Questions tagged [gelfand-duality]

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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 ...
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14 votes
3 answers
1k views

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 ...
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1 answer
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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?
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11 votes
2 answers
236 views

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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5 votes
1 answer
332 views

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 ...
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4 votes
0 answers
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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 ...
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5 votes
1 answer
163 views

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 answers
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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 ...
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8 votes
1 answer
642 views

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 ...
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2 votes
1 answer
271 views

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 $C$, ...
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5 votes
2 answers
841 views

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 ...
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5 votes
1 answer
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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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1 vote
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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 vote
1 answer
193 views

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 ...
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0 answers
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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}\...
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3 votes
0 answers
227 views

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 ...
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1 answer
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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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10 votes
5 answers
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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 ...
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0 votes
1 answer
359 views

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, ...
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36 votes
5 answers
4k views

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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