Questions tagged [gelfand-duality]
The gelfand-duality tag has no usage guidance.
24
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Representing measurable map to compact space as a continuous map
Let $\Omega$ be a measurable space equipped with a $\sigma$-ideal $\mathcal{N}$ (though of as the "null sets"). Define the compact Hausdorff space
$$ \tilde{\Omega} := \mathrm{Spec}(L^\infty(...
- 141
2
votes
1
answer
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Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections
I would be very grateful for any references I might be led to, from a categorical point of view for the functors:
$\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which ...
1
vote
1
answer
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views
Adjunction via Gelfand duality
$\DeclareMathOperator\Hom{Hom}$For which unital $C^{\ast}$-algebras $A$ does it hold that for all compact Hausdorff $S$ we have the bijection:
\begin{align*}
\Hom(A, C(S)) \cong \Hom(S, \Hom (A, \...
- 1,117
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0
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154
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Gelfand's transform for noncommutative $C^*$-algebras
Please excuse me if this is well-known, I am not very familiar with the general theory of $C^*$-algebras.
Let $A$ be a unital separable liminal $C^*$-algebra (in the case I am interested in, ...
- 2,765
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0
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Non-commutative duality II: The two algebras of a groupoid
This question is in a sense the continuation of my previous one, Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?. Reading the great answers and the equally good ...
- 7,487
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3
answers
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Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?
Many interesting C*-algebras can be realized as convolution algebras over a groupoid, an idea introduced in 1980 by Jean Renault (this entry in nLab provides plenty of context to the general approach ...
- 7,487
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votes
1
answer
77
views
Involution in a commutative unital real C* algebra
It follows immediately from Gelfand duality that the involution in a commutative unital real C* algebra is the identity. Is there a direct proof from the axioms of C* algebras?
- 1
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2
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Fundamental group under Gelfand duality
Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^*$-algebras is an equivalence of categories. In other words, all topological ...
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votes
1
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Generalized Gelfand triples
Normally, when we talk about Gelfand triple we have three Hilbert spaces
$$\newcommand{\X}{\mathcal{X}}
\X_+ \subset \X_0 \subset \X_-
$$
such that the subsets are dense, the embedding mappings are ...
4
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0
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Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra
Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra $A$ equivalent to the existence of a countable family $\{\varphi_n\}_{n\in\omega}$ of multiplicative ...
- 4,385
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Are there any non-trivial convergent sequences in the maximal ideal space of the measure algebra?
Consider the measures on the circle, $M(\mathbb T)$, endowed with the convolution product which makes it a unital Banach algebra under the total variation norm. Denote by $\Delta$ the maximal ideal ...
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0
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91
views
Can we express separability of a ray-remainder in terms of the function algebra?
Let $X = [0, 1)$ be a ray and $C(X)$ the algebra of bounded continuous real functions. The spectrum of $C(X)$ is the Stone-Cech compactification $\beta [0,1) $ of the ray. It's easy to see the ...
- 1,761
8
votes
1
answer
685
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Recovering a smooth manifold from its tensor fields
1) Consider the algebra $\mathcal T (M) = \bigoplus \limits _{p, q \ge 0} \mathcal T ^{p, q} (M)$, where $\mathcal T ^{p, q} (M)$ is the space of tensor fields of type $(p,q)$. I should endow it with ...
- 5,105
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1
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Can we Characterise Rings of Continuous Functions?
Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...
- 1,761
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Commutative von Neumann algebras and localizable measure spaces
This is not my subject so I apologize if my question is too obvious or understood from other pages.
I read some pages such as
Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
- 1,324
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votes
1
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855
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Proper continuous image of metrizable space
Motivated by the following post, "Gelfand duality" and the fact that "a Hausdorff continuous image of a compact metric space is metrizable", we ask:
What is a counter example of two locally ...
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0
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Properties of a function from its pullback
Edit: I have now removed the duplication previously referred to. Thank you.
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{...
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1
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Spectrum of a Banach algebra homeomorphic to the spectrum of one of its elements
In G. B. Folland - A Course in Abstract Harmonic Analysis we can read the following
" (1.15) Proposition. Let be $A$ a (complex) commutative unital Banach algebra with unit $e$, let $x_0 \in A$ and ...
user40023
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A noncommutative analogy of the tube lemma
Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to \mathbb{C}\...
- 270
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0
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Tauberian theorem from generalized Gelfand transform
Wiener's theorem gives the necessary and sufficient conditions for the set of translates of a set of functions to be dense in $L^1(\mathbb{R}^n)$, which translates algebraically into a statement about ...
- 1,039
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1
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Dual notion of a local homeomorphism between topological spaces for C*-algebras
Given two locally compact topological spaces $X$ and $Y$, and a local homeomorphism $f : X \to Y$, Gelfand duality gives us a homomorhism $Cf : C_0(Y) \to C_0(X)$. How does the fact that $f$ is a ...
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Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?
In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary ...
- 1,095
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1
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The functor of continuous functions from compact CW-spaces to the reals
The contravariant functor $C(-)$ given by
$$
\hom_{Top}(-,\mathbb{R}):cCW\to Rng
$$
where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, ...
- 2,714
37
votes
5
answers
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Reference for the Gelfand duality theorem for commutative von Neumann algebras
The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann ...
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