Tagged Questions

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

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Ordering: Definition

This was a real question! Given a unital C*-algebra $1\in\mathcal{A}$. For $A\in\mathcal{A}$ denote its spectrum by $\sigma(A)$. Consider the selfadjoints: ...
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why is the direct sum of irreducible representations of a ^$C^*$-algebra faithful?

Let $A$ be a C$^*$-algebra. Let $\mathrm{Irr}(A)=\{[\pi]: \pi \text{ is an irreducible representation of } A\}$ and $\rho\in [\pi]$ if there is an unitary operator $V:H_\pi \to H_\rho$ such that ...
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ker-hull-topology on $Irr(A)$ is the discrete topology ($A$ is a C$^*$-subalgebra of $K(H)$)

Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A} and $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that $V\pi(a)=\rho(a)V$ ...
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$a,b\in A$ are selfadjoint elements of $C^*$-algebras, such that $a\le b$, why is $\|a\|\le \|b\|$

Let $A$ be a unital $C^\ast$-algebra with unit $1_A$. a) Why is $a\le \|a\|1_A$ for all selfadjoint $a\in A$ and b) If $a,b\in A$ are selfadjoint such that $a\le b$, why is $\|a\|\le \|b\|$? I ...
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Is this condition on linear functionals equivalent to boundedness?

Let $T : C_c(X) \rightarrow \mathbb{R}$ be a linear functional satisfying $$\sup \{|T(f)| :\ f \in C_c(X), |f| \leq 1, \text{Supp}(f) \subset K\} < \infty$$ for each compact $K \subset X$. Is ...
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C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators? Equivalently: Let ...
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Why the disk algebra is not a C* algebra.

I'm trying to figure out why the set of bounded analytic functions on the unit disk, A(D), is not a C* algebra. The norm is the sup norm and the involution is $f(z) \to \overline{f(\bar z)}$. I want ...
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Does category theory help in operator algebras?

I'm currently studying the basics of Banach and $C^*$-algebras. Almost all the proofs i've seen so far are very simple but some of them are extremely tricky (in my opinion). This tricky interplay ...
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Almost periodic compactification with the Gelfand-Naimark theorem

Could anyone please help me with a bibliographic reference presenting the almost periodic compactification of a topological group with the aid of the Gelfand-Naimark theorem? Rudin in "Fourier ...
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Operator realizations of algebras

I want to realize the algebra $A_q(\tilde{S}^{n-1})$ as introduced in the acticle of Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf) as bounded operators on a Hilbert space $H$. Can ...
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Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...