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Simon Henry
Simon Henry
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This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.

PS : It is not completely clear from the statement in paper, but they do ask that the functor extend the Gelfand duality on object AND morphisms. So you might hope to obtain such a functor in a way that don't let the morphism of commutative algebra act naturally on the center... But this seems also impossible: if you look at the proof in the paper you only need your functor to be the natural thing on $\mathbb{C}$, $\mathbb{C}^2$ and $\mathbb{C}^3$ are morphisms between them to obtain that the image of $M_3(\mathbb{C})$ is degenerate, and because any functor have to preserve spliting of projection I would bet that no assumptions on morphism is required.

This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.

PS : It is not completely clear from the statement in paper, but they do ask that the functor extend the Gelfand duality on object AND morphisms. So you might hope to obtain such a functor in a way that don't let the morphism of commutative algebra act naturally on the center... But this seems also impossible: if you look at the proof in the paper you only need your functor to be the natural thing on $\mathbb{C}$, $\mathbb{C}^2$ and $\mathbb{C}^3$ are morphisms between them to obtain that the image of $M_3(\mathbb{C})$ is degenerate, and because any functor have to preserve spliting of projection I would bet that no assumptions on morphism is required.

This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.

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Source Link
Simon Henry
Simon Henry
  • 42.4k
  • 5
  • 107
  • 205

This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.

PS : itIt is not completely clear from the statement in the paper if one need a functors, but they do ask that the functor extend the gelfandGelfand duality on object or on object and morphismAND morphisms. So you might hope to obtain such a functor in a way that don't let the obstructionmorphism of commutative algebra act naturally on the center. but it.. But this seems also impossible: if you look at the proof in the paper you only need your functor to be using only very specific commutative algebrathe natural thing on $\mathbb{C}$, $\mathbb{C}^2$ and $\mathbb{C}^3$ are morphisms between them to obtain that the image of $M_3(\mathbb{C})$ is degenerate, and because any functor have to preserve spliting of projection I would bet that no assumptions on morphism is required.

This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.

PS : it is not clear from the statement in the paper if one need a functors that extend the gelfand duality on object or on object and morphism to obtain the obstruction. but it seems to be using only very specific commutative algebra.

This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.

PS : It is not completely clear from the statement in paper, but they do ask that the functor extend the Gelfand duality on object AND morphisms. So you might hope to obtain such a functor in a way that don't let the morphism of commutative algebra act naturally on the center... But this seems also impossible: if you look at the proof in the paper you only need your functor to be the natural thing on $\mathbb{C}$, $\mathbb{C}^2$ and $\mathbb{C}^3$ are morphisms between them to obtain that the image of $M_3(\mathbb{C})$ is degenerate, and because any functor have to preserve spliting of projection I would bet that no assumptions on morphism is required.

Source Link
Simon Henry
Simon Henry
  • 42.4k
  • 5
  • 107
  • 205

This paper proves that there is no functor from the category of C* algebra (and morphisms) to the category of locales/topological spaces/a lot of other things that extend the gelfand duality and send matrix algebra for $n >2$ to some non-empty space. Composing your eventual "center functor" with gelfand spectrum would give such a functor.

PS : it is not clear from the statement in the paper if one need a functors that extend the gelfand duality on object or on object and morphism to obtain the obstruction. but it seems to be using only very specific commutative algebra.