About “names” of von Neuman algebra morphisms
$begingroup$
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
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
$endgroup$
add a comment |
$begingroup$
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
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
$endgroup$
add a comment |
$begingroup$
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
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
$endgroup$
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
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
asked Dec 28 '18 at 19:04
Gabriel PalauGabriel Palau
1106
1106
add a comment |
add a comment |
1 Answer
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$begingroup$
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.
answered Dec 28 '18 at 22:25
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
$begingroup$
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
add a comment |
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$begingroup$
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.
answered Dec 28 '18 at 22:25
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Dec 28 '18 at 22:25
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Dec 28 '18 at 22:25
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
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.
answered Dec 28 '18 at 22:25
Martin ArgeramiMartin Argerami
127k1182183
answered Dec 28 '18 at 22:25
Martin ArgeramiMartin Argerami
127k1182183
answered Dec 28 '18 at 22:25
Martin ArgeramiMartin Argerami
127k1182183
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
add a comment |
$begingroup$
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
add a comment |
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