# 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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### Involution and Gelfand Transform Properties

Let $\mathcal{B}$ be a commutative unital Banach algebra, and let for each $x\in\mathcal{B}$ $\hat{x}$ be the Gelfand transform. I assume that $\mathcal{B}$ has an involution *. I want to show that: ...
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### Universal $C^*$ algebras

It is known that the $C^*$-algebra $\mathcal U$ generated by bilateral shift $\ell^2 (\mathbb Z) \ni e_k \mapsto e_{k+1} \in \ell ^2(\mathbb Z)$, is a universal $C^*$ algebra generated by unitary: for ...
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### about representations of a simple $C^*$-algebra

We know that every simple $C^*$-algebra is primitive, say it has a faithful non-zero irreducible representation. The converse is not necessarily true. An counterexample is just the $B(H)$ when $H$ is ...
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### prove that elements in $K_1(A)$ coincide

Let $A$ be a unital $C^*$-algebra $u\in A$ unitary and $s\in A$ isometry. I already proved that $sus^*+(1-ss^*)$ is an unitary. Why is $[u]_1=[sus^*+(1-ss^*)]_1\in K_1(A)$? Basic definitions: ...
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### Equivalence of some characterization of pure states

I'm looking for a reference or a proof for these well-known facts in $C^*$-algebras theory for which, however, I havent found any clearly written proof of the same type of ones I will sketch. ...
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### Compute the positive part of $K_0(A)$ where $A$ is a simple AF algebra

I'm trying to understand the following example from my lecture notes: Define $A_n=M_{F_n}(\Bbb{C})\oplus M_{F_{n+1}}(\Bbb{C})$ where $F_n$ defined by $F_1=1, \ F_2=2, \ F_{n+2}=F_{n}+F_{n+1}$, i.e., ...
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### homology theory for $C^*$-algebras, map is natural wrt morphisms of short ecact sequences

I want to assure me if I understand the part of the exactness axiom that "$\delta$ is natural" corretly and if not, then my question is: what does it mean? The setting is the following definition: ...
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### Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra for fixed n). Generators of $A_t$ depend continuously ...
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### Frechet derivative of square root on positive elements in some $C^*$-algebra

Let $A$ - is some unital $C^*$ algebra, and $P$ is set of all strictly positive elements in $A$. We can define map $\sqrt{?} : P \to A$ which takes positive element and returns its (unique) strictly ...
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### In a C*-algebra, pure states which share the same kernel are equal

I'm reading C*-Algebras by Jacques Dixmier. And in the proof of 2.9.5, it says Let $A$ be a C*-algebra. If $f$ and $f'$ are two pure states which have the same kernel, then $f=f'$. It should ...
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### Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $\phi : A \to B$ we are to prove these two ...
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### Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
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### Non-commutaive Gelfand-Naimark theorem and dimension of Hilbert space

It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...
### $A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well
Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...