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description of an ideal generated by the projections in a $C^*$ algebra
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If $A$ is a $C^*$ algebra,$P_i$ are projections in $A$,$I$ is the ideal generated by the projections.I think $I$ is the $C^*$ algebra generated by $P_iAP_i$.How to charectarize $I$,is there a precise description of $I$?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 0:55
mathrookie
437211
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up vote
2
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favorite
If $A$ is a $C^*$ algebra,$P_i$ are projections in $A$,$I$ is the ideal generated by the projections.I think $I$ is the $C^*$ algebra generated by $P_iAP_i$.How to charectarize $I$,is there a precise description of $I$?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 0:55
mathrookie
437211
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
If $A$ is a $C^*$ algebra,$P_i$ are projections in $A$,$I$ is the ideal generated by the projections.I think $I$ is the $C^*$ algebra generated by $P_iAP_i$.How to charectarize $I$,is there a precise description of $I$?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 0:55
mathrookie
437211
If $A$ is a $C^*$ algebra,$P_i$ are projections in $A$,$I$ is the ideal generated by the projections.I think $I$ is the $C^*$ algebra generated by $P_iAP_i$.How to charectarize $I$,is there a precise description of $I$?
operator-theory operator-algebras c-star-algebras
asked Jul 22 at 0:55
mathrookie
437211
asked Jul 22 at 0:55
mathrookie
437211
asked Jul 22 at 0:55
mathrookie
437211
asked Jul 22 at 0:55
mathrookie
437211
437211
add a comment |Â
add a comment |Â
1 Answer
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1
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It is not the C$^*$-algebra generated by $P_iAP_i$. For instance take $A=M_2(mathbb C)$, $P=E_11$. The ideal generated by $P$ is $A$, and not $PAP=mathbb C P$.
The ideal generated by elements $x_1,ldots,x_m$ in $A$ is the C$^*$-subalgebra generated by $$ ax_jb: a,bin A, j=1,ldots,m.$$ I don't think you can get anything specific here. Unless the projections are central; in the case the ideal would be $P_1A+cdots+P_mA$.
answered Jul 22 at 22:04
Martin Argerami
116k1071164
1
If the ideal $I$ generated by a infinite set $S$,is ideal $I$ the $C^*$ subalgebra generated by $ axb: a,bin A, xin S$?
– mathrookie
Jul 23 at 0:14
1
Yes, it's actually the closed linear span of that set.
– Martin Argerami
Jul 23 at 2:27
Pro Argerami,I have another question:If $A$ is nonunital,I think the ideal $I$ does not contain $S$ since any $a,bin A$ cannot be $1_A$.Is it correct?
– mathrookie
Jul 25 at 17:47
No, you cannot say that in general. In $c_0$, the ideal generated by $x=(1/n)_n$ does not contain $x$ (because $x=yx$ cannot be satisfied by any $yin c_0$); but the ideal generated by $e_1$ is $mathbb C e_1$, which contains $e_1$.
– Martin Argerami
Jul 25 at 18:15
Quick question: Is it really the usual convention that an element is not necessarily in the ideal it generates? That seems very odd! Do you have a comment on why this is the convention?
– s.harp
Jul 25 at 18:26
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
It is not the C$^*$-algebra generated by $P_iAP_i$. For instance take $A=M_2(mathbb C)$, $P=E_11$. The ideal generated by $P$ is $A$, and not $PAP=mathbb C P$.
The ideal generated by elements $x_1,ldots,x_m$ in $A$ is the C$^*$-subalgebra generated by $$ ax_jb: a,bin A, j=1,ldots,m.$$ I don't think you can get anything specific here. Unless the projections are central; in the case the ideal would be $P_1A+cdots+P_mA$.
answered Jul 22 at 22:04
Martin Argerami
116k1071164
1
If the ideal $I$ generated by a infinite set $S$,is ideal $I$ the $C^*$ subalgebra generated by $ axb: a,bin A, xin S$?
– mathrookie
Jul 23 at 0:14
1
Yes, it's actually the closed linear span of that set.
