MASAS in Type $II_1$ factor

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Let $N$ be a type $II_1$ factor. Does there exist a diffuse abelian sub algebra of $N$ which is not Maximal abelian?

operator-algebras von-neumann-algebras

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edited Jul 27 at 7:52

asked Jul 27 at 6:38

rkmath

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up vote
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down vote

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Let $N$ be a type $II_1$ factor. Does there exist a diffuse abelian sub algebra of $N$ which is not Maximal abelian?

operator-algebras von-neumann-algebras

share|cite|improve this question

edited Jul 27 at 7:52

asked Jul 27 at 6:38

rkmath

607

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Let $N$ be a type $II_1$ factor. Does there exist a diffuse abelian sub algebra of $N$ which is not Maximal abelian?

operator-algebras von-neumann-algebras

share|cite|improve this question

edited Jul 27 at 7:52

asked Jul 27 at 6:38

rkmath

607

Let $N$ be a type $II_1$ factor. Does there exist a diffuse abelian sub algebra of $N$ which is not Maximal abelian?

operator-algebras von-neumann-algebras

share|cite|improve this question

edited Jul 27 at 7:52

asked Jul 27 at 6:38

rkmath

607

share|cite|improve this question

share|cite|improve this question

edited Jul 27 at 7:52

edited Jul 27 at 7:52

edited Jul 27 at 7:52

asked Jul 27 at 6:38

rkmath

607

asked Jul 27 at 6:38

rkmath

607

asked Jul 27 at 6:38

rkmath

607

607

add a comment | 

add a comment | 

1 Answer
1

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oldest

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up vote
3
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In general, a diffuse abelian subalgebra of $N$ may not be a masa of $N$. Choose a projection $p$ in $N$ with trace $1/2$. Take a masa $M$ in $pNp$. There exists a unitary operator $u$ in $N$ such that $u^*pu=p^perp$. Then, $x+u^*xu:xin M$ is diffuse and abelian but not maximal.

share|cite|improve this answer

edited Jul 27 at 8:04

answered Jul 27 at 6:56

Rui Shi

413

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have edited the question. Hope the question is clear now.@ Rui
  – rkmath
  Jul 27 at 7:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please clarify me why $lbrace x+u^*xu :x in M rbrace$ is diffuse abelian but not maximal.
  – rkmath
  Jul 27 at 10:25
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Intuitively, we can write $N=pNpotimes M_2(C)$. If $M$ is a masa in $pNp$, then for every $x$ in $M$, the set of elements beginpmatrix x&0\0&x\ endpmatrix form a diffuse abelian subalgebra $Motimes I_2$ of $N$. It is diffuse, since every projection of this form in the subalgebra is not minimal. Note that $Motimes I_2$ is a proper subalgebra of $Moplus M$. Thus $Motimes I_2$ is not maximal.
  – Rui Shi
  Jul 27 at 13:42
  

  
  
  
  

add a comment | 

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1 Answer
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active

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votes

1 Answer
1

active

oldest

votes

active

oldest

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active

oldest

votes

up vote
3
down vote

In general, a diffuse abelian subalgebra of $N$ may not be a masa of $N$. Choose a projection $p$ in $N$ with trace $1/2$. Take a masa $M$ in $pNp$. There exists a unitary operator $u$ in $N$ such that $u^*pu=p^perp$. Then, $x+u^*xu:xin M$ is diffuse and abelian but not maximal.

share|cite|improve this answer

edited Jul 27 at 8:04

answered Jul 27 at 6:56

Rui Shi

413

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have edited the question. Hope the question is clear now.@ Rui
  – rkmath
  Jul 27 at 7:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please clarify me why $lbrace x+u^*xu :x in M rbrace$ is diffuse abelian but not maximal.
  – rkmath
  Jul 27 at 10:25
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Intuitively, we can write $N=pNpotimes M_2(C)$. If $M$ is a masa in $pNp$, then for every $x$ in $M$, the set of elements beginpmatrix x&0\0&x\ endpmatrix form a diffuse abelian subalgebra $Motimes I_2$ of $N$. It is diffuse, since every projection of this form in the subalgebra is not minimal. Note that $Motimes I_2$ is a proper subalgebra of $Moplus M$. Thus $Motimes I_2$ is not maximal.
  – Rui Shi
  Jul 27 at 13:42
  

  
  
  
  

add a comment | 

up vote
3
down vote

In general, a diffuse abelian subalgebra of $N$ may not be a masa of $N$. Choose a projection $p$ in $N$ with trace $1/2$. Take a masa $M$ in $pNp$. There exists a unitary operator $u$ in $N$ such that $u^*pu=p^perp$. Then, $x+u^*xu:xin M$ is diffuse and abelian but not maximal.

