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splitting lemma in $C^*$ algebras
Clash Royale CLAN TAG#URR8PPP
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In abelian category,there is a splitting lemma. see https://en.wikipedia.org/wiki/Splitting_lemma
I wonder whether the splitting lemma also holds in $C^*$ algebras .Is left split equivalent to right split?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 8:11
mathrookie
437211
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up vote
1
down vote
favorite
In abelian category,there is a splitting lemma. see https://en.wikipedia.org/wiki/Splitting_lemma
I wonder whether the splitting lemma also holds in $C^*$ algebras .Is left split equivalent to right split?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 8:11
mathrookie
437211
This question is very similar to:math.stackexchange.com/questions/2848612/…
– Adrián González-Pérez
Jul 23 at 10:49
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In abelian category,there is a splitting lemma. see https://en.wikipedia.org/wiki/Splitting_lemma
I wonder whether the splitting lemma also holds in $C^*$ algebras .Is left split equivalent to right split?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 8:11
mathrookie
437211
In abelian category,there is a splitting lemma. see https://en.wikipedia.org/wiki/Splitting_lemma
I wonder whether the splitting lemma also holds in $C^*$ algebras .Is left split equivalent to right split?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 8:11
mathrookie
437211
asked Jul 22 at 8:11
mathrookie
437211
asked Jul 22 at 8:11
mathrookie
437211
asked Jul 22 at 8:11
mathrookie
437211
437211
This question is very similar to:math.stackexchange.com/questions/2848612/…
– Adrián González-Pérez
Jul 23 at 10:49
add a comment |Â
This question is very similar to:math.stackexchange.com/questions/2848612/…
– Adrián González-Pérez
Jul 23 at 10:49
This question is very similar to:math.stackexchange.com/questions/2848612/…
– Adrián González-Pérez
Jul 23 at 10:49
This question is very similar to:math.stackexchange.com/questions/2848612/…
– Adrián González-Pérez
Jul 23 at 10:49
add a comment |Â
1 Answer
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No, not even if the C$^*$-algebra is commutative. For instance if you take $A=C_b(mathbb N)$ (bounded sequences), then $I=c_0(mathbb N)$ is an ideal that is not complemented.
answered Jul 22 at 18:53
Martin Argerami
116k1071164
If $A$ has a unit ,$I$ is a finite dimensional ideal,can we decompose it?
– mathrookie
Jul 22 at 19:01
Not sure. I'll think about it.
– Martin Argerami
Jul 22 at 19:08
I have found the answer in Blackdar's book,for a fixed $C^*$ algebra $A$ and $B$,there is always at least extension $E=Aoplus B$.If $B$ is unital,this is the only extension (see Busty 's paper Double centralizers and extension of $C^*$ algeras)
– mathrookie
Jul 23 at 15:41
But that's not what you were asking. If you are given $A$ and $B$, you can form $E=Aoplus B$. Your question is about having $B$ and $E$ and finding $A$.
– Martin Argerami
Jul 23 at 15:46
If $A$ and $A/I$ were given,$E_1=Aoplus A/I$,$E_2=A$ are both extensions,they are $ *$ isomorphic
– mathrookie
Jul 25 at 0:42
 |Â
show 3 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
No, not even if the C$^*$-algebra is commutative. For instance if you take $A=C_b(mathbb N)$ (bounded sequences), then $I=c_0(mathbb N)$ is an ideal that is not complemented.
answered Jul 22 at 18:53
Martin Argerami
116k1071164
If $A$ has a unit ,$I$ is a finite dimensional ideal,can we decompose it?
– mathrookie
Jul 22 at 19:01
Not sure. I'll think about it.
– Martin Argerami
Jul 22 at 19:08
I have found the answer in Blackdar's book,for a fixed $C^*$ algebra $A$ and $B$,there is always at least extension $E=Aoplus B$.If $B$ is unital,this is the only extension (see Busty 's paper Double centralizers and extension of $C^*$ algeras)
– mathrookie
Jul 23 at 15:41
But that's not what you were asking. If you are given $A$ and $B$, you can form $E=Aoplus B$. Your question is about having $B$ and $E$ and finding $A$.
– Martin Argerami
Jul 23 at 15:46
If $A$ and $A/I$ were given,$E_1=Aoplus A/I$,$E_2=A$ are both extensions,they are $ *$ isomorphic
– mathrookie
Jul 25 at 0:42
 |Â
show 3 more comments
up vote
3
down vote
accepted
No, not even if the C$^*$-algebra is commutative. For instance if you take $A=C_b(mathbb N)$ (bounded sequences), then $I=c_0(mathbb N)$ is an ideal that is not complemented.
answered Jul 22 at 18:53
Martin Argerami
116k1071164
If $A$ has a unit ,$I$ is a finite dimensional ideal,can we decompose it?
– mathrookie
Jul 22 at 19:01
Not sure. I'll think about it.
– Martin Argerami
Jul 22 at 19:08
I have found the answer in Blackdar's book,for a fixed $C^*$ algebra $A$ and $B$,there is always at least extension $E=Aoplus B$.If $B$ is unital,this is the only extension (see Busty 's paper Double centralizers and extension of $C^*$ algeras)
– mathrookie
Jul 23 at 15:41
But that's not what you were asking. If you are given $A$ and $B$, you can form $E=Aoplus B$. Your question is about having $B$ and $E$ and finding $A$.
