Questions tagged [gelfand-duality]
The gelfand-duality tag has no usage guidance.
28
questions
2
votes
0
answers
176
views
How can the maximal ideal space of the Fourier Stieltjes algebra be non-separable?
I have been asking a fair few (probably elementary) questions about abstract harmonic analysis lately. By means of explanation, I am just feeling around the subject at the moment and trying to build ...
- 1,915
-2
votes
1
answer
110
views
Mismatch between equivalent definitions of the Bohr compactification of the reals
I feel I'm overlooking something very silly.
The Bohr compactification of $\mathbb R$ has two equivalent definitions.
The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under ...
- 1,915
3
votes
0
answers
103
views
How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
- 1,915
1
vote
0
answers
96
views
When is the "Gelfand Remainder" compact?
Suppose we have a noncompact Hausdorff space $S$ and a Banach algebra $A \subset C^*(S,\mathbb R)$ of the space of real-valued bounded functions on $S$. For niceness let's assume $A$ separates the ...
- 1,915
3
votes
2
answers
265
views
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(...
user103549
2
votes
1
answer
178
views
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
196
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, \...
- 1,377
4
votes
0
answers
166
views
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, ...
- 3,084
3
votes
0
answers
163
views
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 ...
- 7,693
15
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 ...
- 7,693
0
votes
1
answer
96
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
11
votes
2
answers
268
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 ...
- 21.2k
5
votes
1
answer
476
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 ...
4
votes
0
answers
85
views
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 ...
- 4,525
5
votes
1
answer
175
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 ...
- 11.3k