On c-star tensor norms and von Neumann tensor norm
0
$begingroup$
Does on finite tensor products of matrix algebras, all the $C^*$-norms and von Neumann tensor product norm coincide, or rather they are algebraically isomorphic? What about abelian von Neumann algebras considering them as different tensor products do they coincide?
operator-algebras von-neumann-algebras
asked Jan 7 at 13:36
mathlovermathlover
123110
$endgroup$
|
show 1 more comment
0
$begingroup$
Does on finite tensor products of matrix algebras, all the $C^*$-norms and von Neumann tensor product norm coincide, or rather they are algebraically isomorphic? What about abelian von Neumann algebras considering them as different tensor products do they coincide?
operator-algebras von-neumann-algebras
asked Jan 7 at 13:36
mathlovermathlover
123110
$endgroup$
1
$begingroup$
What is "von Neumann tensor product norm"? Plus, if I understood your question correctly, your are asking whether all ($C^ast$-)tensor norms of the algebraic tensor product are the same. Which is obviously false, you have at least the minimal and the maximal one.
$endgroup$
– Adrián González-Pérez
Jan 8 at 10:56
$begingroup$
Yes, you are right. This is the question I was asking.
$endgroup$
– mathlover
Jan 10 at 12:54
1
$begingroup$
@AdriánGonzález-Pérez There are the minimal and maximal $C^ast$-cross norm, but they may very well coincide (they do for finite-dimensional and commutative $C^ast$-algebras). The von Neumann tensor product norm is the same as the minimal one, but the von Neumann tensor product can be bigger than the minimal tensor product (it's the closure in the $sigma$-weak topology instead of the norm topology).
$endgroup$
– MaoWao
Jan 10 at 13:41
$begingroup$
I still have no idea what the question is.
$endgroup$
– Martin Argerami
Jan 10 at 22:11
$begingroup$
@Martin My exact point was on the algebraic tensor product how the norms are related if I take two von Neumann algebras and consider them also abstract $c$-star algebras and putting different $c$ star norms on the algebraic tensor product.
$endgroup$
– mathlover
Jan 11 at 9:10
|
show 1 more comment
0
0
0
0
$begingroup$
Does on finite tensor products of matrix algebras, all the $C^*$-norms and von Neumann tensor product norm coincide, or rather they are algebraically isomorphic? What about abelian von Neumann algebras considering them as different tensor products do they coincide?
operator-algebras von-neumann-algebras
asked Jan 7 at 13:36
mathlovermathlover
123110
$endgroup$
Does on finite tensor products of matrix algebras, all the $C^*$-norms and von Neumann tensor product norm coincide, or rather they are algebraically isomorphic? What about abelian von Neumann algebras considering them as different tensor products do they coincide?
operator-algebras von-neumann-algebras
operator-algebras von-neumann-algebras
asked Jan 7 at 13:36
mathlovermathlover
123110
asked Jan 7 at 13:36
mathlovermathlover
123110
edited Jan 7 at 13:48
mathlover
asked Jan 7 at 13:36
mathlovermathlover
123110
asked Jan 7 at 13:36
mathlovermathlover
123110
asked Jan 7 at 13:36
mathlovermathlover
123110
123110
1
$begingroup$
What is "von Neumann tensor product norm"? Plus, if I understood your question correctly, your are asking whether all ($C^ast$-)tensor norms of the algebraic tensor product are the same. Which is obviously false, you have at least the minimal and the maximal one.
$endgroup$
– Adrián González-Pérez
Jan 8 at 10:56
$begingroup$
Yes, you are right. This is the question I was asking.
$endgroup$
– mathlover
Jan 10 at 12:54
1
$begingroup$
@AdriánGonzález-Pérez There are the minimal and maximal $C^ast$-cross norm, but they may very well coincide (they do for finite-dimensional and commutative $C^ast$-algebras). The von Neumann tensor product norm is the same as the minimal one, but the von Neumann tensor product can be bigger than the minimal tensor product (it's the closure in the $sigma$-weak topology instead of the norm topology).
$endgroup$
– MaoWao
Jan 10 at 13:41
$begingroup$
I still have no idea what the question is.
$endgroup$
– Martin Argerami
Jan 10 at 22:11
$begingroup$
@Martin My exact point was on the algebraic tensor product how the norms are related if I take two von Neumann algebras and consider them also abstract $c$-star algebras and putting different $c$ star norms on the algebraic tensor product.
$endgroup$
– mathlover
Jan 11 at 9:10
|
show 1 more comment
1
$begingroup$
What is "von Neumann tensor product norm"? Plus, if I understood your question correctly, your are asking whether all ($C^ast$-)tensor norms of the algebraic tensor product are the same. Which is obviously false, you have at least the minimal and the maximal one.
$endgroup$
– Adrián González-Pérez
Jan 8 at 10:56
$begingroup$
Yes, you are right. This is the question I was asking.
