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Representations of simple C$^*$-algebras
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I am reading from the following document, and am a bit stumped by footnote 4 on page 5:
https://arxiv.org/pdf/math-ph/0006011.pdf
Actually, I will copy the relevant text because it disappears off the bottom of the page (or at least it does when I view it on my laptop).
"The fact that the CCR-algebra is simple can be seen as the correct generalization of the Stone-von Neumann uniqueness theorem to the case of infinitely many degrees of freedom. Indeed, since $mathcalA(H,sigma)$ is simple, all of its representations are isomorphic. When H is finite-dimensional, this isomorphism is unitarily implementable, entailing the result in [20]."
The object $mathcalA(H,sigma)$ which is mentioned is a C$^*$-algebra.
I have found only one mention of isomorphic representations for C$^*$-algebras, which is Definition 2 in the following document:
http://www.math.ru.nl/~tcrisp/teaching/2017-Cstar-reps/notes/2017-09-25-Cstar-reps-notes.pdf
And my problem is that I can't make any progress with proving the implication hinted at by the footnote, that if $A$ is a C$^*$-algebra, then
$$textA simple Rightarrow text All of it's representations are isomorphic.$$
Well, to me this is what the footnote is hinting at, but it could well be the case that the author just means that this is the case for $mathcalA(H,sigma)$, so I guess I should also be thinking about possible counter examples for the general case. However I'd be the first to admit that my knowledge of representation theory isn't particularly great yet. . .
As usual, I would appreciate any hints/suggestions e.t.c. I'm really quite interested by this footnote, as it seems to say that Slawny's Theorem implies the Stone-Von Neumann Theorem which is something I've not seen anywhere else.
representation-theory mathematical-physics operator-algebras c-star-algebras
asked Jul 26 at 19:46
user505379
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I am reading from the following document, and am a bit stumped by footnote 4 on page 5:
https://arxiv.org/pdf/math-ph/0006011.pdf
Actually, I will copy the relevant text because it disappears off the bottom of the page (or at least it does when I view it on my laptop).
"The fact that the CCR-algebra is simple can be seen as the correct generalization of the Stone-von Neumann uniqueness theorem to the case of infinitely many degrees of freedom. Indeed, since $mathcalA(H,sigma)$ is simple, all of its representations are isomorphic. When H is finite-dimensional, this isomorphism is unitarily implementable, entailing the result in [20]."
The object $mathcalA(H,sigma)$ which is mentioned is a C$^*$-algebra.
I have found only one mention of isomorphic representations for C$^*$-algebras, which is Definition 2 in the following document:
http://www.math.ru.nl/~tcrisp/teaching/2017-Cstar-reps/notes/2017-09-25-Cstar-reps-notes.pdf
And my problem is that I can't make any progress with proving the implication hinted at by the footnote, that if $A$ is a C$^*$-algebra, then
$$textA simple Rightarrow text All of it's representations are isomorphic.$$
Well, to me this is what the footnote is hinting at, but it could well be the case that the author just means that this is the case for $mathcalA(H,sigma)$, so I guess I should also be thinking about possible counter examples for the general case. However I'd be the first to admit that my knowledge of representation theory isn't particularly great yet. . .
As usual, I would appreciate any hints/suggestions e.t.c. I'm really quite interested by this footnote, as it seems to say that Slawny's Theorem implies the Stone-Von Neumann Theorem which is something I've not seen anywhere else.
representation-theory mathematical-physics operator-algebras c-star-algebras
asked Jul 26 at 19:46
user505379
888
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am reading from the following document, and am a bit stumped by footnote 4 on page 5:
https://arxiv.org/pdf/math-ph/0006011.pdf
Actually, I will copy the relevant text because it disappears off the bottom of the page (or at least it does when I view it on my laptop).
"The fact that the CCR-algebra is simple can be seen as the correct generalization of the Stone-von Neumann uniqueness theorem to the case of infinitely many degrees of freedom. Indeed, since $mathcalA(H,sigma)$ is simple, all of its representations are isomorphic. When H is finite-dimensional, this isomorphism is unitarily implementable, entailing the result in [20]."
The object $mathcalA(H,sigma)$ which is mentioned is a C$^*$-algebra.
