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Questions tagged [c-star-algebras]

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

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$.

For bounded operators on a given Hilbert space, C-algebras characterize topologically closed subalgebras of ${\mathcal B}({\mathcal H})$ (in operator norm), also closed under taking the adjoint operator. C-algebras are at the heart of and are extensively used in .

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Algebra of matrices -- equivalence of norms

Let $A = M_n(\mathbb C)$. Then it is possible to endow this $\ast$-algebra with several different norms (see here): $$ \|a\|_1 = \max_j \sum_i |a_{ij}|$$ $$ \|a\|_\infty = \max_i \sum_j |a_{ij}|$$ and the operator norm: $$ \|a\| =…
user167889
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Positive elements in star algebras

Let $A$ be a $C^\ast$-algebra. Is it possible to prove that if $a \ge 0$ then $ab, ba \ge 0$ if and only if $b \ge 0$?
user167889
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Norm on unitisation is submultiplicative

Let $A$ be a $C^\ast$-algebra and let $\widetilde{A}$ denote its unitisation. Define a norm on $\widetilde{A}$ as $\|(a,\lambda)\| = \sup_{\|b\|=1} \|ab + \lambda b\|$. I could show that $\|(a,\lambda)^\ast (a,\lambda)\| = \|(a,\lambda)\|^2$ using…
user167889
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On a condition similar to the star algebra

Recently I have been reading about star algebras. In particular, $C^\ast$-algebras. It seems that the condition $\|a^\ast a\| = \|a\|^2$ is quite strong and much is known about $C^\ast$-algebras. I was wondering if it makes sense to consider the…
user167889
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Why is the natural map from maximal to reduced C star algebra surjective?

In the book "Kazhdan' property (T)" the third book in this link, page 438, One sentence is "the regular representation defines a surjective *-homomorphism $\lambda_G: C^{*}(G)\to C^{*}_\text{red}(G)$". I understand the existence of such…
ougao
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In a C*-algebra, does $|xyz| \leq |xz|^\frac{1}{2} | y| \ | xz|^\frac{1}{2}$ hold?

Let $x,y,z$ be elements of a C*-algebra $A$. Does the inequality $$|xyz| \leq |xz|^\frac{1}{2} | y| \ | xz|^\frac{1}{2}$$ hold in the positive cone of $A$? Here $|x| := (x^*x)^\frac{1}{2}$. If $y= 1$, this is an equality. If $z=1$, this is $…
Mike F
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How to understand the well-supported element in the $C^*$-algebra?

An element $x$ in the $C^*$-algebra $A$ is well-supported if there is a $p\in A$ with $x=xp$ and $x^*x$ invertible in $pAp$. That is the definition, but I cannot catch the key of it. Maybe you can show me some examples, which one is well-supported,…
Strongart
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C$^*$-Algebra Beginner's Questions

I am starting with $C^*$ algebras. There are some notations that I don't understand. Please help me. What does the identity representation of $C^*$ algebras mean? Let $A$ be $C^*$ algebra generated by the irreducible operator A and the identity.…
user120127
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Is a projection infinite iff its range is infinite-dimensional?

A projection is said to be infinite if it is Murray-vN equivalent to some subprojection. In other words, there exists $q,a\in A$ such that $q\leq p$, $p=aa^*$ and $q=a^*a$. Is this condition equivalent to the projection having infinite-dimensional…
Muddana
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Positive definiteness of the unitalization

Let $A$ be a $C^*$-algebra. Consider the unitalization $A^+$ with underlying vector space $A^+=A\oplus \mathbb{C}$. The norm on $A^+$ is defined as $||a+\lambda||_{A^+}=\mathrm{sup}_{||b||\leq 1}\ ||ab+\lambda b||_A$. In order to show that this is…
Roland
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Let $A$ be a $C^\ast$-algebra and $f\colon A\to\mathbb{C}$ a positive linear functional. Suppose $a\leq b^{[1]}$, is it true that $f(a)\leq f(b)$?

Let $A$ be a $C^\ast$-algebra and $f\colon A\to\mathbb{C}$ a positive$^[2]$ linear functional. Suppose $a\leq b^{[1]}$, is it true that $f(a)\leq f(b)$? This is used in a proof, but for approximate identities, maybe they are needed. $^{[1]}$ Note…
Rough_Manifolds
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Support of a subalgebra of a $C^*$ algebra

Let $A$ be a finite dimensional $C^*$ algebra and let $I$ be a two sided ideal in $A$. What is meant by the notion of support of $A/I$. I have heard of support of a self adjoint operators.Can any one please provide the reference for this?
budi
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If $a$ is invertible in a unital C$ ^*$-algebra, then $a^{-1} \in C^*(a)$

This is Exercise 1.6(iii) from Intro. to K-Theory for $C^*$-Algebras by Rørdam et al. If $A$ is a unital $C^*$-algebra, and $a \in A$ is invertible, then I want to show that $a^{-1} \in C^*(a)$. What I know is: $a$ is invertible if and only if…
FratSourced
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Can every bounded functional be written as a linear combination of two bounded hermitian functionals?

Let $A$ be a $C^{\ast}$-algebra. Let $\rho$ be a bounded linear functional on $A.$ Does there exist bounded hermitian functionals $\rho_1$ and $\rho_2$ on $A$ such that $\rho = \rho_1 + i \rho_2\ $? Any suggestion in this regard would be…
Anil Bagchi.
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Denote by $\text{Ped}(A)$ the Pedersen ideal of $A$ and assume $a\in A^+$ is a full element. Is $\text{Ped}(\overline{aAa})=\overline{aAa}$?

Pedersen ideal is the minimal hereditary dense ideal. An element $a$ is full if the ideal generated by $a$ is dense. Assume $a\in \text{Ped}(A)$ is a positive full element. Let $A_1=\overline{aAa}$. Is $\text{Ped}(A_1)=A_1$? Brown's theorem: Assume…
Yuz
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