What does the enveloping von Neumann algebra functor do to locally compact Hausdorff spaces?
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Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is a property that the weak operator topology can detect, if $A$ were Abelian, then $A^{**}$ would be as well.
Now, in the Abelian case, $Acong C_0(Omega)$ and $A^{**}cong C(dotOmega)$ (with $dotOmega$ necessarily a totally disconnected compact Hausdorff space). What does the functor that takes $Omega$ to $dotOmega$ look like? Is it adjoint to the forgetful functor from the category of totally disconnected Hausdorff spaces to LCHaus? Is there some explicit description of $dotOmega$ based on $Omega$?
c-star-algebras von-neumann-algebras
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
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add a comment |
$begingroup$
Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is a property that the weak operator topology can detect, if $A$ were Abelian, then $A^{**}$ would be as well.
Now, in the Abelian case, $Acong C_0(Omega)$ and $A^{**}cong C(dotOmega)$ (with $dotOmega$ necessarily a totally disconnected compact Hausdorff space). What does the functor that takes $Omega$ to $dotOmega$ look like? Is it adjoint to the forgetful functor from the category of totally disconnected Hausdorff spaces to LCHaus? Is there some explicit description of $dotOmega$ based on $Omega$?
c-star-algebras von-neumann-algebras
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
$endgroup$
1
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It's definitely not adjoint to the forgetful functor from totally disconnected Hausdorff spaces--that adjoint doesn't exist.
$endgroup$
– Eric Wofsey
Dec 24 '18 at 22:52
2
$begingroup$
It computes the hyperstonean cover of Ω. See the work of Dixmier.
$endgroup$
– Dmitri Pavlov
Dec 25 '18 at 0:54
add a comment |
$begingroup$
Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is a property that the weak operator topology can detect, if $A$ were Abelian, then $A^{**}$ would be as well.
Now, in the Abelian case, $Acong C_0(Omega)$ and $A^{**}cong C(dotOmega)$ (with $dotOmega$ necessarily a totally disconnected compact Hausdorff space). What does the functor that takes $Omega$ to $dotOmega$ look like? Is it adjoint to the forgetful functor from the category of totally disconnected Hausdorff spaces to LCHaus? Is there some explicit description of $dotOmega$ based on $Omega$?
c-star-algebras von-neumann-algebras
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
$endgroup$
Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is a property that the weak operator topology can detect, if $A$ were Abelian, then $A^{**}$ would be as well.
Now, in the Abelian case, $Acong C_0(Omega)$ and $A^{**}cong C(dotOmega)$ (with $dotOmega$ necessarily a totally disconnected compact Hausdorff space). What does the functor that takes $Omega$ to $dotOmega$ look like? Is it adjoint to the forgetful functor from the category of totally disconnected Hausdorff spaces to LCHaus? Is there some explicit description of $dotOmega$ based on $Omega$?
c-star-algebras von-neumann-algebras
c-star-algebras von-neumann-algebras
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
asked Dec 24 '18 at 21:40
Ashwin TrisalAshwin Trisal
1,2891516
1,2891516
1
$begingroup$
It's definitely not adjoint to the forgetful functor from totally disconnected Hausdorff spaces--that adjoint doesn't exist.
$endgroup$
– Eric Wofsey
Dec 24 '18 at 22:52
2
$begingroup$
It computes the hyperstonean cover of Ω. See the work of Dixmier.
$endgroup$
– Dmitri Pavlov
Dec 25 '18 at 0:54
add a comment |
1
$begingroup$
It's definitely not adjoint to the forgetful functor from totally disconnected Hausdorff spaces--that adjoint doesn't exist.
$endgroup$
– Eric Wofsey
Dec 24 '18 at 22:52
2
$begingroup$
It computes the hyperstonean cover of Ω. See the work of Dixmier.
$endgroup$
– Dmitri Pavlov
Dec 25 '18 at 0:54
1
1
$begingroup$
It's definitely not adjoint to the forgetful functor from totally disconnected Hausdorff spaces--that adjoint doesn't exist.
$endgroup$
– Eric Wofsey
Dec 24 '18 at 22:52
$begingroup$
It's definitely not adjoint to the forgetful functor from totally disconnected Hausdorff spaces--that adjoint doesn't exist.
$endgroup$
– Eric Wofsey
Dec 24 '18 at 22:52
2
2
$begingroup$
It computes the hyperstonean cover of Ω. See the work of Dixmier.
$endgroup$
– Dmitri Pavlov
Dec 25 '18 at 0:54
$begingroup$
It computes the hyperstonean cover of Ω. See the work of Dixmier.
$endgroup$
– Dmitri Pavlov
Dec 25 '18 at 0:54
add a comment |
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$begingroup$
It's definitely not adjoint to the forgetful functor from totally disconnected Hausdorff spaces--that adjoint doesn't exist.
$endgroup$
– Eric Wofsey
Dec 24 '18 at 22:52
2
$begingroup$
It computes the hyperstonean cover of Ω. See the work of Dixmier.
$endgroup$
– Dmitri Pavlov
Dec 25 '18 at 0:54