a norm on a complex matrix
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Suppose $A=prod_n M_{k(n)}(mathbb{C})$,where $M_{k_n}(mathbb{C})$ is the space of all $k(n) times k_n$ complex matrices,does there exist a norm $| |_0$ on $A$ (which is different from the operator norm)satisfying the following condition:
when $|x_n| to 0,|x_n|_0 to 0,$ when $|y_n|_0 to 0,tr(y_n)to 0 $,where $| |$ is the operator norm,$x=(x_n),y=(y_n) in prod_n M_{k(n)}(mathbb{C}),tr()$ is the unique faithful tracial state on $M_{k_n}(mathbb{C})$.
operator-theory matrix-calculus operator-algebras c-star-algebras
asked Nov 14 at 17:33
mathrookie
697512
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up vote
0
down vote
favorite
Suppose $A=prod_n M_{k(n)}(mathbb{C})$,where $M_{k_n}(mathbb{C})$ is the space of all $k(n) times k_n$ complex matrices,does there exist a norm $| |_0$ on $A$ (which is different from the operator norm)satisfying the following condition:
when $|x_n| to 0,|x_n|_0 to 0,$ when $|y_n|_0 to 0,tr(y_n)to 0 $,where $| |$ is the operator norm,$x=(x_n),y=(y_n) in prod_n M_{k(n)}(mathbb{C}),tr()$ is the unique faithful tracial state on $M_{k_n}(mathbb{C})$.
operator-theory matrix-calculus operator-algebras c-star-algebras
asked Nov 14 at 17:33
mathrookie
697512
2
I don't know what the "unique faithful tracial state" is, but all norms on $M_n(mathbb C)$ are equivalent. Thus, if one norm with the desired properties exists, then all norms satisfy that property.
– Giuseppe Negro
Nov 14 at 17:36
@Giuseppe Negro,I have reedited the question
– mathrookie
Nov 14 at 18:33
3
@mathrookie, could you try to put a little more effort into your questions? A few little things like capitalization and spaces after punctuation marks would greatly improve their readability. Also, $M_k(mathbb{C})$ is not a complex $ktimes k$ matrix, but the space of all complex $ktimes k$ matrices.
– MaoWao
Nov 15 at 11:03
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $A=prod_n M_{k(n)}(mathbb{C})$,where $M_{k_n}(mathbb{C})$ is the space of all $k(n) times k_n$ complex matrices,does there exist a norm $| |_0$ on $A$ (which is different from the operator norm)satisfying the following condition:
when $|x_n| to 0,|x_n|_0 to 0,$ when $|y_n|_0 to 0,tr(y_n)to 0 $,where $| |$ is the operator norm,$x=(x_n),y=(y_n) in prod_n M_{k(n)}(mathbb{C}),tr()$ is the unique faithful tracial state on $M_{k_n}(mathbb{C})$.
operator-theory matrix-calculus operator-algebras c-star-algebras
asked Nov 14 at 17:33
mathrookie
697512
Suppose $A=prod_n M_{k(n)}(mathbb{C})$,where $M_{k_n}(mathbb{C})$ is the space of all $k(n) times k_n$ complex matrices,does there exist a norm $| |_0$ on $A$ (which is different from the operator norm)satisfying the following condition:
when $|x_n| to 0,|x_n|_0 to 0,$ when $|y_n|_0 to 0,tr(y_n)to 0 $,where $| |$ is the operator norm,$x=(x_n),y=(y_n) in prod_n M_{k(n)}(mathbb{C}),tr()$ is the unique faithful tracial state on $M_{k_n}(mathbb{C})$.
operator-theory matrix-calculus operator-algebras c-star-algebras
operator-theory matrix-calculus operator-algebras c-star-algebras
asked Nov 14 at 17:33
mathrookie
697512
asked Nov 14 at 17:33
mathrookie
697512
edited Nov 15 at 13:20
asked Nov 14 at 17:33
mathrookie
697512
asked Nov 14 at 17:33
mathrookie
697512
asked Nov 14 at 17:33
mathrookie
697512
697512
2
I don't know what the "unique faithful tracial state" is, but all norms on $M_n(mathbb C)$ are equivalent. Thus, if one norm with the desired properties exists, then all norms satisfy that property.
– Giuseppe Negro
Nov 14 at 17:36
@Giuseppe Negro,I have reedited the question
– mathrookie
Nov 14 at 18:33
3
@mathrookie, could you try to put a little more effort into your questions? A few little things like capitalization and spaces after punctuation marks would greatly improve their readability. Also, $M_k(mathbb{C})$ is not a complex $ktimes k$ matrix, but the space of all complex $ktimes k$ matrices.
– MaoWao
Nov 15 at 11:03
add a comment |
2
I don't know what the "unique faithful tracial state" is, but all norms on $M_n(mathbb C)$ are equivalent. Thus, if one norm with the desired properties exists, then all norms satisfy that property.
– Giuseppe Negro
Nov 14 at 17:36
@Giuseppe Negro,I have reedited the question
– mathrookie
Nov 14 at 18:33
3
@mathrookie, could you try to put a little more effort into your questions? A few little things like capitalization and spaces after punctuation marks would greatly improve their readability. Also, $M_k(mathbb{C})$ is not a complex $ktimes k$ matrix, but the space of all complex $ktimes k$ matrices.
– MaoWao
Nov 15 at 11:03
2
2
I don't know what the "unique faithful tracial state" is, but all norms on $M_n(mathbb C)$ are equivalent. Thus, if one norm with the desired properties exists, then all norms satisfy that property.
– Giuseppe Negro
Nov 14 at 17:36
I don't know what the "unique faithful tracial state" is, but all norms on $M_n(mathbb C)$ are equivalent. Thus, if one norm with the desired properties exists, then all norms satisfy that property.
– Giuseppe Negro
Nov 14 at 17:36
@Giuseppe Negro,I have reedited the question
– mathrookie
Nov 14 at 18:33
@Giuseppe Negro,I have reedited the question
– mathrookie
Nov 14 at 18:33
3
3
@mathrookie, could you try to put a little more effort into your questions? A few little things like capitalization and spaces after punctuation marks would greatly improve their readability. Also, $M_k(mathbb{C})$ is not a complex $ktimes k$ matrix, but the space of all complex $ktimes k$ matrices.
– MaoWao
Nov 15 at 11:03
@mathrookie, could you try to put a little more effort into your questions? A few little things like capitalization and spaces after punctuation marks would greatly improve their readability. Also, $M_k(mathbb{C})$ is not a complex $ktimes k$ matrix, but the space of all complex $ktimes k$ matrices.
– MaoWao
Nov 15 at 11:03
add a comment |
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I don't know what the "unique faithful tracial state" is, but all norms on $M_n(mathbb C)$ are equivalent. Thus, if one norm with the desired properties exists, then all norms satisfy that property.
– Giuseppe Negro
Nov 14 at 17:36
@Giuseppe Negro,I have reedited the question
– mathrookie
Nov 14 at 18:33
3
@mathrookie, could you try to put a little more effort into your questions? A few little things like capitalization and spaces after punctuation marks would greatly improve their readability. Also, $M_k(mathbb{C})$ is not a complex $ktimes k$ matrix, but the space of all complex $ktimes k$ matrices.
– MaoWao
Nov 15 at 11:03