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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).

Gelfand duality is a duality between commutative $C^*$-algebras $A$ and compact Hausdorff spaces $X$ is the same: it assigns to $X$ the space of continuous functions (which vanish at infinity) from $X$ to $C$, the complex numbers. Conversely, the space $X$ can be reconstructed from $A$ as the spectrum of $A$.

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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 geometrical/topological context and vice versa. I…
user829347
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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 that we have a…
Andromeda
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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
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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 Gelfand duality to the full subcategory of Stonean…
FML
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Real Gelfand dual

By Gelfand duality, a hausdorff locally compact space $X$ is homeomorphic to the maximal ideals of the complex-valued continuous functions over $X$ that vanish at infinity. Does this still hold if we consider instead real-valued continuous functions…
V. Semeria
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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 commutative $C^\ast$-algebra, then the Gelfand…
stoic-santiago
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About the dual of a vector-valued weak-* Lebesgue space

Here, $\Omega\subseteq\mathbb{R}^n$ is an open, bounded set (as nice as you want). It is well known that $L^1(\Omega)^* = L^\infty(\Omega)$. Moreover, it is also true that if we endow $L^\infty(\Omega)$ with the weak-* topology (let us denote this…
rod
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Riesz representation theorem for Banach-valued functions

Riesz representation theorem states that, if $\Omega$ is a locally compact Hausdorff space, then $C_0(\Omega)^* = \mathcal{M}(\Omega)$ (or better, they are isomorphic as normed spaces), where $C_0(\Omega)$ is the set of all continuous functions that…
rod
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Gelfand duality says $C_b^{\mathbb R}(X,\mathbb C) \simeq C_b(Y,\mathbb R)$ for some $Y$. But what is $Y$?

The real-valued version of Gelfand duality (ala Johnstone's "Stone Spaces") says every real Banach Algebra is equivalent to the algebra $C_b(Y,\mathbb R)$ of bounded real functions from some compact Hausdorff space $Y$. Edit: No it doesn't. It only…
Daron
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Representation theory to prove harmonic decomposition

I am not very familiar with representation theory, but I keep seeing it applied in very different contexts. I read applications of representation theory for decompositions of space of polynomials, like the harmonic decomposition or, more generally,…
Jules Binet
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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 space $X^*$ such that $$L^m=\{x\in…
Eric
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