Mixing
Existence of faithful normal state
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Does there always exist faithful normal state on center of von Neumann algebra?
Further, in type II$_1$ von Neumann algebras are tracial states coming from the center valued trace?
operator-algebras von-neumann-algebras
edited Jul 21 at 12:45
Martin Argerami
116k1071164
asked Jul 19 at 20:22
mathlover
10518
add a comment |Â
up vote
1
down vote
favorite
Does there always exist faithful normal state on center of von Neumann algebra?
Further, in type II$_1$ von Neumann algebras are tracial states coming from the center valued trace?
operator-algebras von-neumann-algebras
edited Jul 21 at 12:45
Martin Argerami
116k1071164
asked Jul 19 at 20:22
mathlover
10518
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Does there always exist faithful normal state on center of von Neumann algebra?
Further, in type II$_1$ von Neumann algebras are tracial states coming from the center valued trace?
operator-algebras von-neumann-algebras
edited Jul 21 at 12:45
Martin Argerami
116k1071164
asked Jul 19 at 20:22
mathlover
10518
Does there always exist faithful normal state on center of von Neumann algebra?
Further, in type II$_1$ von Neumann algebras are tracial states coming from the center valued trace?
operator-algebras von-neumann-algebras
edited Jul 21 at 12:45
Martin Argerami
116k1071164
asked Jul 19 at 20:22
mathlover
10518
edited Jul 21 at 12:45
Martin Argerami
116k1071164
edited Jul 21 at 12:45
Martin Argerami
116k1071164
edited Jul 21 at 12:45
Martin Argerami
116k1071164
116k1071164
asked Jul 19 at 20:22
mathlover
10518
asked Jul 19 at 20:22
mathlover
10518
asked Jul 19 at 20:22
mathlover
10518
10518
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer.
As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have
$$
M=bigoplus_n A_notimes N_n,
$$
where $A_n$ are abelian and $N_n$ are II$_1$-factors, the center-valued trace is induced by
$$
Phi(bigoplus_n a_notimes x_n)=bigoplus_n a_notimes 1.
$$
Tracial states are obtained by taking $f_nin S(A_n)$ and $tau_n$ the unique trace on $N_n$, and letting
$$
phi(bigoplus_n a_notimes x_n)=sum_n f_n(a_n)tau(x_n).
$$
I cannot immediately say in what sense $phi$ "comes" from $Phi$.
answered Jul 21 at 12:44
Martin Argerami
116k1071164
Meaning is if $rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $phi$ such that $phi(A)=rho(A)I$
– mathlover
Jul 21 at 18:31
The center-valued trace is unique.
– Martin Argerami
Jul 21 at 19:24
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer.
As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have
$$
M=bigoplus_n A_notimes N_n,
$$
where $A_n$ are abelian and $N_n$ are II$_1$-factors, the center-valued trace is induced by
$$
Phi(bigoplus_n a_notimes x_n)=bigoplus_n a_notimes 1.
$$
Tracial states are obtained by taking $f_nin S(A_n)$ and $tau_n$ the unique trace on $N_n$, and letting
$$
phi(bigoplus_n a_notimes x_n)=sum_n f_n(a_n)tau(x_n).
$$
I cannot immediately say in what sense $phi$ "comes" from $Phi$.
answered Jul 21 at 12:44
Martin Argerami
116k1071164
Meaning is if $rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $phi$ such that $phi(A)=rho(A)I$
– mathlover
Jul 21 at 18:31
The center-valued trace is unique.
– Martin Argerami
Jul 21 at 19:24
add a comment |Â
up vote
1
down vote
accepted
The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer.
As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have
$$
M=bigoplus_n A_notimes N_n,
$$
where $A_n$ are abelian and $N_n$ are II$_1$-factors, the center-valued trace is induced by
$$
Phi(bigoplus_n a_notimes x_n)=bigoplus_n a_notimes 1.
$$
Tracial states are obtained by taking $f_nin S(A_n)$ and $tau_n$ the unique trace on $N_n$, and letting
$$
phi(bigoplus_n a_notimes x_n)=sum_n f_n(a_n)tau(x_n).
$$
I cannot immediately say in what sense $phi$ "comes" from $Phi$.
answered Jul 21 at 12:44
Martin Argerami
116k1071164
Meaning is if $rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $phi$ such that $phi(A)=rho(A)I$
– mathlover
Jul 21 at 18:31
The center-valued trace is unique.
– Martin Argerami
Jul 21 at 19:24
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer.
As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have
$$
M=bigoplus_n A_notimes N_n,
$$
where $A_n$ are abelian and $N_n$ are II$_1$-factors, the center-valued trace is induced by
$$
Phi(bigoplus_n a_notimes x_n)=bigoplus_n a_notimes 1.
$$
Tracial states are obtained by taking $f_nin S(A_n)$ and $tau_n$ the unique trace on $N_n$, and letting
$$
phi(bigoplus_n a_notimes x_n)=sum_n f_n(a_n)tau(x_n).
$$
I cannot immediately say in what sense $phi$ "comes" from $Phi$.
answered Jul 21 at 12:44
Martin Argerami
116k1071164
The first question is the same as asking if every abelian von Neumann algebra has a faithful normal state. The answer is easily no, by taking for instance the example in this answer.
As for all tracial states coming from the center-valued trace, it is not entirely clear to me what you mean. If you have
$$
M=bigoplus_n A_notimes N_n,
$$
where $A_n$ are abelian and $N_n$ are II$_1$-factors, the center-valued trace is induced by
$$
Phi(bigoplus_n a_notimes x_n)=bigoplus_n a_notimes 1.
$$
Tracial states are obtained by taking $f_nin S(A_n)$ and $tau_n$ the unique trace on $N_n$, and letting
$$
phi(bigoplus_n a_notimes x_n)=sum_n f_n(a_n)tau(x_n).
$$
I cannot immediately say in what sense $phi$ "comes" from $Phi$.
answered Jul 21 at 12:44
Martin Argerami
116k1071164
answered Jul 21 at 12:44
Martin Argerami
116k1071164
answered Jul 21 at 12:44
Martin Argerami
116k1071164
answered Jul 21 at 12:44
Martin Argerami
116k1071164
116k1071164
Meaning is if $rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $phi$ such that $phi(A)=rho(A)I$
– mathlover
Jul 21 at 18:31
The center-valued trace is unique.
– Martin Argerami
Jul 21 at 19:24
add a comment |Â
Meaning is if $rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $phi$ such that $phi(A)=rho(A)I$
– mathlover
Jul 21 at 18:31
The center-valued trace is unique.
– Martin Argerami
Jul 21 at 19:24
Meaning is if $rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $phi$ such that $phi(A)=rho(A)I$
– mathlover
Jul 21 at 18:31
Meaning is if $rho$ is a tracial state in von Neumann algebra(factor), does there exist centervalued trace $phi$ such that $phi(A)=rho(A)I$
– mathlover
Jul 21 at 18:31
The center-valued trace is unique.
– Martin Argerami
Jul 21 at 19:24
The center-valued trace is unique.
– Martin Argerami
Jul 21 at 19:24
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2857022%2fexistence-of-faithful-normal-state%23new-answer', 'question_page');
);
Post as a guest
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
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
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