– Martin Argerami
Jul 23 at 2:27
Pro Argerami,I have another question:If $A$ is nonunital,I think the ideal $I$ does not contain $S$ since any $a,bin A$ cannot be $1_A$.Is it correct?
– mathrookie
Jul 25 at 17:47
No, you cannot say that in general. In $c_0$, the ideal generated by $x=(1/n)_n$ does not contain $x$ (because $x=yx$ cannot be satisfied by any $yin c_0$); but the ideal generated by $e_1$ is $mathbb C e_1$, which contains $e_1$.
– Martin Argerami
Jul 25 at 18:15
Quick question: Is it really the usual convention that an element is not necessarily in the ideal it generates? That seems very odd! Do you have a comment on why this is the convention?
– s.harp
Jul 25 at 18:26
 |Â
show 1 more comment
up vote
1
down vote
accepted
It is not the C$^*$-algebra generated by $P_iAP_i$. For instance take $A=M_2(mathbb C)$, $P=E_11$. The ideal generated by $P$ is $A$, and not $PAP=mathbb C P$.
The ideal generated by elements $x_1,ldots,x_m$ in $A$ is the C$^*$-subalgebra generated by $$ ax_jb: a,bin A, j=1,ldots,m.$$ I don't think you can get anything specific here. Unless the projections are central; in the case the ideal would be $P_1A+cdots+P_mA$.
answered Jul 22 at 22:04
Martin Argerami
116k1071164
1
If the ideal $I$ generated by a infinite set $S$,is ideal $I$ the $C^*$ subalgebra generated by $ axb: a,bin A, xin S$?
– mathrookie
Jul 23 at 0:14
1
Yes, it's actually the closed linear span of that set.
– Martin Argerami
Jul 23 at 2:27
Pro Argerami,I have another question:If $A$ is nonunital,I think the ideal $I$ does not contain $S$ since any $a,bin A$ cannot be $1_A$.Is it correct?
– mathrookie
Jul 25 at 17:47
No, you cannot say that in general. In $c_0$, the ideal generated by $x=(1/n)_n$ does not contain $x$ (because $x=yx$ cannot be satisfied by any $yin c_0$); but the ideal generated by $e_1$ is $mathbb C e_1$, which contains $e_1$.
– Martin Argerami
Jul 25 at 18:15
Quick question: Is it really the usual convention that an element is not necessarily in the ideal it generates? That seems very odd! Do you have a comment on why this is the convention?
– s.harp
Jul 25 at 18:26
 |Â
show 1 more comment
up vote
1
down vote
accepted
up vote
1
down vote
accepted
It is not the C$^*$-algebra generated by $P_iAP_i$. For instance take $A=M_2(mathbb C)$, $P=E_11$. The ideal generated by $P$ is $A$, and not $PAP=mathbb C P$.
The ideal generated by elements $x_1,ldots,x_m$ in $A$ is the C$^*$-subalgebra generated by $$ ax_jb: a,bin A, j=1,ldots,m.$$ I don't think you can get anything specific here. Unless the projections are central; in the case the ideal would be $P_1A+cdots+P_mA$.
answered Jul 22 at 22:04
Martin Argerami
116k1071164
It is not the C$^*$-algebra generated by $P_iAP_i$. For instance take $A=M_2(mathbb C)$, $P=E_11$. The ideal generated by $P$ is $A$, and not $PAP=mathbb C P$.
The ideal generated by elements $x_1,ldots,x_m$ in $A$ is the C$^*$-subalgebra generated by $$ ax_jb: a,bin A, j=1,ldots,m.$$ I don't think you can get anything specific here. Unless the projections are central; in the case the ideal would be $P_1A+cdots+P_mA$.
answered Jul 22 at 22:04
Martin Argerami
116k1071164
edited Jul 25 at 18:11
answered Jul 22 at 22:04
Martin Argerami
116k1071164
answered Jul 22 at 22:04
Martin Argerami
116k1071164
answered Jul 22 at 22:04
Martin Argerami
116k1071164
116k1071164
1
If the ideal $I$ generated by a infinite set $S$,is ideal $I$ the $C^*$ subalgebra generated by $ axb: a,bin A, xin S$?
