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GNS representation doubt
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Let $M$ be a von Neumann algebra inside $B(mathcalH)$, if we take $xi$ $in mathcalH$, consider vector state $omega_xi$,
$(i)$ Does there exist GNS Hilbert space coming from $M$ such that whose transported state will be $omega_xi$?
$(ii)$ Furthermore are GNS Hilbert spaces are always separable?
$(iii)$ what is the importance of doing GNS construction just to only get cyclic vector, can anybody help me out connecting more ideas with GNS construction?
functional-analysis operator-theory operator-algebras c-star-algebras von-neumann-algebras
edited Jul 17 at 11:58
Omnomnomnom
121k784170
asked Jul 17 at 10:59
mathlover
11218
add a comment |Â
up vote
1
down vote
favorite
Let $M$ be a von Neumann algebra inside $B(mathcalH)$, if we take $xi$ $in mathcalH$, consider vector state $omega_xi$,
$(i)$ Does there exist GNS Hilbert space coming from $M$ such that whose transported state will be $omega_xi$?
$(ii)$ Furthermore are GNS Hilbert spaces are always separable?
$(iii)$ what is the importance of doing GNS construction just to only get cyclic vector, can anybody help me out connecting more ideas with GNS construction?
functional-analysis operator-theory operator-algebras c-star-algebras von-neumann-algebras
edited Jul 17 at 11:58
Omnomnomnom
121k784170
asked Jul 17 at 10:59
mathlover
11218
For (ii), von Neumann algebra almost never represent onto separable Hilbert spaces (finite-dimensionality is necessary I reckon). (i) I am unsure of what the questions is, what is the transported state you mentioned? (iii) Are you only interested in von Neumann algebras or C*-algebras in general?
– Munk
Jul 20 at 1:30
Yes I am interested in vN algebras
– mathlover
Jul 20 at 4:47
Transported meaning is $phi=omega_xicircpi$
– mathlover
Jul 20 at 5:31
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $M$ be a von Neumann algebra inside $B(mathcalH)$, if we take $xi$ $in mathcalH$, consider vector state $omega_xi$,
$(i)$ Does there exist GNS Hilbert space coming from $M$ such that whose transported state will be $omega_xi$?
$(ii)$ Furthermore are GNS Hilbert spaces are always separable?
$(iii)$ what is the importance of doing GNS construction just to only get cyclic vector, can anybody help me out connecting more ideas with GNS construction?
functional-analysis operator-theory operator-algebras c-star-algebras von-neumann-algebras
edited Jul 17 at 11:58
Omnomnomnom
121k784170
asked Jul 17 at 10:59
mathlover
11218
Let $M$ be a von Neumann algebra inside $B(mathcalH)$, if we take $xi$ $in mathcalH$, consider vector state $omega_xi$,
$(i)$ Does there exist GNS Hilbert space coming from $M$ such that whose transported state will be $omega_xi$?
$(ii)$ Furthermore are GNS Hilbert spaces are always separable?
$(iii)$ what is the importance of doing GNS construction just to only get cyclic vector, can anybody help me out connecting more ideas with GNS construction?
functional-analysis operator-theory operator-algebras c-star-algebras von-neumann-algebras
edited Jul 17 at 11:58
Omnomnomnom
121k784170
asked Jul 17 at 10:59
mathlover
11218
edited Jul 17 at 11:58
Omnomnomnom
121k784170
edited Jul 17 at 11:58
Omnomnomnom
121k784170
edited Jul 17 at 11:58
Omnomnomnom
121k784170
121k784170
asked Jul 17 at 10:59
mathlover
11218
asked Jul 17 at 10:59
mathlover
11218
asked Jul 17 at 10:59
mathlover
11218
11218
For (ii), von Neumann algebra almost never represent onto separable Hilbert spaces (finite-dimensionality is necessary I reckon). (i) I am unsure of what the questions is, what is the transported state you mentioned? (iii) Are you only interested in von Neumann algebras or C*-algebras in general?
– Munk
Jul 20 at 1:30
Yes I am interested in vN algebras
– mathlover
Jul 20 at 4:47
Transported meaning is $phi=omega_xicircpi$
– mathlover
Jul 20 at 5:31
add a comment |Â
For (ii), von Neumann algebra almost never represent onto separable Hilbert spaces (finite-dimensionality is necessary I reckon). (i) I am unsure of what the questions is, what is the transported state you mentioned? (iii) Are you only interested in von Neumann algebras or C*-algebras in general?
