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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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unitization of a unital $C^*$-algebra

I have a little question about unitization of a $C^*$-algebra. If $A$ is a non-unital $C^*$-algebra, set $A_1=A\oplus\mathbb{C}$ as vector spaces and define a multiplication, involution and a norm in a usual way, $$(a,\lambda)\cdot(b,\mu)=(ab+\mu…
google-hupf
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Example of a *-homomorphism that is faithful on a dense *-subalgebra, but not everwhere

Let $A,B$ be C*-algebras and let $\varphi: A \to B$ be a $*$-homomorphism. Suppose that $\ker( \varphi) \cap D = \{0\}$ where $D$ is a dense $*$-subalgebra of $A$. Does it follow that $\varphi$ is injective? I'm pretty sure the answer is "no". At…
Mike F
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Invertibility in subalgebra

I have some trouble proving the following statement: Let $A$ be a self-adjoint element of a $C^*$-algebra $\mathcal{B}$ and let $\mathcal{A}$ denote the unital subalgebra of $\mathcal{B}$ that is generated by 1 and $A$. If $A$ is invertible in…
ranja
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How to think of functional calculus for C*-algebras

I was wondering if it were possible to think about functional calculus for C*-algebras in terms of actual "extension of functions". More specifically, let $A$ be a C*-algebra, $x\in A$ a normal element. Then Gelfand duality gives a *-morphism…
Aaron Wild
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Does antisymmetricity of $\preceq$ fail in general C*-algebras?

For projections $p,q$, write $p\preceq q$ if there is $u$ such that $u^*u=p$ and $uu^*\leq q$. Write $p\sim q$ if $u^*u=p$ and $uu^*=q$. It is a theorem that $\preceq $ is a partial order in a Von Neumann algebra. But does antisymmetricity hold in…
Sui
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$*$-homomorphism between matrix algebras

Let $\theta:M_3(\mathbb{C}) \to M_7(\mathbb{C})$ be a $*$-homomorphism. Since matrix algebras over a field are simple, we know that $\theta$ must be injective. After reading this I conclude that, up to unitary equivalence, $$ \theta(A) =…
ragrigg
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Detail in the algebraic proof of the polar decomposition

Most of the time one finds the proof for bounded operators on a Hilbert space, but Sakai in his book "C*-algebras and W*-agebras" gives a purely algbraic one, (Thm 1.12.1 p.27-28, partially at his link): Instead of taking directly $\sqrt{a^*a}$ as…
Noix07
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If $\sum_{j=1}^n v_j^*v_j=1_A=\sum_{j=1}^n v_jv_j^*$ in a C* algebra A, then $\sum_{j=1}^n v_j$ is unitary.

Let $v_1,v_2, \cdot v_n$ be partial isometries in a unital $C^*$-algebra $A$ and suppose that $$\sum_{j=1}^n v_j^*v_j=1_A=\sum_{j=1}^n v_jv_j*$$ Show that $\sum_{j=1}^n v_j$ is unitary. This a exercise from the book "An Introduction to K-theory…
Arindam
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What exactly is meant by a semicontinuous, semifinite trace on a C* algebra?

I see phrases like "$\tau$ is a semicontinuous, semifinite trace on a C* algebra $A$" thrown around a lot in papers without any extra qualification. So, I gather this terminology is standard? I've never been exactly sure what it means though. I am…
Mike F
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product $ab$ of postive elements $a,b$ is again positive, if $ab=ba$.

Let $A$ be a $C^*$-algebra, $a,b\in A$ positive elements (this means self-adjoint and the spectrum lies in $[0,\infty)$). In general, $ab$ isn't positive, for example consider the matrices $a=\begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix},…
google-hupf
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Commutativity criteria for $C^*$-algebras

If for any $x, y\in A$ a $C^*$-algebra, $0\leq x\leq y\implies x^2\leq y^2$, then it is true that A is commutative ? It is easy to show that the implication $0\leq x\leq y\implies x^2\leq y^2$ is not true in general, for example in…
Patissot
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Equivalence between the GNS representation of two different positive linear functionals

Let $\varphi $ be a positive linear functional on $C^*$-algebra $A$ and let $(\pi _{\varphi},H_\varphi ,\xi)$ be the associated GNS representation. Let $\psi \in A_+^*$. Show that the two next propositions are equivalent : i) there exists $\eta \in…
Patissot
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Please can you check my proof of the spectral mapping theorem?

I would like someone to help me check if I understand this proof and for this reason I would like to give the proof here in my own words. The statement I am proving is this: Let $A$ be a unital $C^\ast$-algebra and $a \in A$ normal. Then for all…
user167889
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Simplicity of Cuntz algebra

It is well known that the Cuntz algebra $\mathcal{O}_N$ (for fixed $N$) is simple. Is there any easy way to exhibit it (as a Banach algebra only) as a quotient of a Banach algebra modulo some maximal ideal? Of course, I am not interested in the…
Batykaf
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Why do we call it dual C*-algebra?

In the book: Higson N, Roe J. Analytic K-homology[M]. Oxford University Press, 2000. The Definition 5.1 is about dual of a C*-algebra, In simple term, embedding a C*-algebra a into a operator algebra B(H) of Hilbert space H, the dual of C*-algebra…
Strongart
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