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Extension of an automorphism to its predual

1

$begingroup$

Let $M$ is a von Neumann algebra equipped with state $varphi$ , $alpha$ $in$ $Aut(M)$ preserving state $varphi$, then does $alpha$ extends to $alpha_1:L^1(M,varphi)rightarrow L^1(M, varphi)$?

operator-algebras von-neumann-algebras

share|cite|improve this question

edited Mar 27 at 6:44

mathlover

asked Mar 27 at 5:45

mathlovermathlover

123110

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer is yes if the automorphism is normal. Which i think is automatic if $varphi$ is normal.
  $endgroup$
  – Adrián González-Pérez
  Mar 27 at 9:38
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $M$ is a von Neumann algebra equipped with state $varphi$ , $alpha$ $in$ $Aut(M)$ preserving state $varphi$, then does $alpha$ extends to $alpha_1:L^1(M,varphi)rightarrow L^1(M, varphi)$?

operator-algebras von-neumann-algebras

share|cite|improve this question

edited Mar 27 at 6:44

mathlover

asked Mar 27 at 5:45

mathlovermathlover

123110

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer is yes if the automorphism is normal. Which i think is automatic if $varphi$ is normal.
  $endgroup$
  – Adrián González-Pérez
  Mar 27 at 9:38
  

  
  
  
  

add a comment | 

$begingroup$

Let $M$ is a von Neumann algebra equipped with state $varphi$ , $alpha$ $in$ $Aut(M)$ preserving state $varphi$, then does $alpha$ extends to $alpha_1:L^1(M,varphi)rightarrow L^1(M, varphi)$?

operator-algebras von-neumann-algebras

share|cite|improve this question

edited Mar 27 at 6:44

mathlover

asked Mar 27 at 5:45

mathlovermathlover

123110

$endgroup$

share|cite|improve this question

edited Mar 27 at 6:44

mathlover

asked Mar 27 at 5:45

mathlovermathlover

123110

share|cite|improve this question

edited Mar 27 at 6:44

mathlover

asked Mar 27 at 5:45

mathlovermathlover

123110

asked Mar 27 at 5:45

mathlovermathlover

123110

asked Mar 27 at 5:45

mathlovermathlover

123110

1 Answer
1

active

oldest

votes

2

$begingroup$

The space L$^1(M,φ)$ is independent of the choice of the weight φ.
It is known as the predual of M, and is also denoted by $M_*$.

In Sakai's approach to von Neumann algebras,
von Neumann algebras are defined as C*-algebras that admit a predual
and morphisms of von Neumann algebras are defined as morphisms of C*-algebras that
admit a predual.
So the answer to your question is tautologically true.
See Sakai's book “C*-algebras and W*-algebras”.

share|cite|improve this answer

answered Mar 27 at 13:46

Dmitri PavlovDmitri Pavlov

58826

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add a comment | 

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2

$begingroup$

The space L$^1(M,φ)$ is independent of the choice of the weight φ.
It is known as the predual of M, and is also denoted by $M_*$.

In Sakai's approach to von Neumann algebras,
von Neumann algebras are defined as C*-algebras that admit a predual
and morphisms of von Neumann algebras are defined as morphisms of C*-algebras that
admit a predual.
So the answer to your question is tautologically true.
See Sakai's book “C*-algebras and W*-algebras”.

share|cite|improve this answer

answered Mar 27 at 13:46

Dmitri PavlovDmitri Pavlov

58826

$endgroup$

add a comment | 

2

$begingroup$

The space L$^1(M,φ)$ is independent of the choice of the weight φ.
It is known as the predual of M, and is also denoted by $M_*$.

In Sakai's approach to von Neumann algebras,
von Neumann algebras are defined as C*-algebras that admit a predual
and morphisms of von Neumann algebras are defined as morphisms of C*-algebras that
admit a predual.
So the answer to your question is tautologically true.
See Sakai's book “C*-algebras and W*-algebras”.

share|cite|improve this answer

answered Mar 27 at 13:46

Dmitri PavlovDmitri Pavlov

58826

$endgroup$

add a comment | 

$begingroup$

The space L$^1(M,φ)$ is independent of the choice of the weight φ.
It is known as the predual of M, and is also denoted by $M_*$.

In Sakai's approach to von Neumann algebras,
von Neumann algebras are defined as C*-algebras that admit a predual
and morphisms of von Neumann algebras are defined as morphisms of C*-algebras that
admit a predual.
So the answer to your question is tautologically true.
See Sakai's book “C*-algebras and W*-algebras”.

share|cite|improve this answer

answered Mar 27 at 13:46

Dmitri PavlovDmitri Pavlov

58826

$endgroup$

share|cite|improve this answer

answered Mar 27 at 13:46

Dmitri PavlovDmitri Pavlov

58826

answered Mar 27 at 13:46

Dmitri PavlovDmitri Pavlov

58826

answered Mar 27 at 13:46

Dmitri PavlovDmitri Pavlov

58826