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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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How do you prove that every AF C*-algebra is finite?

A C*-algebra $A$ is finite if $s^*s=1$ implies $ss^*=1$. A C*-algebra $A$ is AF if: for all $a_1,\ldots,a_n\in A$ and $\varepsilon>0$, there exists a finite-dimensional C*-subalgebra $B_n$ of $A$ and elements $b_1,\ldots,b_n\in B$ so that…
Ehsaan
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In a C*-algebra $A$, $x$ is self-adjoint iff $\lim_{t\to 0}(1/t)(\Vert 1-itx\Vert-1)=0$.

The question I am having trouble with is the following: Let $A$ be a C$^*$-algebra. Show that an element $x$ of $A$ is self-adjoint iff $\lim_{t\to 0}(1/t)(\Vert 1-itx\Vert-1)=0$. (Hint: If $h\in A$ is self-adjoint, then $\exp(ith)=1+ith+o(t)$ is…
Bill
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What is a support projection in a $C^*$-algebra?

Let $A$ a $C^*$-algebra and consider $a\in A$ self adjoint and $ax=xa$ for all $x\in A$. I want to know: -what is the support projection of $a$? -what is the definition of the support projection of an element in a $C^*$-algebra in genereal? I used…
asaed
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positive elements in the unitization of a $C^*$-algebra

Let $A$ be a $C^*$-algebra and consider it's unitization $A_1$ whose underlying vector space is the direct sum $A\oplus \mathbb{C}$. I want to know how does positive elements in $A_1$ look like. I tried to find out it with the definition (maybe an…
tau
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countable unitary element in a separable C*-algebra

How do I show that the set of unitary equivalence classes of projections is countable in a unital separable $C^*$-algebra? So I tried to show that the set of unitary elements in $C^*$-algebra is countable, but it was not successful. Thanks.
Andrew39
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Irreducible representations of $C(T,B(X))$

Let $T$ be a compact topological space, $X$ a finite-dimensional Hilbert space, $B(X)$ the algebra of operators in $X$, and $C(T,B(X))$ the $C^*$-algebra of continuous maps from $T$ into $B(X)$ (with the poinwise algebraic operations and the uniform…
Sergei Akbarov
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non-degenerate representation of a C*-algebra A is a direct sum of cyclic representations of A.

I am studying chapter 5 of the book Murphy. In the proof of Theorem 5.1.3 is at the bottom of my many questions. I'm Thanks for help in understanding prove. Thank advance. Theorem 5.1.3 Let $(H,\varphi)$ be a non-degenerate representation of a…
reza
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Unitaries $u$ span $A$ linearly?

I can't understand this paragraph in my book: If $a$ is a self-adjoint element of the closed unit ball of a unital $C^\ast$-algebra $A$ then $1-a^2$ is positive and $u=a + i\sqrt{1-a^2}$ and $v = a - i\sqrt{1-a^2}$ are unitaries such that $a =…
user167889
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How can I argue that this is an isomorphism?

Let $A$ be a unital $C^\ast$ algebra and let $B$ be a (not necessarily unital) $C^\ast$-subalgebra such that $B \oplus \mathbb C = A$. I want to argue that the map $\varphi : \widetilde{B} \to A$, $(b,\lambda) \mapsto b + \lambda$ is an…
user167889
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Choosing a subalgebra that separates points

Recall the (complex) Stone Weierstrass theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the $\ast$-algebra of continuous maps $X \to \mathbb C$. Then any $\ast$-subalgebra of $C(X)$ is dense in $C(X)$ if and only of $A$ separates…
user167889
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$\ell^p$ with pointwise multiplication -- example of $C^\ast$ algebra?

I was trying to think of some examples of $C^\ast$-algebras and I think $\ell^p$ with pointwise multiplication would be a good example. My reasoning is that if $a_n, b_n$ are in $\ell^p$ then eventually $|a_n b_n| \le |a_n|$ so this is closed with…
user167889
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Is $\|T^2\|=\|T S\|$

Let $A$ be a C$^\ast$-algebra and let $S,T: A \to A$ be bounded linear operators such that $\|T\|=\|S\|$. Is it true that $\|T^2\| = \|ST\|=\|TS\|=\|S^2\|$? I believe not but if not I don't understand why the last equation holds here: If $a \in A$…
user167889
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Non-isomorphic coarsely equivalent spaces

I've been reading about uniform Roe algebras lately and in particular the rigidity problem. I know that in general $C_u^*(X)$ and $C_u^*(Y)$ being *-isomorphic doesn't necessrily imply that $X$ and $Y$ are isomorphic as coarse spaces, but rather…
ham_ham01
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Positive part of an element of $C_0(X)$

I'm currently working on the Isem24 $C^*$-Algebras notes and there's a question I cannot answer. Let $A$ be a commutative $C^*$-Algebra, let be $x$ and $h$ two self-adjoint, positive elements of $A$ such that $h \geq x$. Let's denote by $x_+$ and…
ZirKeNx7
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States given by Dirac measure

Let $G$ be an infinite cyclic group. Then we have $C^*(G)\cong C^*(\mathbb{Z})\cong C(S^1)$. Now let $\delta_{x_0}$ be the state corresponding to the Dirac measure at $x_0 \in S^1$ given by $\delta_{x_0}(f)=f(x_0)$ which is evaluation at $x_0$. How…
budi
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