– Martin Argerami
Jul 23 at 15:46
If $A$ and $A/I$ were given,$E_1=Aoplus A/I$,$E_2=A$ are both extensions,they are $ *$ isomorphic
– mathrookie
Jul 25 at 0:42
 |Â
show 3 more comments
up vote
3
down vote
accepted
up vote
3
down vote
accepted
No, not even if the C$^*$-algebra is commutative. For instance if you take $A=C_b(mathbb N)$ (bounded sequences), then $I=c_0(mathbb N)$ is an ideal that is not complemented.
answered Jul 22 at 18:53
Martin Argerami
116k1071164
No, not even if the C$^*$-algebra is commutative. For instance if you take $A=C_b(mathbb N)$ (bounded sequences), then $I=c_0(mathbb N)$ is an ideal that is not complemented.
answered Jul 22 at 18:53
Martin Argerami
116k1071164
answered Jul 22 at 18:53
Martin Argerami
116k1071164
answered Jul 22 at 18:53
Martin Argerami
116k1071164
answered Jul 22 at 18:53
Martin Argerami
116k1071164
116k1071164
If $A$ has a unit ,$I$ is a finite dimensional ideal,can we decompose it?
– mathrookie
Jul 22 at 19:01
Not sure. I'll think about it.
– Martin Argerami
Jul 22 at 19:08
I have found the answer in Blackdar's book,for a fixed $C^*$ algebra $A$ and $B$,there is always at least extension $E=Aoplus B$.If $B$ is unital,this is the only extension (see Busty 's paper Double centralizers and extension of $C^*$ algeras)
– mathrookie
Jul 23 at 15:41
But that's not what you were asking. If you are given $A$ and $B$, you can form $E=Aoplus B$. Your question is about having $B$ and $E$ and finding $A$.
– Martin Argerami
Jul 23 at 15:46
If $A$ and $A/I$ were given,$E_1=Aoplus A/I$,$E_2=A$ are both extensions,they are $ *$ isomorphic
– mathrookie
Jul 25 at 0:42
 |Â
show 3 more comments
If $A$ has a unit ,$I$ is a finite dimensional ideal,can we decompose it?
– mathrookie
Jul 22 at 19:01
Not sure. I'll think about it.
– Martin Argerami
Jul 22 at 19:08
I have found the answer in Blackdar's book,for a fixed $C^*$ algebra $A$ and $B$,there is always at least extension $E=Aoplus B$.If $B$ is unital,this is the only extension (see Busty 's paper Double centralizers and extension of $C^*$ algeras)
– mathrookie
Jul 23 at 15:41
But that's not what you were asking. If you are given $A$ and $B$, you can form $E=Aoplus B$. Your question is about having $B$ and $E$ and finding $A$.
– Martin Argerami
Jul 23 at 15:46
If $A$ and $A/I$ were given,$E_1=Aoplus A/I$,$E_2=A$ are both extensions,they are $ *$ isomorphic
– mathrookie
Jul 25 at 0:42
If $A$ has a unit ,$I$ is a finite dimensional ideal,can we decompose it?
– mathrookie
Jul 22 at 19:01
If $A$ has a unit ,$I$ is a finite dimensional ideal,can we decompose it?
– mathrookie
Jul 22 at 19:01
Not sure. I'll think about it.
– Martin Argerami
Jul 22 at 19:08
Not sure. I'll think about it.
– Martin Argerami
Jul 22 at 19:08
I have found the answer in Blackdar's book,for a fixed $C^*$ algebra $A$ and $B$,there is always at least extension $E=Aoplus B$.If $B$ is unital,this is the only extension (see Busty 's paper Double centralizers and extension of $C^*$ algeras)
– mathrookie
Jul 23 at 15:41
I have found the answer in Blackdar's book,for a fixed $C^*$ algebra $A$ and $B$,there is always at least extension $E=Aoplus B$.If $B$ is unital,this is the only extension (see Busty 's paper Double centralizers and extension of $C^*$ algeras)
– mathrookie
Jul 23 at 15:41
But that's not what you were asking. If you are given $A$ and $B$, you can form $E=Aoplus B$. Your question is about having $B$ and $E$ and finding $A$.
– Martin Argerami
Jul 23 at 15:46
But that's not what you were asking. If you are given $A$ and $B$, you can form $E=Aoplus B$. Your question is about having $B$ and $E$ and finding $A$.
– Martin Argerami
Jul 23 at 15:46
If $A$ and $A/I$ were given,$E_1=Aoplus A/I$,$E_2=A$ are both extensions,they are $ *$ isomorphic
– mathrookie
Jul 25 at 0:42
If $A$ and $A/I$ were given,$E_1=Aoplus A/I$,$E_2=A$ are both extensions,they are $ *$ isomorphic
– mathrookie
Jul 25 at 0:42
 |Â
show 3 more comments
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This question is very similar to:math.stackexchange.com/questions/2848612/…
– Adrián González-Pérez
Jul 23 at 10:49