f(x)geq0$ if $xgeq0$. Note that $xgeq0$ not only means that $sigma(x)subset[0,infty)$ but also that $x$ is selfadjoint.



          Completely positive means that $f^{(n)}=fotimes I_n:Motimes M_n(mathbb C)to Notimes M_n(mathbb C)$ is positive for all $ninmathbb N$. That is if $Xin M_n(M)$ is positive, then $[f(X_{kj})]_{k,j}in M_n(N)$ is positive.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators.
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 3:52










          • $begingroup$
            Supremum="least upper bound". If you have an order, you have notion of supremum.
            $endgroup$
            – Martin Argerami
            Dec 29 '18 at 3:53










          • $begingroup$
            yes, you are right. Thanks again
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 4:01














          1












          1








          1





          $begingroup$

          You are right about unital.



          Normal means that $f$ respect suprema of bounded nets of selfadjoints. That is, if ${a_j}subset M$ and $a=sup a_j$, then $f(a)=sup f(a_j)$. This is the same as saying that $f$ is sot-sot continuous.



          Positive means that $f(x)geq0$ if $xgeq0$. Note that $xgeq0$ not only means that $sigma(x)subset[0,infty)$ but also that $x$ is selfadjoint.



          Completely positive means that $f^{(n)}=fotimes I_n:Motimes M_n(mathbb C)to Notimes M_n(mathbb C)$ is positive for all $ninmathbb N$. That is if $Xin M_n(M)$ is positive, then $[f(X_{kj})]_{k,j}in M_n(N)$ is positive.






          share|cite|improve this answer









          $endgroup$



          You are right about unital.



          Normal means that $f$ respect suprema of bounded nets of selfadjoints. That is, if ${a_j}subset M$ and $a=sup a_j$, then $f(a)=sup f(a_j)$. This is the same as saying that $f$ is sot-sot continuous.



          Positive means that $f(x)geq0$ if $xgeq0$. Note that $xgeq0$ not only means that $sigma(x)subset[0,infty)$ but also that $x$ is selfadjoint.



          Completely positive means that $f^{(n)}=fotimes I_n:Motimes M_n(mathbb C)to Notimes M_n(mathbb C)$ is positive for all $ninmathbb N$. That is if $Xin M_n(M)$ is positive, then $[f(X_{kj})]_{k,j}in M_n(N)$ is positive.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 28 '18 at 22:25









          Martin ArgeramiMartin Argerami

          127k1182183




          127k1182183












          • $begingroup$
            Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators.
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 3:52










          • $begingroup$
            Supremum="least upper bound". If you have an order, you have notion of supremum.
            $endgroup$
            – Martin Argerami
            Dec 29 '18 at 3:53










          • $begingroup$
            yes, you are right. Thanks again
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 4:01


















          • $begingroup$
            Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators.
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 3:52










          • $begingroup$
            Supremum="least upper bound". If you have an order, you have notion of supremum.
            $endgroup$
            – Martin Argerami
            Dec 29 '18 at 3:53










          • $begingroup$
            yes, you are right. Thanks again
            $endgroup$
            – Gabriel Palau
            Dec 29 '18 at 4:01
















          $begingroup$
          Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators.
          $endgroup$
          – Gabriel Palau
          Dec 29 '18 at 3:52




          $begingroup$
          Thanks @Martin Argerami, Your answer was very usfull, but I have a new question. The point is that I have never heard the consept of supremum in the context of operators, so I was a bit shocked when I read the definition of normal, but then you say that that is equivalent to sot-sot continuity, and I can understand that. However, I stayed with the doubt about the sup of operators.
          $endgroup$
          – Gabriel Palau
          Dec 29 '18 at 3:52












          $begingroup$
          Supremum="least upper bound". If you have an order, you have notion of supremum.
          $endgroup$
          – Martin Argerami
          Dec 29 '18 at 3:53




          $begingroup$
          Supremum="least upper bound". If you have an order, you have notion of supremum.
          $endgroup$
          – Martin Argerami
          Dec 29 '18 at 3:53












          $begingroup$
          yes, you are right. Thanks again
          $endgroup$
          – Gabriel Palau
          Dec 29 '18 at 4:01




          $begingroup$
          yes, you are right. Thanks again
          $endgroup$
          – Gabriel Palau
          Dec 29 '18 at 4:01


















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