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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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Sum of two positive elements is positive in a $C^*$ algebra is positive

I think this is easy to see, using Gelfand Transform. Using the transform we can see $spec(a)=\{\phi(a)| \phi\in Spec(A)\}$ thus $spec(a+b)=\{\phi(a)+\phi(b)|\phi\in Spec(A)\}$ Now if both $a$ and $b$ are positive then clearly $\phi(a)$ and…
Sorfosh
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Dimension of state space

Let A be a $\mathcal{C}$* algebra. We define state , say $\phi$ on A ( linear functional on A) such that f is positive and $\phi$( 1)= 1 . I'm trying to prove the following : If A is isomorphic to C iff order of state space is 1. The usual part…
Leo
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Definition of BDF sum of extensions

The sum of two extension is defined in the following way: Let $\mathcal{B}$ and $\mathcal{A}$ be two separable C*-algebras with $\mathcal{B}$ stable and $\phi,\psi: \mathcal{A} \rightarrow \mathcal{M(B)/B}$ be two extensions of $\mathcal{B}$ by…
Arindam
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Example of an bounded , self-inverse, anti-multiplicative, anti-linear operator on banach algebra

Actually I want an involution on Banach algebra B such that ||x*|| $\le$ c.||x|| $\forall$ x in B and some c > 1 $\exists$ y in B such that ||y*|| $\ne$ ||y||
Manabendra Giri
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Separable reducing subspace of a representation

I'm looking for a hint or a reference to understand what's going on in the following problem. Suppose that $A$ is a unital $C^*$-algebra, and let $\pi : A \rightarrow B(H)$ be a representation. Problem. If $A$ is separable and $\pi$ is injective,…
ragrigg
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Bicommutant of self-adjoint subset of an involutive algebra

1.1.9. Let $A$ be an involutive algebra and $M$ a self-adjoint subset of $A$... If the elements of $M$ commute pairwise, then $M\subset M'$, so that $M'\supset M''$ and $M''$ is commutative... Source: Dixmier's Here $M'$ and $M''$ are the…
Frenzy Li
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An application to Zorn's lemma

Let $S$ be a subset of $A_+$ the positive elements of a $C^\ast$-algebra $A$ which is weakly compact. I want to show that $S$ has a minimal and maximal elements. I know that $S$ has a partial order since its a subset of $A_+$ we say $a\leq b$ for…
adham
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for every finite dimensional $C^*$-algebra there is a faithful, non-degenerate representation-> is $\dim H_1<\infty$?

Let $A$ be a $C^*$-algebra, $A$ finite dimensional. Then there is a faithful, non-degenerate representation of $A$. How to prove it?. Take an irreducible representation $\pi_1:A\to L(H_1)$ of $A$ with $H_1=\overline{\pi(A)x}$ for a fixed $x\in…
banach-c
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does this theorem imply that every $C^*$-algebra has an approximate unit?

I read that every $C^*$-algebra has a approximate unit. But we proved only the following theorem in lecture: Let $A$ be a $C^*$-algebra, $I\subseteq A$ an ideal which is dense in $A$. Then there is an approximate unit…
google-hupf
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$b \le \|b\|$ even when $b$ is not normal or self-adjoint?

It is a theorem in $C^\ast$ algebras that if $0\le a \le b$ then $\|a\|\le \|b\|$. The proof given in this book (page 47) starts by asserting that $b \le \|b\|$ because we can use the Gelfand transform on the $C^\ast$ algebra generated by $1$ and…
user167889
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Why is the restriction of a character here non zero?

Let $A$ be a unital $C^\ast$-algebra, let $a$ be normal, $B$ the $\ast$-subalgebra generated by $1$ and $a$ and $f\in C(\sigma (a))$. Let $C$ be the $\ast$-algebra generated by $1$ and $f(a)$. If $\tau \in \Omega (B)$ why is $\tau\mid_C$ non-zero?
user167889
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Is this really the argument used here?

Let $A$ be some normed vector space and let $A^\ast$ denote its dual. Then if for all $\varphi \in A^\ast$ we have $\varphi (a) = 0$ then $a=0$. This is an argument that is frequently used. Consider this theorem: If $A$ is a unital $C^\ast$ algebra…
user167889
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What is the result that implies $\ell^1$ is isomorphic to $C(K)$

In this answer here t.b. writes that $\ell^1(\mathbb Z)$ would then have to be isomorphic to a space of the form $C(K)$ with $K$ compact (metrizable and infinite). What is the result they are referring to?
user167889
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