I have found only one mention of isomorphic representations for C$^*$-algebras, which is Definition 2 in the following document:
http://www.math.ru.nl/~tcrisp/teaching/2017-Cstar-reps/notes/2017-09-25-Cstar-reps-notes.pdf
And my problem is that I can't make any progress with proving the implication hinted at by the footnote, that if $A$ is a C$^*$-algebra, then
$$textA simple Rightarrow text All of it's representations are isomorphic.$$
Well, to me this is what the footnote is hinting at, but it could well be the case that the author just means that this is the case for $mathcalA(H,sigma)$, so I guess I should also be thinking about possible counter examples for the general case. However I'd be the first to admit that my knowledge of representation theory isn't particularly great yet. . .
As usual, I would appreciate any hints/suggestions e.t.c. I'm really quite interested by this footnote, as it seems to say that Slawny's Theorem implies the Stone-Von Neumann Theorem which is something I've not seen anywhere else.
representation-theory mathematical-physics operator-algebras c-star-algebras
asked Jul 26 at 19:46
user505379
888
I am reading from the following document, and am a bit stumped by footnote 4 on page 5:
https://arxiv.org/pdf/math-ph/0006011.pdf
Actually, I will copy the relevant text because it disappears off the bottom of the page (or at least it does when I view it on my laptop).
"The fact that the CCR-algebra is simple can be seen as the correct generalization of the Stone-von Neumann uniqueness theorem to the case of infinitely many degrees of freedom. Indeed, since $mathcalA(H,sigma)$ is simple, all of its representations are isomorphic. When H is finite-dimensional, this isomorphism is unitarily implementable, entailing the result in [20]."
The object $mathcalA(H,sigma)$ which is mentioned is a C$^*$-algebra.
I have found only one mention of isomorphic representations for C$^*$-algebras, which is Definition 2 in the following document:
http://www.math.ru.nl/~tcrisp/teaching/2017-Cstar-reps/notes/2017-09-25-Cstar-reps-notes.pdf
And my problem is that I can't make any progress with proving the implication hinted at by the footnote, that if $A$ is a C$^*$-algebra, then
$$textA simple Rightarrow text All of it's representations are isomorphic.$$
Well, to me this is what the footnote is hinting at, but it could well be the case that the author just means that this is the case for $mathcalA(H,sigma)$, so I guess I should also be thinking about possible counter examples for the general case. However I'd be the first to admit that my knowledge of representation theory isn't particularly great yet. . .
As usual, I would appreciate any hints/suggestions e.t.c. I'm really quite interested by this footnote, as it seems to say that Slawny's Theorem implies the Stone-Von Neumann Theorem which is something I've not seen anywhere else.
representation-theory mathematical-physics operator-algebras c-star-algebras
asked Jul 26 at 19:46
user505379
888
asked Jul 26 at 19:46
user505379
888
asked Jul 26 at 19:46
user505379
888
asked Jul 26 at 19:46
user505379
888
888
add a comment |Â
add a comment |Â
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It is not true that simplicity implies isomorphism of representations (it may be true for your algebra, that I don't know what it is).
I cannot immediately think of an elementary example, but here are a couple.
Take the free groups $mathbb F_2$ and $mathbb F_3$ and their reduced C$^*$-algebras. The canonical homomorphism $mathbb F_3tomathbb F_2$ lifts to a $*$-epimomorphism $C_r^*(mathbb F_3)to C_r^*(mathbb F_2)$. Compare with the identity representation $C_r^*(mathbb F_3)to C_r^*(mathbb F_3)$ and the fact that $C_r^*(mathbb F_3)$ and $C_r^*(mathbb F_2)$ are not isomorphic (this is due to Pimsner-Voiculescu if I'm not wrong), and that both are simple.
Take the Cuntz algebras $mathbb O_2$ and $mathbb O_3$. It is well-known that they are not isomorphic, and $mathbb O_3$ embeds in $mathbb O_2$ (all exact algebras embed in it). So again we get two representations of $mathbb O_3$ (one into itself, one into $mathbb O_2$) that are not isomorphic. And Cuntz algebras are simple.
answered Jul 27 at 3:45
Martin Argerami
115k1071164
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
It is not true that simplicity implies isomorphism of representations (it may be true for your algebra, that I don't know what it is).
I cannot immediately think of an elementary example, but here are a couple.
Take the free groups $mathbb F_2$ and $mathbb F_3$ and their reduced C$^*$-algebras. The canonical homomorphism $mathbb F_3tomathbb F_2$ lifts to a $*$-epimomorphism $C_r^*(mathbb F_3)to C_r^*(mathbb F_2)$. Compare with the identity representation $C_r^*(mathbb F_3)to C_r^*(mathbb F_3)$ and the fact that $C_r^*(mathbb F_3)$ and $C_r^*(mathbb F_2)$ are not isomorphic (this is due to Pimsner-Voiculescu if I'm not wrong), and that both are simple.
