unitization of an essential ideal [closed]
$begingroup$
Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?
operator-theory operator-algebras c-star-algebras
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
$endgroup$
closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, metamorphy, Davide Giraudo, amWhy, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?
operator-theory operator-algebras c-star-algebras
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
$endgroup$
closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, metamorphy, Davide Giraudo, amWhy, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?
operator-theory operator-algebras c-star-algebras
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
$endgroup$
Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $tilde{I}=Ibigoplus Bbb C$ is an essential ideal of unitization $Bbigoplus Bbb C$ of $B$?
operator-theory operator-algebras c-star-algebras
operator-theory operator-algebras c-star-algebras
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
asked Dec 19 '18 at 5:05
mathrookiemathrookie
918512
918512
closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, metamorphy, Davide Giraudo, amWhy, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Saad, metamorphy, Davide Giraudo, amWhy, Leucippus Dec 27 '18 at 0:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Saad, metamorphy, Davide Giraudo, amWhy, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:
(see Theorem 3.1.8, [1])
When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.
Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.
[1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:
(see Theorem 3.1.8, [1])
When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.
Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.
[1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
$endgroup$
add a comment |
$begingroup$
If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:
(see Theorem 3.1.8, [1])
When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.
Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.
[1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
$endgroup$
add a comment |
$begingroup$
If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:
(see Theorem 3.1.8, [1])
When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.
Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.
[1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
$endgroup$
If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:
(see Theorem 3.1.8, [1])
When $I$ is unital, the $n$ is an *-isomorphism, which implies that $tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.
Back to your question, if $tilde{I}$ is an essential ideal of $Boplus mathbb{C}$, then $tilde{I}=Boplusmathbb{C}$, a contradiction.
[1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
answered Dec 19 '18 at 12:39
C.DingC.Ding
1,3911321
1,3911321
add a comment |
add a comment |