– Munk
Jul 20 at 1:30
Yes I am interested in vN algebras
– mathlover
Jul 20 at 4:47
Transported meaning is $phi=omega_xicircpi$
– mathlover
Jul 20 at 5:31
For (ii), von Neumann algebra almost never represent onto separable Hilbert spaces (finite-dimensionality is necessary I reckon). (i) I am unsure of what the questions is, what is the transported state you mentioned? (iii) Are you only interested in von Neumann algebras or C*-algebras in general?
– Munk
Jul 20 at 1:30
For (ii), von Neumann algebra almost never represent onto separable Hilbert spaces (finite-dimensionality is necessary I reckon). (i) I am unsure of what the questions is, what is the transported state you mentioned? (iii) Are you only interested in von Neumann algebras or C*-algebras in general?
– Munk
Jul 20 at 1:30
Yes I am interested in vN algebras
– mathlover
Jul 20 at 4:47
Yes I am interested in vN algebras
– mathlover
Jul 20 at 4:47
Transported meaning is $phi=omega_xicircpi$
– mathlover
Jul 20 at 5:31
Transported meaning is $phi=omega_xicircpi$
– mathlover
Jul 20 at 5:31
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
If $M$ is sot/wot separable (i.e., most known and used von Neumann algebras out there) and $phi$ is normal, then the Hilbert space $H_phi$ is separable. To see this, let $x_nsubset M$ be a dense sequence; then for any $xin M$ there exists a bounded subnet $x_n_j$ ($|x_n_j|leq c$) with $psi(x_n_j)to psi(x)$ for all normal states $psi$ (the "bounded" is obtained via Kaplansky on the balls of radius $n$). Fix $varepsilon>0$ and $etain H_phi$. Then there exists $xin M$ with $|eta-hat x|<varepsilon$. So
beginalign
limsup_j |hat x_n_j-eta|_phi
&leqlimsup_j|hat x_n_j-hat x|_phi+|hat x-eta|_phi\ \
&=limsup_jphi((x_n_j-x)^*(x_n_j-x))+varepsilon\ \
&leq (c+|x|)limsup_jphi(x_n_j-x)+varepsilon\ \
&=varepsilon.
endalign
As $varepsilon$ was arbitrary, $hat x_n_jtoeta$, and $H_phi$ is separable.
The original importance of the GNS construction is that it allows you to start with an abstract C$^*$-algebra, and construct a representation on a Hilbert space. By adding the GNS representations over all (classes of) states, one gets a faithful representation of your abstract C$^*$-algebra as bounded operators on a Hilbert space. And the construction is explicit; in particular, having a cyclic vector allows one to define operators explicitly; a nice example of this is the proof that a non-degenerate representation from an ideal extends to the whole algebra (see Lemma I.9.14 in Davidson's C$^*$-Algebra By Example for details).
As for the "transport" question, I'm not sure what you are after. If you start with any state and do GNS, you get a vector state. If you start with a vector state and do GNS with it, nothing too remarkable happens.
answered Jul 20 at 18:52
Martin Argerami
116k1071164
More specifically can we say any two cyclic representations of $C^*$ algebra is unitary equivalent as a consequence of GNS @Martin?
– mathlover
Jul 21 at 9:24
No, not at all. That would make all GNS representations equivalent.
– Martin Argerami
Jul 21 at 12:30
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
If $M$ is sot/wot separable (i.e., most known and used von Neumann algebras out there) and $phi$ is normal, then the Hilbert space $H_phi$ is separable. To see this, let $x_nsubset M$ be a dense sequence; then for any $xin M$ there exists a bounded subnet $x_n_j$ ($|x_n_j|leq c$) with $psi(x_n_j)to psi(x)$ for all normal states $psi$ (the "bounded" is obtained via Kaplansky on the balls of radius $n$). Fix $varepsilon>0$ and $etain H_phi$. Then there exists $xin M$ with $|eta-hat x|<varepsilon$. So
beginalign
limsup_j |hat x_n_j-eta|_phi
&leqlimsup_j|hat x_n_j-hat x|_phi+|hat x-eta|_phi\ \
&=limsup_jphi((x_n_j-x)^*(x_n_j-x))+varepsilon\ \
&leq (c+|x|)limsup_jphi(x_n_j-x)+varepsilon\ \
&=varepsilon.
endalign
As $varepsilon$ was arbitrary, $hat x_n_jtoeta$, and $H_phi$ is separable.
