Questions tagged [gelfand-duality]

For questions related to Gelfand duality (duality between spaces and their algebras of functions for the case of compact topological spaces and commutative C-star algebras).

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Proof of the Gelfand-Naimark Theorem

I am reading a proof of the Gelfand-Naimark theorem in Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff. I have questions about some of the steps. Theorem. If $A$ is a ...
stoic-santiago's user avatar
4 votes
1 answer
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Spectrum of a direct product of commutative $C^*$-algebras.

Let $\{A_i: i \in I\}$ be a collection of commutative $C^*$-algebras. Given a commutative $C^*$-algebra $A$, denote its spectrum (consisting of non-zero algebra morphisms) by $\Omega(A)$. It is true ...
Andromeda's user avatar
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The relation between linear functionals in the dual space and codimension of a subspace

I want to prove a statement that a subspace $L^m$ of X is of codimension at most m if and only if there exists linear functionals $\lambda_1,\cdots,\lambda_m$ : $X\rightarrow\mathbb{R}$ in the dual ...
Eric's user avatar
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20 votes
1 answer
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How to understand what a 'noncommutative space' is

Whilst acquainting myself with the fundamentals of $C^*$-algebras and their $K$-theory, I read about how Gelfand Duality allows various $C^*$-algebraic concepts to be directly translated into a ...
user829347's user avatar
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4 votes
0 answers
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Can one deduce Gelfand Duality from Isbell Duality?

I was reading about Isbell Duality in nLab. It gives an example of how we have an adjunction from $TAlg$ to presheaves. But it is not clear to me how one can deduce the Gelfand Duality from this.
W. Zhan's user avatar
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2 votes
1 answer
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Gelfand duality restricted to the category of Stone spaces

It is well known that the category of unital commutative $C^\ast$-algebras is dually equivalent to the category $\mathbf{KHaus}$ of compact Hausdorff spaces. I have found that restricting the ...
FML's user avatar
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