share|cite|improve this answer

edited Jul 27 at 8:04

answered Jul 27 at 6:56

Rui Shi

413

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have edited the question. Hope the question is clear now.@ Rui
  – rkmath
  Jul 27 at 7:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please clarify me why $lbrace x+u^*xu :x in M rbrace$ is diffuse abelian but not maximal.
  – rkmath
  Jul 27 at 10:25
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Intuitively, we can write $N=pNpotimes M_2(C)$. If $M$ is a masa in $pNp$, then for every $x$ in $M$, the set of elements beginpmatrix x&0\0&x\ endpmatrix form a diffuse abelian subalgebra $Motimes I_2$ of $N$. It is diffuse, since every projection of this form in the subalgebra is not minimal. Note that $Motimes I_2$ is a proper subalgebra of $Moplus M$. Thus $Motimes I_2$ is not maximal.
  – Rui Shi
  Jul 27 at 13:42
  

  
  
  
  

add a comment | 

up vote
3
down vote

up vote
3
down vote

In general, a diffuse abelian subalgebra of $N$ may not be a masa of $N$. Choose a projection $p$ in $N$ with trace $1/2$. Take a masa $M$ in $pNp$. There exists a unitary operator $u$ in $N$ such that $u^*pu=p^perp$. Then, $x+u^*xu:xin M$ is diffuse and abelian but not maximal.

share|cite|improve this answer

edited Jul 27 at 8:04

answered Jul 27 at 6:56

Rui Shi

413

In general, a diffuse abelian subalgebra of $N$ may not be a masa of $N$. Choose a projection $p$ in $N$ with trace $1/2$. Take a masa $M$ in $pNp$. There exists a unitary operator $u$ in $N$ such that $u^*pu=p^perp$. Then, $x+u^*xu:xin M$ is diffuse and abelian but not maximal.

share|cite|improve this answer

edited Jul 27 at 8:04

answered Jul 27 at 6:56

Rui Shi

413

share|cite|improve this answer

share|cite|improve this answer

edited Jul 27 at 8:04

edited Jul 27 at 8:04

edited Jul 27 at 8:04

answered Jul 27 at 6:56

Rui Shi

413

answered Jul 27 at 6:56

Rui Shi

413

answered Jul 27 at 6:56

Rui Shi

413

413

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have edited the question. Hope the question is clear now.@ Rui
  – rkmath
  Jul 27 at 7:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please clarify me why $lbrace x+u^*xu :x in M rbrace$ is diffuse abelian but not maximal.
  – rkmath
  Jul 27 at 10:25
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Intuitively, we can write $N=pNpotimes M_2(C)$. If $M$ is a masa in $pNp$, then for every $x$ in $M$, the set of elements beginpmatrix x&0\0&x\ endpmatrix form a diffuse abelian subalgebra $Motimes I_2$ of $N$. It is diffuse, since every projection of this form in the subalgebra is not minimal. Note that $Motimes I_2$ is a proper subalgebra of $Moplus M$. Thus $Motimes I_2$ is not maximal.
  – Rui Shi
  Jul 27 at 13:42
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have edited the question. Hope the question is clear now.@ Rui
  – rkmath
  Jul 27 at 7:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please clarify me why $lbrace x+u^*xu :x in M rbrace$ is diffuse abelian but not maximal.
  – rkmath
  Jul 27 at 10:25
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Intuitively, we can write $N=pNpotimes M_2(C)$. If $M$ is a masa in $pNp$, then for every $x$ in $M$, the set of elements beginpmatrix x&0\0&x\ endpmatrix form a diffuse abelian subalgebra $Motimes I_2$ of $N$. It is diffuse, since every projection of this form in the subalgebra is not minimal. Note that $Motimes I_2$ is a proper subalgebra of $Moplus M$. Thus $Motimes I_2$ is not maximal.
  – Rui Shi
  Jul 27 at 13:42
  

  
  
  
  

I have edited the question. Hope the question is clear now.@ Rui
– rkmath
Jul 27 at 7:55

I have edited the question. Hope the question is clear now.@ Rui
– rkmath
Jul 27 at 7:55

Please clarify me why $lbrace x+u^*xu :x in M rbrace$ is diffuse abelian but not maximal.
– rkmath
Jul 27 at 10:25

Please clarify me why $lbrace x+u^*xu :x in M rbrace$ is diffuse abelian but not maximal.
– rkmath
Jul 27 at 10:25

2

2

Intuitively, we can write $N=pNpotimes M_2(C)$. If $M$ is a masa in $pNp$, then for every $x$ in $M$, the set of elements beginpmatrix x&0\0&x\ endpmatrix form a diffuse abelian subalgebra $Motimes I_2$ of $N$. It is diffuse, since every projection of this form in the subalgebra is not minimal. Note that $Motimes I_2$ is a proper subalgebra of $Moplus M$. Thus $Motimes I_2$ is not maximal.
– Rui Shi
Jul 27 at 13:42

Intuitively, we can write $N=pNpotimes M_2(C)$. If $M$ is a masa in $pNp$, then for every $x$ in $M$, the set of elements beginpmatrix x&0\0&x\ endpmatrix form a diffuse abelian subalgebra $Motimes I_2$ of $N$. It is diffuse, since every projection of this form in the subalgebra is not minimal. Note that $Motimes I_2$ is a proper subalgebra of $Moplus M$. Thus $Motimes I_2$ is not maximal.
– Rui Shi
Jul 27 at 13:42

add a comment | 

 

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