The original importance of the GNS construction is that it allows you to start with an abstract C$^*$-algebra, and construct a representation on a Hilbert space. By adding the GNS representations over all (classes of) states, one gets a faithful representation of your abstract C$^*$-algebra as bounded operators on a Hilbert space. And the construction is explicit; in particular, having a cyclic vector allows one to define operators explicitly; a nice example of this is the proof that a non-degenerate representation from an ideal extends to the whole algebra (see Lemma I.9.14 in Davidson's C$^*$-Algebra By Example for details).
As for the "transport" question, I'm not sure what you are after. If you start with any state and do GNS, you get a vector state. If you start with a vector state and do GNS with it, nothing too remarkable happens.
answered Jul 20 at 18:52
Martin Argerami
116k1071164
More specifically can we say any two cyclic representations of $C^*$ algebra is unitary equivalent as a consequence of GNS @Martin?
– mathlover
Jul 21 at 9:24
No, not at all. That would make all GNS representations equivalent.
– Martin Argerami
Jul 21 at 12:30
add a comment |Â
up vote
1
down vote
accepted
If $M$ is sot/wot separable (i.e., most known and used von Neumann algebras out there) and $phi$ is normal, then the Hilbert space $H_phi$ is separable. To see this, let $x_nsubset M$ be a dense sequence; then for any $xin M$ there exists a bounded subnet $x_n_j$ ($|x_n_j|leq c$) with $psi(x_n_j)to psi(x)$ for all normal states $psi$ (the "bounded" is obtained via Kaplansky on the balls of radius $n$). Fix $varepsilon>0$ and $etain H_phi$. Then there exists $xin M$ with $|eta-hat x|<varepsilon$. So
beginalign
limsup_j |hat x_n_j-eta|_phi
&leqlimsup_j|hat x_n_j-hat x|_phi+|hat x-eta|_phi\ \
&=limsup_jphi((x_n_j-x)^*(x_n_j-x))+varepsilon\ \
&leq (c+|x|)limsup_jphi(x_n_j-x)+varepsilon\ \
&=varepsilon.
endalign
As $varepsilon$ was arbitrary, $hat x_n_jtoeta$, and $H_phi$ is separable.
The original importance of the GNS construction is that it allows you to start with an abstract C$^*$-algebra, and construct a representation on a Hilbert space. By adding the GNS representations over all (classes of) states, one gets a faithful representation of your abstract C$^*$-algebra as bounded operators on a Hilbert space. And the construction is explicit; in particular, having a cyclic vector allows one to define operators explicitly; a nice example of this is the proof that a non-degenerate representation from an ideal extends to the whole algebra (see Lemma I.9.14 in Davidson's C$^*$-Algebra By Example for details).
As for the "transport" question, I'm not sure what you are after. If you start with any state and do GNS, you get a vector state. If you start with a vector state and do GNS with it, nothing too remarkable happens.
answered Jul 20 at 18:52
Martin Argerami
116k1071164
More specifically can we say any two cyclic representations of $C^*$ algebra is unitary equivalent as a consequence of GNS @Martin?
– mathlover
Jul 21 at 9:24
No, not at all. That would make all GNS representations equivalent.
– Martin Argerami
Jul 21 at 12:30
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
If $M$ is sot/wot separable (i.e., most known and used von Neumann algebras out there) and $phi$ is normal, then the Hilbert space $H_phi$ is separable. To see this, let $x_nsubset M$ be a dense sequence; then for any $xin M$ there exists a bounded subnet $x_n_j$ ($|x_n_j|leq c$) with $psi(x_n_j)to psi(x)$ for all normal states $psi$ (the "bounded" is obtained via Kaplansky on the balls of radius $n$). Fix $varepsilon>0$ and $etain H_phi$. Then there exists $xin M$ with $|eta-hat x|<varepsilon$. So
beginalign
limsup_j |hat x_n_j-eta|_phi
&leqlimsup_j|hat x_n_j-hat x|_phi+|hat x-eta|_phi\ \
&=limsup_jphi((x_n_j-x)^*(x_n_j-x))+varepsilon\ \
&leq (c+|x|)limsup_jphi(x_n_j-x)+varepsilon\ \
&=varepsilon.
endalign
As $varepsilon$ was arbitrary, $hat x_n_jtoeta$, and $H_phi$ is separable.
