Importance of pure states in $C^{*}$ algebras
$begingroup$
In a tensor product of $C^{*}$-algebras I have seen some proofs are used by the argument of norms by pure states. Why pure states are essential to study related to vN algebras and $C^{*}$-algebras?
von-neumann-algebras
asked Jan 15 at 7:14
mathlovermathlover
119110
$endgroup$
add a comment |
$begingroup$
In a tensor product of $C^{*}$-algebras I have seen some proofs are used by the argument of norms by pure states. Why pure states are essential to study related to vN algebras and $C^{*}$-algebras?
von-neumann-algebras
asked Jan 15 at 7:14
mathlovermathlover
119110
$endgroup$
add a comment |
$begingroup$
In a tensor product of $C^{*}$-algebras I have seen some proofs are used by the argument of norms by pure states. Why pure states are essential to study related to vN algebras and $C^{*}$-algebras?
von-neumann-algebras
asked Jan 15 at 7:14
mathlovermathlover
119110
$endgroup$
In a tensor product of $C^{*}$-algebras I have seen some proofs are used by the argument of norms by pure states. Why pure states are essential to study related to vN algebras and $C^{*}$-algebras?
von-neumann-algebras
von-neumann-algebras
asked Jan 15 at 7:14
mathlovermathlover
119110
asked Jan 15 at 7:14
mathlovermathlover
119110
asked Jan 15 at 7:14
mathlovermathlover
119110
asked Jan 15 at 7:14
mathlovermathlover
119110
asked Jan 15 at 7:14
mathlovermathlover
119110
119110
add a comment |
add a comment |
1 Answer
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The question is fairly broad, but here are a couple things:
the pure states are the extreme points of the state space. So any property that holds for pure states and survives convex combinations and pointiwse limits, holds for all states.
pure states are "minimal", in the sense that if $psileqphi$ with both states and $phi$ pure, then $psi=c,phi$ for some $c>0$.
the GNS representation of a pure state is irreducible.
Without thinking of a specific case, this is as far as I can go.
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The question is fairly broad, but here are a couple things:
the pure states are the extreme points of the state space. So any property that holds for pure states and survives convex combinations and pointiwse limits, holds for all states.
pure states are "minimal", in the sense that if $psileqphi$ with both states and $phi$ pure, then $psi=c,phi$ for some $c>0$.
the GNS representation of a pure state is irreducible.
Without thinking of a specific case, this is as far as I can go.
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
$endgroup$
add a comment |
$begingroup$
The question is fairly broad, but here are a couple things:
the pure states are the extreme points of the state space. So any property that holds for pure states and survives convex combinations and pointiwse limits, holds for all states.
pure states are "minimal", in the sense that if $psileqphi$ with both states and $phi$ pure, then $psi=c,phi$ for some $c>0$.
the GNS representation of a pure state is irreducible.
Without thinking of a specific case, this is as far as I can go.
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
$endgroup$
add a comment |
$begingroup$
The question is fairly broad, but here are a couple things:
the pure states are the extreme points of the state space. So any property that holds for pure states and survives convex combinations and pointiwse limits, holds for all states.
pure states are "minimal", in the sense that if $psileqphi$ with both states and $phi$ pure, then $psi=c,phi$ for some $c>0$.
the GNS representation of a pure state is irreducible.
Without thinking of a specific case, this is as far as I can go.
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
$endgroup$
The question is fairly broad, but here are a couple things:
the pure states are the extreme points of the state space. So any property that holds for pure states and survives convex combinations and pointiwse limits, holds for all states.
pure states are "minimal", in the sense that if $psileqphi$ with both states and $phi$ pure, then $psi=c,phi$ for some $c>0$.
the GNS representation of a pure state is irreducible.
Without thinking of a specific case, this is as far as I can go.
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
answered Jan 15 at 17:51
Martin ArgeramiMartin Argerami
126k1182181
126k1182181
add a comment |
add a comment |
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