central projections
$begingroup$
Suppose $A$ is a separable $C^*$ algebra(not necessarily unital),and let $I$ be the ideal generated by central projections in $A$,does there exist nonzero pairwise projections in the quotient $A/I$?
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
$endgroup$
add a comment |
$begingroup$
Suppose $A$ is a separable $C^*$ algebra(not necessarily unital),and let $I$ be the ideal generated by central projections in $A$,does there exist nonzero pairwise projections in the quotient $A/I$?
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
$endgroup$
$begingroup$
Pairwise what ?
$endgroup$
– user42761
Dec 5 '18 at 21:00
add a comment |
$begingroup$
Suppose $A$ is a separable $C^*$ algebra(not necessarily unital),and let $I$ be the ideal generated by central projections in $A$,does there exist nonzero pairwise projections in the quotient $A/I$?
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
$endgroup$
Suppose $A$ is a separable $C^*$ algebra(not necessarily unital),and let $I$ be the ideal generated by central projections in $A$,does there exist nonzero pairwise projections in the quotient $A/I$?
operator-theory operator-algebras c-star-algebras
operator-theory operator-algebras c-star-algebras
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
asked Dec 4 '18 at 17:51
mathrookiemathrookie
918512
918512
$begingroup$
Pairwise what ?
$endgroup$
– user42761
Dec 5 '18 at 21:00
add a comment |
$begingroup$
Pairwise what ?
$endgroup$
– user42761
Dec 5 '18 at 21:00
$begingroup$
Pairwise what ?
$endgroup$
– user42761
Dec 5 '18 at 21:00
$begingroup$
Pairwise what ?
$endgroup$
– user42761
Dec 5 '18 at 21:00
add a comment |
1 Answer
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$begingroup$
No. Take $A=C_0(mathbb R)$, then the only (central) projection is $0$, so $I={0}$ and $A/I=A$, which has no nonzero projections.
At the other end, take $A=K(H)$, then $I={0}$, and $K(H)/I=K(H)$ has lots and lots of projections.
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
$begingroup$
Another question:Given a $C^*$ algebra $A$,how to find all the central projections in $A$?or how to determine the number of central projections?
$endgroup$
– mathrookie
Dec 4 '18 at 21:12
$begingroup$
That depends on what algebra you are dealing with. In general, the question is meaningless.
$endgroup$
– Martin Argerami
Dec 4 '18 at 21:32
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No. Take $A=C_0(mathbb R)$, then the only (central) projection is $0$, so $I={0}$ and $A/I=A$, which has no nonzero projections.
At the other end, take $A=K(H)$, then $I={0}$, and $K(H)/I=K(H)$ has lots and lots of projections.
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
$begingroup$
Another question:Given a $C^*$ algebra $A$,how to find all the central projections in $A$?or how to determine the number of central projections?
$endgroup$
– mathrookie
Dec 4 '18 at 21:12
$begingroup$
That depends on what algebra you are dealing with. In general, the question is meaningless.
$endgroup$
– Martin Argerami
Dec 4 '18 at 21:32
add a comment |
$begingroup$
No. Take $A=C_0(mathbb R)$, then the only (central) projection is $0$, so $I={0}$ and $A/I=A$, which has no nonzero projections.
At the other end, take $A=K(H)$, then $I={0}$, and $K(H)/I=K(H)$ has lots and lots of projections.
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
$begingroup$
Another question:Given a $C^*$ algebra $A$,how to find all the central projections in $A$?or how to determine the number of central projections?
$endgroup$
– mathrookie
Dec 4 '18 at 21:12
$begingroup$
That depends on what algebra you are dealing with. In general, the question is meaningless.
$endgroup$
– Martin Argerami
Dec 4 '18 at 21:32
add a comment |
$begingroup$
No. Take $A=C_0(mathbb R)$, then the only (central) projection is $0$, so $I={0}$ and $A/I=A$, which has no nonzero projections.
At the other end, take $A=K(H)$, then $I={0}$, and $K(H)/I=K(H)$ has lots and lots of projections.
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
$endgroup$
No. Take $A=C_0(mathbb R)$, then the only (central) projection is $0$, so $I={0}$ and $A/I=A$, which has no nonzero projections.
At the other end, take $A=K(H)$, then $I={0}$, and $K(H)/I=K(H)$ has lots and lots of projections.
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
answered Dec 4 '18 at 19:24
Martin ArgeramiMartin Argerami
127k1182183
127k1182183
$begingroup$
Another question:Given a $C^*$ algebra $A$,how to find all the central projections in $A$?or how to determine the number of central projections?
$endgroup$
– mathrookie
Dec 4 '18 at 21:12
$begingroup$
That depends on what algebra you are dealing with. In general, the question is meaningless.
$endgroup$
– Martin Argerami
Dec 4 '18 at 21:32
add a comment |
$begingroup$
Another question:Given a $C^*$ algebra $A$,how to find all the central projections in $A$?or how to determine the number of central projections?
$endgroup$
– mathrookie
Dec 4 '18 at 21:12
$begingroup$
That depends on what algebra you are dealing with. In general, the question is meaningless.
$endgroup$
– Martin Argerami
Dec 4 '18 at 21:32
$begingroup$
Another question:Given a $C^*$ algebra $A$,how to find all the central projections in $A$?or how to determine the number of central projections?
$endgroup$
– mathrookie
Dec 4 '18 at 21:12
$begingroup$
Another question:Given a $C^*$ algebra $A$,how to find all the central projections in $A$?or how to determine the number of central projections?
$endgroup$
– mathrookie
Dec 4 '18 at 21:12
$begingroup$
That depends on what algebra you are dealing with. In general, the question is meaningless.
$endgroup$
– Martin Argerami
Dec 4 '18 at 21:32
$begingroup$
That depends on what algebra you are dealing with. In general, the question is meaningless.
$endgroup$
– Martin Argerami
Dec 4 '18 at 21:32
add a comment |
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$begingroup$
Pairwise what ?
$endgroup$
– user42761
Dec 5 '18 at 21:00