Revisions to Gelfand-Naimark from the category-theoretic point of view

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edited Feb 15 at 10:11

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I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let $spec(\mathcal A)$$\operatorname{spec}(\mathcal A)$ denote the spectrum of $A$ and $C(X)$ the algebra of complex continuous functions on $X$. Then spec and C define contravariant functors from commC* alg to CompHausTop, which (correct me if i'm wrong) establish an equivalence between the two categories. Gelfand-Naimark theorem has a non-commutative analogue, which is based on the so-called GNS construction and which shows that every non commutative $C^*$ -algebras has a faithful isometric * -representation on a Hilbert space $H$. In this case I can't see an analogue of the preceding equivalence of categories, which is equally meaningful. Does it exist?

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let $spec(\mathcal A)$ denote the spectrum of $A$ and $C(X)$ the algebra of complex continuous functions on $X$. Then spec and C define contravariant functors from commC* alg to CompHausTop, which (correct me if i'm wrong) establish an equivalence between the two categories. Gelfand-Naimark theorem has a non-commutative analogue, which is based on the so-called GNS construction and which shows that every non commutative $C^*$ -algebras has a faithful isometric * -representation on a Hilbert space $H$. In this case I can't see an analogue of the preceding equivalence of categories, which is equally meaningful. Does it exist?

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let $\operatorname{spec}(\mathcal A)$ denote the spectrum of $A$ and $C(X)$ the algebra of complex continuous functions on $X$. Then spec and C define contravariant functors from commC* alg to CompHausTop, which (correct me if i'm wrong) establish an equivalence between the two categories. Gelfand-Naimark theorem has a non-commutative analogue, which is based on the so-called GNS construction and which shows that every non commutative $C^*$ -algebras has a faithful isometric * -representation on a Hilbert space $H$. In this case I can't see an analogue of the preceding equivalence of categories, which is equally meaningful. Does it exist?

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edit approved Feb 15 at 10:11

Esmond

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I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra$C^*$-algebra (with unit) A$\mathcal{A}$ and the C* algebra$C^*$ -algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let spec(A)$spec(\mathcal A)$ denote the spectrum of A$A$ and C(X)$C(X)$ the algebra of complex continuous functions on X$X$. Then spec and C define contravariant functors from commC* alg to CompHausTop, which (correct me if i'm wrong) establish an equivalence between the two categories. Gelfand-Naimark theorem has a non-commutative analogue, which is based on the so-called GNS construction and which shows that every non commutative C* algebras$C^*$ -algebras has a faithful isometric * -representation on a Hilbert space H$H$. In this case I can't see an analogue of the preceding equivalence of categories, which is equally meaningful. Does it exist?

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let spec(A) denote the spectrum of A and C(X) the algebra of complex continuous functions on X. Then spec and C define contravariant functors from commC* alg to CompHausTop, which (correct me if i'm wrong) establish an equivalence between the two categories. Gelfand-Naimark theorem has a non-commutative analogue, which is based on the so-called GNS construction and which shows that every non commutative C* algebras has a faithful isometric *-representation on a Hilbert space H. In this case I can't see an analogue of the preceding equivalence of categories, which is equally meaningful. Does it exist?

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let $spec(\mathcal A)$ denote the spectrum of $A$ and $C(X)$ the algebra of complex continuous functions on $X$. Then spec and C define contravariant functors from commC* alg to CompHausTop, which (correct me if i'm wrong) establish an equivalence between the two categories. Gelfand-Naimark theorem has a non-commutative analogue, which is based on the so-called GNS construction and which shows that every non commutative $C^*$ -algebras has a faithful isometric * -representation on a Hilbert space $H$. In this case I can't see an analogue of the preceding equivalence of categories, which is equally meaningful. Does it exist?