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Proving Uniqueness of CCR Algebras
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I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.
I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:
https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf
I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.
Proof:
Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.
Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by
$$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$
Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.
This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.
(1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?
(2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?
(3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.
Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.
functional-analysis mathematical-physics operator-algebras
asked Jul 18 at 14:36
user505379
9019
add a comment |Â
up vote
5
down vote
favorite
I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.
I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:
https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf
I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.
Proof:
Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.
Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by
$$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$
Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.
This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.
(1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?
(2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?
(3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.
Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.
functional-analysis mathematical-physics operator-algebras
asked Jul 18 at 14:36
user505379
9019
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.
I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:
https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf
I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.
Proof:
Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.
Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by
$$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$
Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.
This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.
(1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?
(2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?
(3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.
Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.
functional-analysis mathematical-physics operator-algebras
asked Jul 18 at 14:36
user505379
9019
I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.
I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:
https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf
I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.
Proof:
Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $omega$), we must show that the $^*$-isomorphism $pi:langle W_1(V)ranglerightarrowlangle W_2(V)rangle$ between the $^*$-algebras $langle W_1(V)rangle$ and $langle W_2(V)rangle$ extends to an isometry between $A_1$ and $A_2$$^(1)$.
Then the norm $|x|=|pi(x)|_2$ is introduced on $A_1$, and it noted that $|pi(x)|_2leq |x|_textmax$, where $|cdot|_textmax$ is defined in Lemma 9 by
$$|x|_textmax=suptext is a C^*text norm on langle W_1(V)rangle.$$
Then it is concluded that $varphi$ extends to a $^*$-homomorphism $overlinelangle W_1(V)rangle^textmaxrightarrow A_2$$^(2)$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^(3)$.
This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.
(1) Checking this will be sufficient because we could then do the same for the inverse $varphi^-1$ to get the required map between $A_1$ and $A_2$?
(2) The bound $|pi(x)|_2leq |x|_textmax$ shows that $pi$ is continuous, so $pi$ extends to $overlinelangle W_1(V)rangle^textmax$ by continuity and density of $langle W_1(V)rangle$?
(3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $overlinelangle W_1(V)rangle^textmax$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.
Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated.
functional-analysis mathematical-physics operator-algebras
asked Jul 18 at 14:36
user505379
9019
edited Aug 9 at 14:30
asked Jul 18 at 14:36
user505379
9019
asked Jul 18 at 14:36
user505379
9019
asked Jul 18 at 14:36
user505379
9019
9019
add a comment |Â
add a comment |Â
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I've had a good old think about this, and may have filled in some of the blanks.
For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.
Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition
$$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$
(where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).
Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.
Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).
I am still a bit stuck on the last part.
I think that I'm going to take a break from thinking about this, and return in a day or two.
answered Jul 19 at 15:35
user505379
9019
Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
– user505379
Jul 24 at 14:36
Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
– user505379
Aug 9 at 15:39
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
I've had a good old think about this, and may have filled in some of the blanks.
For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.
Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition
$$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$
(where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).
Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.
Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).
I am still a bit stuck on the last part.
I think that I'm going to take a break from thinking about this, and return in a day or two.
answered Jul 19 at 15:35
user505379
9019
Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
– user505379
Jul 24 at 14:36
Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
– user505379
Aug 9 at 15:39
add a comment |Â
up vote
0
down vote
I've had a good old think about this, and may have filled in some of the blanks.
For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.
Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition
$$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$
(where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).
Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.
Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).
I am still a bit stuck on the last part.
I think that I'm going to take a break from thinking about this, and return in a day or two.
answered Jul 19 at 15:35
user505379
9019
Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
– user505379
Jul 24 at 14:36
Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
– user505379
Aug 9 at 15:39
add a comment |Â
up vote
0
down vote
up vote
0
down vote
I've had a good old think about this, and may have filled in some of the blanks.
For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.
Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition
$$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$
(where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).
Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.
Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).
I am still a bit stuck on the last part.
I think that I'm going to take a break from thinking about this, and return in a day or two.
answered Jul 19 at 15:35
user505379
9019
I've had a good old think about this, and may have filled in some of the blanks.
For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $W(u)text; uin V$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $W(u)text; uin V$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $W(u)text; uin V$.
Consider the $^*$-isomorphism $pi:langle W_1(V) ranglerightarrowlangle W_2(V) rangle$. The set $langle W_2(V) rangle$ is dense in $A_2$, and the image of the composition
$$langle W_1(V) rangle oversetpirightarrowlangle W_2(V) rangle oversetihookrightarrow A_2,$$
(where $i$ denotes inclusion), is $langle W_2(V) rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism).
Defining $|x|:=|pi(x)|_2$ gives a C$^*$-norm on $langle W_1(V) rangle$ so there is the bound $|pi(x)|_2=|x|leq |x|_textmax$ which shows that the composition above is continuous and we may extend it to a map $varphi:overlinelangle W_1(V) rangle^textmaxrightarrow A_2$. Now we invoke Lemma 10 to deduce that $varphi$ is injective and hence isometric.
Since $varphi$ extends $icirc pi$, the image of $varphi$ must contain $langle W_2(V) rangle$, but since $varphi$ is an isometry into a complete metric space, its image must be closed, therefore $varphi$ is surjective, so is an isomorphism. (Presumably the extension $varphi$ is also a $^*$-homomorphism too).
I am still a bit stuck on the last part.
I think that I'm going to take a break from thinking about this, and return in a day or two.
answered Jul 19 at 15:35
user505379
9019
edited Jul 20 at 16:59
answered Jul 19 at 15:35
user505379
9019
answered Jul 19 at 15:35
user505379
9019
answered Jul 19 at 15:35
user505379
9019
9019
Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
– user505379
Jul 24 at 14:36
Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
– user505379
Aug 9 at 15:39
add a comment |Â
Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
– user505379
Jul 24 at 14:36
Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
– user505379
Aug 9 at 15:39
Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
– user505379
Jul 24 at 14:36
Hmm, for the last part I think that it may have been obvious the whole time, but I was just failing to spot it. I think that we just observe that $(overlinelangle W_1(V) rangle^textmax,W)$ satisfies the definition of a CCR-representation, so that $overlinelangle W_1(V) rangle^textmax$ is a CCR algebra. Then replacing $A_2$ by this in the above would give imply the full result.
– user505379
Jul 24 at 14:36
Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
– user505379
Aug 9 at 15:39
Well I finished working through this proof, then looked back at their Lemma 10 and realised that I don't understand their argument there either... Oh well.
– user505379
Aug 9 at 15:39
add a comment |Â
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