Mixing
Extension of a von Neumann algebra by a von Neumann algebra
up vote
2
down vote
favorite
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0to Ato Cto Bto 0$$
Assume that $A,B$ are generated by their projections. Is $C$ necessarily generated by its projections, too?
Assume that $A,B$ are von Neumann algebras, is $C$ necessarily a von Neumann algebra, too?
Does the last question has an obvious answer when $A,B$ (hence $C$) are commutative algebras?
measure-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
Ali Taghavi
224329
add a comment |
up vote
2
down vote
favorite
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0to Ato Cto Bto 0$$
Assume that $A,B$ are generated by their projections. Is $C$ necessarily generated by its projections, too?
Assume that $A,B$ are von Neumann algebras, is $C$ necessarily a von Neumann algebra, too?
Does the last question has an obvious answer when $A,B$ (hence $C$) are commutative algebras?
measure-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
Ali Taghavi
224329
I don't know what you mean by a short exact sequence of unital algebras. If $C to B$ is a nontrivial surjection of unital algebras, its kernel is a nontrivial ideal of $C$, which is at best a nonunital algebra in general, and if it has a unit the map $A to C$ won't preserve units.
– Qiaochu Yuan
14 hours ago
@Qiaochu In the question we do not assume that $Ato C$ preserve unit.
– Ali Taghavi
14 hours ago
@QiaochuYuan The reason that we emphasize on the word "Unital" is that a von neumann algebra is actually a unital algebra since it is equal to its double commutant. So by unital consideration, we do not assume that morphisms are unital morphism. However in the question we consider abstract von Neumann algebras the definition in page 42 of this paper dmitripavlov.org/scans/guichardet.pdf
– Ali Taghavi
2 hours ago
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0to Ato Cto Bto 0$$
Assume that $A,B$ are generated by their projections. Is $C$ necessarily generated by its projections, too?
Assume that $A,B$ are von Neumann algebras, is $C$ necessarily a von Neumann algebra, too?
Does the last question has an obvious answer when $A,B$ (hence $C$) are commutative algebras?
measure-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
Ali Taghavi
224329
Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:
$$0to Ato Cto Bto 0$$
Assume that $A,B$ are generated by their projections. Is $C$ necessarily generated by its projections, too?
Assume that $A,B$ are von Neumann algebras, is $C$ necessarily a von Neumann algebra, too?
Does the last question has an obvious answer when $A,B$ (hence $C$) are commutative algebras?
measure-theory operator-algebras c-star-algebras von-neumann-algebras
measure-theory operator-algebras c-star-algebras von-neumann-algebras
asked yesterday
Ali Taghavi
224329
asked yesterday
Ali Taghavi
224329
edited 3 hours ago
asked yesterday
Ali Taghavi
224329
asked yesterday
Ali Taghavi
224329
asked yesterday
Ali Taghavi
224329
224329
I don't know what you mean by a short exact sequence of unital algebras. If $C to B$ is a nontrivial surjection of unital algebras, its kernel is a nontrivial ideal of $C$, which is at best a nonunital algebra in general, and if it has a unit the map $A to C$ won't preserve units.
– Qiaochu Yuan
14 hours ago
@Qiaochu In the question we do not assume that $Ato C$ preserve unit.
– Ali Taghavi
14 hours ago
@QiaochuYuan The reason that we emphasize on the word "Unital" is that a von neumann algebra is actually a unital algebra since it is equal to its double commutant. So by unital consideration, we do not assume that morphisms are unital morphism. However in the question we consider abstract von Neumann algebras the definition in page 42 of this paper dmitripavlov.org/scans/guichardet.pdf
– Ali Taghavi
2 hours ago
add a comment |
I don't know what you mean by a short exact sequence of unital algebras. If $C to B$ is a nontrivial surjection of unital algebras, its kernel is a nontrivial ideal of $C$, which is at best a nonunital algebra in general, and if it has a unit the map $A to C$ won't preserve units.
– Qiaochu Yuan
14 hours ago
@Qiaochu In the question we do not assume that $Ato C$ preserve unit.
– Ali Taghavi
14 hours ago
@QiaochuYuan The reason that we emphasize on the word "Unital" is that a von neumann algebra is actually a unital algebra since it is equal to its double commutant. So by unital consideration, we do not assume that morphisms are unital morphism. However in the question we consider abstract von Neumann algebras the definition in page 42 of this paper dmitripavlov.org/scans/guichardet.pdf
– Ali Taghavi
2 hours ago
I don't know what you mean by a short exact sequence of unital algebras. If $C to B$ is a nontrivial surjection of unital algebras, its kernel is a nontrivial ideal of $C$, which is at best a nonunital algebra in general, and if it has a unit the map $A to C$ won't preserve units.
– Qiaochu Yuan
14 hours ago
I don't know what you mean by a short exact sequence of unital algebras. If $C to B$ is a nontrivial surjection of unital algebras, its kernel is a nontrivial ideal of $C$, which is at best a nonunital algebra in general, and if it has a unit the map $A to C$ won't preserve units.
– Qiaochu Yuan
14 hours ago
@Qiaochu In the question we do not assume that $Ato C$ preserve unit.
– Ali Taghavi
14 hours ago
@Qiaochu In the question we do not assume that $Ato C$ preserve unit.
– Ali Taghavi
14 hours ago
@QiaochuYuan The reason that we emphasize on the word "Unital" is that a von neumann algebra is actually a unital algebra since it is equal to its double commutant. So by unital consideration, we do not assume that morphisms are unital morphism. However in the question we consider abstract von Neumann algebras the definition in page 42 of this paper dmitripavlov.org/scans/guichardet.pdf
– Ali Taghavi
2 hours ago
@QiaochuYuan The reason that we emphasize on the word "Unital" is that a von neumann algebra is actually a unital algebra since it is equal to its double commutant. So by unital consideration, we do not assume that morphisms are unital morphism. However in the question we consider abstract von Neumann algebras the definition in page 42 of this paper dmitripavlov.org/scans/guichardet.pdf
– Ali Taghavi
2 hours ago
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003201%2fextension-of-a-von-neumann-algebra-by-a-von-neumann-algebra%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
I don't know what you mean by a short exact sequence of unital algebras. If $C to B$ is a nontrivial surjection of unital algebras, its kernel is a nontrivial ideal of $C$, which is at best a nonunital algebra in general, and if it has a unit the map $A to C$ won't preserve units.
– Qiaochu Yuan
14 hours ago
@Qiaochu In the question we do not assume that $Ato C$ preserve unit.
– Ali Taghavi
14 hours ago
@QiaochuYuan The reason that we emphasize on the word "Unital" is that a von neumann algebra is actually a unital algebra since it is equal to its double commutant. So by unital consideration, we do not assume that morphisms are unital morphism. However in the question we consider abstract von Neumann algebras the definition in page 42 of this paper dmitripavlov.org/scans/guichardet.pdf
– Ali Taghavi
2 hours ago