The original importance of the GNS construction is that it allows you to start with an abstract C$^*$-algebra, and construct a representation on a Hilbert space. By adding the GNS representations over all (classes of) states, one gets a faithful representation of your abstract C$^*$-algebra as bounded operators on a Hilbert space. And the construction is explicit; in particular, having a cyclic vector allows one to define operators explicitly; a nice example of this is the proof that a non-degenerate representation from an ideal extends to the whole algebra (see Lemma I.9.14 in Davidson's C$^*$-Algebra By Example for details).
As for the "transport" question, I'm not sure what you are after. If you start with any state and do GNS, you get a vector state. If you start with a vector state and do GNS with it, nothing too remarkable happens.
answered Jul 20 at 18:52
Martin Argerami
116k1071164
If $M$ is sot/wot separable (i.e., most known and used von Neumann algebras out there) and $phi$ is normal, then the Hilbert space $H_phi$ is separable. To see this, let $x_nsubset M$ be a dense sequence; then for any $xin M$ there exists a bounded subnet $x_n_j$ ($|x_n_j|leq c$) with $psi(x_n_j)to psi(x)$ for all normal states $psi$ (the "bounded" is obtained via Kaplansky on the balls of radius $n$). Fix $varepsilon>0$ and $etain H_phi$. Then there exists $xin M$ with $|eta-hat x|<varepsilon$. So
beginalign
limsup_j |hat x_n_j-eta|_phi
&leqlimsup_j|hat x_n_j-hat x|_phi+|hat x-eta|_phi\ \
&=limsup_jphi((x_n_j-x)^*(x_n_j-x))+varepsilon\ \
&leq (c+|x|)limsup_jphi(x_n_j-x)+varepsilon\ \
&=varepsilon.
endalign
As $varepsilon$ was arbitrary, $hat x_n_jtoeta$, and $H_phi$ is separable.
The original importance of the GNS construction is that it allows you to start with an abstract C$^*$-algebra, and construct a representation on a Hilbert space. By adding the GNS representations over all (classes of) states, one gets a faithful representation of your abstract C$^*$-algebra as bounded operators on a Hilbert space. And the construction is explicit; in particular, having a cyclic vector allows one to define operators explicitly; a nice example of this is the proof that a non-degenerate representation from an ideal extends to the whole algebra (see Lemma I.9.14 in Davidson's C$^*$-Algebra By Example for details).
As for the "transport" question, I'm not sure what you are after. If you start with any state and do GNS, you get a vector state. If you start with a vector state and do GNS with it, nothing too remarkable happens.
answered Jul 20 at 18:52
Martin Argerami
116k1071164
answered Jul 20 at 18:52
Martin Argerami
116k1071164
answered Jul 20 at 18:52
Martin Argerami
116k1071164
answered Jul 20 at 18:52
Martin Argerami
116k1071164
116k1071164
More specifically can we say any two cyclic representations of $C^*$ algebra is unitary equivalent as a consequence of GNS @Martin?
– mathlover
Jul 21 at 9:24
No, not at all. That would make all GNS representations equivalent.
– Martin Argerami
Jul 21 at 12:30
add a comment |Â
More specifically can we say any two cyclic representations of $C^*$ algebra is unitary equivalent as a consequence of GNS @Martin?
– mathlover
Jul 21 at 9:24
No, not at all. That would make all GNS representations equivalent.
– Martin Argerami
Jul 21 at 12:30
More specifically can we say any two cyclic representations of $C^*$ algebra is unitary equivalent as a consequence of GNS @Martin?
– mathlover
Jul 21 at 9:24
More specifically can we say any two cyclic representations of $C^*$ algebra is unitary equivalent as a consequence of GNS @Martin?
– mathlover
Jul 21 at 9:24
No, not at all. That would make all GNS representations equivalent.
– Martin Argerami
Jul 21 at 12:30
No, not at all. That would make all GNS representations equivalent.
– Martin Argerami
Jul 21 at 12:30
add a comment |Â
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For (ii), von Neumann algebra almost never represent onto separable Hilbert spaces (finite-dimensionality is necessary I reckon). (i) I am unsure of what the questions is, what is the transported state you mentioned? (iii) Are you only interested in von Neumann algebras or C*-algebras in general?
– Munk
Jul 20 at 1:30
Yes I am interested in vN algebras
– mathlover
Jul 20 at 4:47
Transported meaning is $phi=omega_xicircpi$
– mathlover
Jul 20 at 5:31