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Questions tagged [gelfand-representation]

Gelfand representation is a way of representing commutative Banach algebras as algebras of continuous functions.

Gelfand representation can be related to functional analysis, Banach algebras and $C^*$-algebras.

74 questions
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What space corresponds to the localisation of the ring of continuous functions?

Suppose $A$ is a commutative Banach algebra. By Gelfand duality there is a compactum $X$ such that $A = C(X)$ is the ring of continuous functions. The space $X$ can be recovered as the space of characters on $A$. That is to say multiplicative linear…
Daron
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Learning roadmap and prerequisites for Isbell duality

I'm looking for a roadmap to learning about Isbell duality. I know a reasonable amount about several of the "specific" dualities (Gelfand duality, AffSch - CRing, frames - locales, etc), especially affine schemes. However, I'm having trouble finding…
Jon H
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Where does Gelfand Theory fail for non-commutative algebras.

I'm trying to get my head around Gelfand theory, and I can't seem to find the subtleties between commutative and non-commutative algebras. Why is there not a one-to-one correspondence between maximal ideals of a non-commutative algebra and the…
Jacob Denson
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Approximate unit on $C_0(X)$ converges uniformly on compact sets

Let $(e_{\lambda})_{\lambda \in \Lambda} \subseteq C_0(X)$ with $X$ locally compact an (increasing) approximate unit. I assume that for every compact $K\subseteq X$ we have $\left\Vert 1-e_{\lambda}\right\Vert _{K\text{, }\infty}\rightarrow0$ and I…
worldreporter
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An example which disproves $\|p(T)\|_{op} = \|p\|_\sup$ in arbitrary Banach spaces (Gelfand theory).

Let me go straight to the question and then write some discussion about it. Question: Does there exist a Banach space $X$, an operator $T:X\rightarrow X$ with operator norm $\|T\|_{op}\leq 1$ and a sequence of polynomials $p_n:\mathbb{C}\rightarrow…
Yanko
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When exactly is the character space of a Banach algebra empty?

It is well-known that the character space, (i.e. the set of multiplicative characters) of a commutative, unital Banach algebra is non-empty. But is there a complete characterization of when exactly the character space of Banach algebra is empty? Or…
TuringTester69
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Unital homomorphism to semisimple Banach algebra is automatically continuous

I need to prove that any unital homomorphism $\phi: A \to B$, where $A$ is unital Banach algebra and $B$ is semisimple Banach algebra is continuous. The definition of "semisimple" I know is that the kernel of Gelfand transform which is the same as…
Invincible
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Is the range of the Gelfand transform closed?

Let $A$ be a commutative unital Banach algebra. Consider the Gelfand map $\Gamma:A\longrightarrow C(M_A)$, $\Gamma(a)=\hat{a}$, where $M_A$ is the Character space of $A$. Is the image of the Gelfand map, $\Gamma(A)$ closed in $C(M_A)$ with the…
user346635
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Character Space of $C^1[0,1]$ and Gelfand Representation

I have recently been working through some exercises in Murphy's "C*-Algebras and Operator Theory," and I am having some trouble with Exercise 10 in Chapter 1. The exercise is as follows: Let $A = C^1[0,1]$. Let $x: [0,1] \longrightarrow \mathbb{C}$…
LSK21
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What is the Gelfand-Naimark representation of functions that don't vanish at infinity?

The Gelfand-Naimark theorem says that every commutative C*-algebra is isometrically isomorphic to $C_0(X)$, the set of continuous functions $f:X\rightarrow\mathbb{C}$ that vanish at infinity, for some locally compact Hausdorff space $X$. What…
Jahan Claes
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Gelfand transform on functions

The Gelfand transformation identifies function spaces $C_0(X)$ for locally compact Haussdorff $X$ with commutative $C^*$ Algebras. Additionally there is a statement that if $f: X \to Y$ is a proper and continuous map, this induces a $*$-morphism…
s.harp
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Norm of GNS representation vs its state

Let $A$ be a $C^*$-algebra, $\phi$ a state on $A,$ and $\pi$ the GNS representation associated with $\phi.$ Can it be said in general that $\sup_{a\in A}\frac{\|\pi(a)\|^2}{\phi(a^*a)}=M<\infty$? If not, can we find increasing, continuous…
Miles Gould
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The spectrum of an abelian von Neumann algebra

I am trying to understand the spectrum of an abelian Von Neumann algebra $A$. Let $\Omega(A)$ denote this spectrum with the weak$^*$-topology, and $\Omega_{\mathrm{disc}}(A)$ is isolated points in $\Omega(A).$ Is there a way to calculate exactly for…
Miles Gould
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Show that $\text {Ran} \left (P \left (\{\lambda \} \right ) \right )= E_{\lambda}$ for any $\lambda \in \sigma (A).$

Let $H$ be a complex Hilbert space and $\mathcal A$ be a commutative unital $C^{\ast}$-subalgebra of $B (H).$ Then the Gelfand transform gives a isometric-$\ast$-isomorphism from $\mathcal A$ onto $C \left (\Sigma \right ),$ where $\Sigma$ is the…
Anacardium
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How to describe the Gelfand transform of the Banach algebra of complex Borel measures on the real line?

Let $M$ be the Banach algebra of all complex Borel measures on $\mathbb{R}$. To be clear, Norm: $\| \mu \| = |\mu|(\mathbb{R})$, where $|\mu|(E)$ is the total variance. Product: $(\mu \ast \lambda)(E) = (\mu \times \lambda)(\{(x,y):x+y \in…
user614535
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