– mathrookie
Jul 23 at 0:14
1
Yes, it's actually the closed linear span of that set.
– Martin Argerami
Jul 23 at 2:27
Pro Argerami,I have another question:If $A$ is nonunital,I think the ideal $I$ does not contain $S$ since any $a,bin A$ cannot be $1_A$.Is it correct?
– mathrookie
Jul 25 at 17:47
No, you cannot say that in general. In $c_0$, the ideal generated by $x=(1/n)_n$ does not contain $x$ (because $x=yx$ cannot be satisfied by any $yin c_0$); but the ideal generated by $e_1$ is $mathbb C e_1$, which contains $e_1$.
– Martin Argerami
Jul 25 at 18:15
Quick question: Is it really the usual convention that an element is not necessarily in the ideal it generates? That seems very odd! Do you have a comment on why this is the convention?
– s.harp
Jul 25 at 18:26
 |Â
show 1 more comment
1
If the ideal $I$ generated by a infinite set $S$,is ideal $I$ the $C^*$ subalgebra generated by $ axb: a,bin A, xin S$?
– mathrookie
Jul 23 at 0:14
1
Yes, it's actually the closed linear span of that set.
– Martin Argerami
Jul 23 at 2:27
Pro Argerami,I have another question:If $A$ is nonunital,I think the ideal $I$ does not contain $S$ since any $a,bin A$ cannot be $1_A$.Is it correct?
– mathrookie
Jul 25 at 17:47
No, you cannot say that in general. In $c_0$, the ideal generated by $x=(1/n)_n$ does not contain $x$ (because $x=yx$ cannot be satisfied by any $yin c_0$); but the ideal generated by $e_1$ is $mathbb C e_1$, which contains $e_1$.
– Martin Argerami
Jul 25 at 18:15
Quick question: Is it really the usual convention that an element is not necessarily in the ideal it generates? That seems very odd! Do you have a comment on why this is the convention?
– s.harp
Jul 25 at 18:26
1
1
If the ideal $I$ generated by a infinite set $S$,is ideal $I$ the $C^*$ subalgebra generated by $ axb: a,bin A, xin S$?
– mathrookie
Jul 23 at 0:14
If the ideal $I$ generated by a infinite set $S$,is ideal $I$ the $C^*$ subalgebra generated by $ axb: a,bin A, xin S$?
– mathrookie
Jul 23 at 0:14
1
1
Yes, it's actually the closed linear span of that set.
– Martin Argerami
Jul 23 at 2:27
Yes, it's actually the closed linear span of that set.
– Martin Argerami
Jul 23 at 2:27
Pro Argerami,I have another question:If $A$ is nonunital,I think the ideal $I$ does not contain $S$ since any $a,bin A$ cannot be $1_A$.Is it correct?
– mathrookie
Jul 25 at 17:47
Pro Argerami,I have another question:If $A$ is nonunital,I think the ideal $I$ does not contain $S$ since any $a,bin A$ cannot be $1_A$.Is it correct?
– mathrookie
Jul 25 at 17:47
No, you cannot say that in general. In $c_0$, the ideal generated by $x=(1/n)_n$ does not contain $x$ (because $x=yx$ cannot be satisfied by any $yin c_0$); but the ideal generated by $e_1$ is $mathbb C e_1$, which contains $e_1$.
– Martin Argerami
Jul 25 at 18:15
No, you cannot say that in general. In $c_0$, the ideal generated by $x=(1/n)_n$ does not contain $x$ (because $x=yx$ cannot be satisfied by any $yin c_0$); but the ideal generated by $e_1$ is $mathbb C e_1$, which contains $e_1$.
– Martin Argerami
Jul 25 at 18:15
Quick question: Is it really the usual convention that an element is not necessarily in the ideal it generates? That seems very odd! Do you have a comment on why this is the convention?
– s.harp
Jul 25 at 18:26
Quick question: Is it really the usual convention that an element is not necessarily in the ideal it generates? That seems very odd! Do you have a comment on why this is the convention?
– s.harp
Jul 25 at 18:26
 |Â
show 1 more comment
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