Take the Cuntz algebras $mathbb O_2$ and $mathbb O_3$. It is well-known that they are not isomorphic, and $mathbb O_3$ embeds in $mathbb O_2$ (all exact algebras embed in it). So again we get two representations of $mathbb O_3$ (one into itself, one into $mathbb O_2$) that are not isomorphic. And Cuntz algebras are simple.
answered Jul 27 at 3:45
Martin Argerami
115k1071164
add a comment |Â
up vote
2
down vote
It is not true that simplicity implies isomorphism of representations (it may be true for your algebra, that I don't know what it is).
I cannot immediately think of an elementary example, but here are a couple.
Take the free groups $mathbb F_2$ and $mathbb F_3$ and their reduced C$^*$-algebras. The canonical homomorphism $mathbb F_3tomathbb F_2$ lifts to a $*$-epimomorphism $C_r^*(mathbb F_3)to C_r^*(mathbb F_2)$. Compare with the identity representation $C_r^*(mathbb F_3)to C_r^*(mathbb F_3)$ and the fact that $C_r^*(mathbb F_3)$ and $C_r^*(mathbb F_2)$ are not isomorphic (this is due to Pimsner-Voiculescu if I'm not wrong), and that both are simple.
Take the Cuntz algebras $mathbb O_2$ and $mathbb O_3$. It is well-known that they are not isomorphic, and $mathbb O_3$ embeds in $mathbb O_2$ (all exact algebras embed in it). So again we get two representations of $mathbb O_3$ (one into itself, one into $mathbb O_2$) that are not isomorphic. And Cuntz algebras are simple.
answered Jul 27 at 3:45
Martin Argerami
115k1071164
add a comment |Â
up vote
2
down vote
up vote
2
down vote
It is not true that simplicity implies isomorphism of representations (it may be true for your algebra, that I don't know what it is).
I cannot immediately think of an elementary example, but here are a couple.
Take the free groups $mathbb F_2$ and $mathbb F_3$ and their reduced C$^*$-algebras. The canonical homomorphism $mathbb F_3tomathbb F_2$ lifts to a $*$-epimomorphism $C_r^*(mathbb F_3)to C_r^*(mathbb F_2)$. Compare with the identity representation $C_r^*(mathbb F_3)to C_r^*(mathbb F_3)$ and the fact that $C_r^*(mathbb F_3)$ and $C_r^*(mathbb F_2)$ are not isomorphic (this is due to Pimsner-Voiculescu if I'm not wrong), and that both are simple.
Take the Cuntz algebras $mathbb O_2$ and $mathbb O_3$. It is well-known that they are not isomorphic, and $mathbb O_3$ embeds in $mathbb O_2$ (all exact algebras embed in it). So again we get two representations of $mathbb O_3$ (one into itself, one into $mathbb O_2$) that are not isomorphic. And Cuntz algebras are simple.
answered Jul 27 at 3:45
Martin Argerami
115k1071164
It is not true that simplicity implies isomorphism of representations (it may be true for your algebra, that I don't know what it is).
I cannot immediately think of an elementary example, but here are a couple.
Take the free groups $mathbb F_2$ and $mathbb F_3$ and their reduced C$^*$-algebras. The canonical homomorphism $mathbb F_3tomathbb F_2$ lifts to a $*$-epimomorphism $C_r^*(mathbb F_3)to C_r^*(mathbb F_2)$. Compare with the identity representation $C_r^*(mathbb F_3)to C_r^*(mathbb F_3)$ and the fact that $C_r^*(mathbb F_3)$ and $C_r^*(mathbb F_2)$ are not isomorphic (this is due to Pimsner-Voiculescu if I'm not wrong), and that both are simple.
Take the Cuntz algebras $mathbb O_2$ and $mathbb O_3$. It is well-known that they are not isomorphic, and $mathbb O_3$ embeds in $mathbb O_2$ (all exact algebras embed in it). So again we get two representations of $mathbb O_3$ (one into itself, one into $mathbb O_2$) that are not isomorphic. And Cuntz algebras are simple.
answered Jul 27 at 3:45
Martin Argerami
115k1071164
edited Jul 27 at 13:18
answered Jul 27 at 3:45
Martin Argerami
115k1071164
answered Jul 27 at 3:45
Martin Argerami
115k1071164
answered Jul 27 at 3:45
Martin Argerami
115k1071164
115k1071164
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