$endgroup$
– mathlover
Jan 10 at 12:54
1
$begingroup$
@AdriánGonzález-Pérez There are the minimal and maximal $C^ast$-cross norm, but they may very well coincide (they do for finite-dimensional and commutative $C^ast$-algebras). The von Neumann tensor product norm is the same as the minimal one, but the von Neumann tensor product can be bigger than the minimal tensor product (it's the closure in the $sigma$-weak topology instead of the norm topology).
$endgroup$
– MaoWao
Jan 10 at 13:41
$begingroup$
I still have no idea what the question is.
$endgroup$
– Martin Argerami
Jan 10 at 22:11
$begingroup$
@Martin My exact point was on the algebraic tensor product how the norms are related if I take two von Neumann algebras and consider them also abstract $c$-star algebras and putting different $c$ star norms on the algebraic tensor product.
$endgroup$
– mathlover
Jan 11 at 9:10
1
1
$begingroup$
What is "von Neumann tensor product norm"? Plus, if I understood your question correctly, your are asking whether all ($C^ast$-)tensor norms of the algebraic tensor product are the same. Which is obviously false, you have at least the minimal and the maximal one.
$endgroup$
– Adrián González-Pérez
Jan 8 at 10:56
$begingroup$
What is "von Neumann tensor product norm"? Plus, if I understood your question correctly, your are asking whether all ($C^ast$-)tensor norms of the algebraic tensor product are the same. Which is obviously false, you have at least the minimal and the maximal one.
$endgroup$
– Adrián González-Pérez
Jan 8 at 10:56
$begingroup$
Yes, you are right. This is the question I was asking.
$endgroup$
– mathlover
Jan 10 at 12:54
$begingroup$
Yes, you are right. This is the question I was asking.
$endgroup$
– mathlover
Jan 10 at 12:54
1
1
$begingroup$
@AdriánGonzález-Pérez There are the minimal and maximal $C^ast$-cross norm, but they may very well coincide (they do for finite-dimensional and commutative $C^ast$-algebras). The von Neumann tensor product norm is the same as the minimal one, but the von Neumann tensor product can be bigger than the minimal tensor product (it's the closure in the $sigma$-weak topology instead of the norm topology).
$endgroup$
– MaoWao
Jan 10 at 13:41
$begingroup$
@AdriánGonzález-Pérez There are the minimal and maximal $C^ast$-cross norm, but they may very well coincide (they do for finite-dimensional and commutative $C^ast$-algebras). The von Neumann tensor product norm is the same as the minimal one, but the von Neumann tensor product can be bigger than the minimal tensor product (it's the closure in the $sigma$-weak topology instead of the norm topology).
$endgroup$
– MaoWao
Jan 10 at 13:41
$begingroup$
I still have no idea what the question is.
$endgroup$
– Martin Argerami
Jan 10 at 22:11
$begingroup$
I still have no idea what the question is.
$endgroup$
– Martin Argerami
Jan 10 at 22:11
$begingroup$
@Martin My exact point was on the algebraic tensor product how the norms are related if I take two von Neumann algebras and consider them also abstract $c$-star algebras and putting different $c$ star norms on the algebraic tensor product.
$endgroup$
– mathlover
Jan 11 at 9:10
$begingroup$
@Martin My exact point was on the algebraic tensor product how the norms are related if I take two von Neumann algebras and consider them also abstract $c$-star algebras and putting different $c$ star norms on the algebraic tensor product.
$endgroup$
– mathlover
Jan 11 at 9:10
|
show 1 more comment
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1
$begingroup$
What is "von Neumann tensor product norm"? Plus, if I understood your question correctly, your are asking whether all ($C^ast$-)tensor norms of the algebraic tensor product are the same. Which is obviously false, you have at least the minimal and the maximal one.
$endgroup$
– Adrián González-Pérez
Jan 8 at 10:56
$begingroup$
Yes, you are right. This is the question I was asking.
$endgroup$
– mathlover
Jan 10 at 12:54
1
$begingroup$
@AdriánGonzález-Pérez There are the minimal and maximal $C^ast$-cross norm, but they may very well coincide (they do for finite-dimensional and commutative $C^ast$-algebras). The von Neumann tensor product norm is the same as the minimal one, but the von Neumann tensor product can be bigger than the minimal tensor product (it's the closure in the $sigma$-weak topology instead of the norm topology).
$endgroup$
– MaoWao
Jan 10 at 13:41
$begingroup$
I still have no idea what the question is.
$endgroup$
– Martin Argerami
Jan 10 at 22:11
$begingroup$
@Martin My exact point was on the algebraic tensor product how the norms are related if I take two von Neumann algebras and consider them also abstract $c$-star algebras and putting different $c$ star norms on the algebraic tensor product.
$endgroup$
– mathlover
Jan 11 at 9:10