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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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Norm of group algebra in $C^*_r(G)$

There is something that is nagging me, I thought that the operator norm of the group ring inside the $C^*_r(G)$ was the same as it's $l^2$ norm but this seems to not be true, clearly the $l^2$ norm is less than the operator norm by evaluating by the…
sirjoe
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spectrum of a sequence

suppose $A$ is a C* algebra and I consider a sequence of non invertible elements $a_n$ which is bounded below and bounded above in norm. I'm wanting to show that it is not possible to pick a sequence $c_n$ of elements in $A$ whose norms converge to…
sirjoe
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From separable C*-algebra $\mathcal{A}$ a *-representation into $B(l_2)$

Given a separable $C^*$-algebra $\mathcal{A}$ can we construct a $*$-representation $\phi: \mathcal{A} \rightarrow \mathcal{B}(l_2)$ such that for every $x \neq 0$, ideal generated by $\phi(x)$ is $\mathcal{B}(l_2)$? I only guess that the image of…
Arindam
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Approximate units in AF algebras

I am trying to build an approximate unit whose elements have finite spectrum in any AF-algebra. Suppose $A$ is a C*-algebra which is built as an inductive sequence $(A_n,\lambda_n)$ of finite dimensional C*-algebras. If we consider the maps $\phi^n…
sirjoe
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Example of Approximately Inner automorphism which is not inner

Apart from inductive limit constructions such as the Carr Algebra (Where you could just take direct sums of unitaries) I can't think of an example of a C*-algebra and an approximately inner automorphism of it which is not inner. I was wondering if…
sirjoe
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The inner structure of finite-dimensional $C^*$-algebras

This is just to complete the questions about the inner structure of finite-dimensional $C^*$-algebras. Please correct the statements and help me with the appropriate links if it is not very difficult. Maybe I am wrong somewhere. Let $H$ be a Hilbert…
AAK
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Is it possible $\lVert a\rVert =\lVert 1+a\rVert $ in a C$^*$-algebra?

If $A$ is a unital C$^*$-algebra and $a\in A$, Is it possible $ \lVert a \rVert =\lVert 1+a \rVert $ for an $a\geq 0$ ? I think it's trivial that it's not possible but I can't prove it for even $ A=Mat_{n\times n}(\mathbb{C})! $
Darman
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A problem in the proof of the irrational rotational algebra has unique trace

I'm reading Davidson's book C* Algebras by Example.in the proof of the irrational rotational algebra has unique trace. he define a automorphism: $$\rho_{\lambda,\mu}(U)=\lambda U, \rho_{\lambda,\mu}(V)=\mu V, $$ where $U,V$ are the generator of…
knot
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Let $A=\mathbb C[x] $ prove there is no norm on $A$ in which it is a C* algebra

Let $A=\mathbb C[x] $ prove there is no norm on A in which it makes a C* algebra. i think this is true because the spec(a) is infinity for any $a\in A$ ? but im not sure how to prove it. I did try verifying the axioms for a norm but im not sure what…
Faust
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Image of finite element of AF algebra is finite?

Let $A = \overline{\bigcup A_n}$, $B = \overline{\bigcup B_n}$ be AF algebras with the same Bratelli diagram. Then there is an isomorphism $\phi : A \rightarrow B$. Let $x \in \bigcup A_n$ (say, $x \in A_k$ for some $k$). Is it always the case that…
vagnard
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The absolute value of a bounded linear functional on a C*-algebra

Let us consider the commutative C*-algebra $C_0(\Omega)$. Let $\mu$ be a complex Radon measure on $\Omega$. By the Riesz representation theorem, $\mu$ may be considered as a bounded linear functional on $C_0(\Omega)$. The polar decomposition says…
ABB
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does $\|x\|=\sqrt{\|a\|^2+\|b\|^2}$ hold in an arbritrary $C^*$algebra?

$A$ $C^*$-algebra, $x\in A$ can be written as $x=a+ib$ with $a$ and $b$ self-adjoint. Then $\|x\|=\sqrt{\|a\|^2+\|b\|^2}$ is not true in general, right? If $A=M_2(\mathbb{C})$ maybe you can find a counterexample, $\|a\|$ and $\|b\|$ is the spectral…
algebra
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On cyclic representations of a C*-algebras

I feel the following assertion is true but have no evidence to prove: There exists an infinite dimensional C*-algebra such any cyclic representation $\pi$ of $A$ is finite dimensional! Probably $\bigoplus_{1}^{\infty}M_2(\mathbb{C})$ works.
ABB
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irreducible representations and pure states of $M_n(\mathbb{C})$

We consider the $C^\ast$-algebra $A=M_n(\mathbb{C})$, which can be seen as $L(\mathbb{C}^n)$. Prove that: (1) $id:A\to A$ is a irreducible $\ast$-representation of $A$ (2) all irreducible $\ast$-representations $\pi:A\to L(H_{\pi})$ of $A$ are…
banach-c
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Let $A$ be a $C^\ast$-algebra, $a\in A$. Equivalent: $a\ge 0 \iff$ for all states $\varphi\in S(A)$ is $\varphi(a)\ge 0$

Let $A$ be a $C^\ast$-algebra, $a\in A$. Equivalent: $a\ge 0 \iff$ for all states $\varphi\in S(A)$ is $\varphi(a)\ge 0$. First of all $S(A)$ is the state space of $A$, i.e. all positive linear functional with norm 1. The direction $\Rightarrow$ is…
banach-c
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