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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Ordering: Definition

This was a real question! Given a unital C*-algebra $1\in\mathcal{A}$. For $A\in\mathcal{A}$ denote its spectrum by $\sigma(A)$. Consider the selfadjoints: ...
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1answer
17 views

why is the direct sum of irreducible representations of a ^$C^*$-algebra faithful?

Let $A$ be a C$^*$-algebra. Let $\mathrm{Irr}(A)=\{[\pi]: \pi \text{ is an irreducible representation of } A\}$ and $\rho\in [\pi]$ if there is an unitary operator $V:H_\pi \to H_\rho$ such that ...
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1answer
35 views

ker-hull-topology on $Irr(A)$ is the discrete topology ($A$ is a C$^*$-subalgebra of $K(H)$)

Let $A$ be a C$^*$-algebra. Let $Irr(A)=\{[\pi]: \pi$ is an irreducible representation of A} and $\rho\in [\pi]$ if there is an unitary operator $V:H_{\pi}\to H_{\rho}$ such that $V\pi(a)=\rho(a)V$ ...
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$a,b\in A$ are selfadjoint elements of $C^*$-algebras, such that $a\le b$, why is $\|a\|\le \|b\|$

Let $A$ be a unital $C^\ast$-algebra with unit $1_A$. a) Why is $a\le \|a\|1_A$ for all selfadjoint $a\in A$ and b) If $a,b\in A$ are selfadjoint such that $a\le b$, why is $\|a\|\le \|b\|$? I ...
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1answer
22 views

Is this condition on linear functionals equivalent to boundedness?

Let $T : C_c(X) \rightarrow \mathbb{R}$ be a linear functional satisfying $$\sup \{|T(f)| :\ f \in C_c(X), |f| \leq 1, \text{Supp}(f) \subset K\} < \infty$$ for each compact $K \subset X$. Is ...
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1answer
52 views

C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators? Equivalently: Let ...
2
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1answer
107 views

Why the disk algebra is not a C* algebra.

I'm trying to figure out why the set of bounded analytic functions on the unit disk, A(D), is not a C* algebra. The norm is the sup norm and the involution is $f(z) \to \overline{f(\bar z)}$. I want ...
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1answer
300 views

Does category theory help in operator algebras?

I'm currently studying the basics of Banach and $C^*$-algebras. Almost all the proofs i've seen so far are very simple but some of them are extremely tricky (in my opinion). This tricky interplay ...
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0answers
31 views

Almost periodic compactification with the Gelfand-Naimark theorem

Could anyone please help me with a bibliographic reference presenting the almost periodic compactification of a topological group with the aid of the Gelfand-Naimark theorem? Rudin in "Fourier ...
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0answers
59 views

Operator realizations of algebras

I want to realize the algebra $A_q(\tilde{S}^{n-1})$ as introduced in the acticle of Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf) as bounded operators on a Hilbert space $H$. Can ...
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2answers
89 views

Prove that $A\geq I$ implies that $A$ is invertible.

Here's the question: Let $A$ be a positive operator on a (possibly infinite dimensional) Hilbert space. Let $I$ denote the identity operator. Suppose that $A \geq I$, which is to say that $A - ...
2
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1answer
52 views

universal $C^*$-algebra isomorphic to $K(l^2(I))$

Let $I\neq \emptyset$ be a set, $X=\{e_{ij}:i,j\in I\}$ with the relations $$R=\{e_{ij}e_{kl}=\begin{cases}e_{il}, & \text{if }j=k,\\ 0, & \text{ else} \end{cases},\;\; e_{ij}^*=e_{ji}:\; ...
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1answer
52 views

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 $A$ ...
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1answer
50 views

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 ...
2
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1answer
53 views

Unit of a purely infinite, simple C*-algebra

Suppose that we have a purely infinite, simple C*-algebra with unit $1$. Can we find two projections $p,q$ both equivalent to the identity such that $1=p+q$ and $pq=0$? Well, there are two ...
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1answer
14 views

$u_n=a( \frac{1}{n}+a)^{-1} $ is an approximate unit of $A$ if $a$ is strictly positive

Let $A$ be a $C^\ast$-algebra, $a\in A$ such that $\varphi(a)>0$ for all states $\varphi$ on $A$. How to prove, that $(u_n)_{n\in\mathbb{N}}$ with $u_n=a( \frac{1}{n}+a)^{-1} $ is an approximate ...
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1answer
24 views

strictly positive element in a $C^*$-algebra. Where is the mistake in the proof?

Let $a\in A$ a stricly positive element (this means: for all states $\varphi$ of $A$ is $\varphi(a)>0$), let $u_n=a(\frac{1}{n}+a)^{-1}$, $n\in\mathbb{N}$ . Claim: for all $b\in A$, for all states ...
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2answers
33 views

A question about corners of $C^\ast$-algebras

Let $\mathcal{A}$ be a $C^\ast$-algebra, $p\in M_n(\mathcal{A})$ a projection, is there a $k\in\mathbb{N}$ such that $pM_n(\mathcal{A})p\cong M_k(\mathcal{A})$ ? Thanks a lot!
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1answer
26 views

C*-algebra generated by a non invertible normal element

Let $A$ be a C*-algebra and $x\in A$ be a non-invertible normal element. By functional calculus, we know $$C^*(x,1)\simeq C(\sigma(x))$$ Where $\sigma(x)$ means the spectrum of $x$. I need to ...
3
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1answer
79 views

When can $*$-algebras be turned into $C^*$-algebas?

Let $A$ be a (not necessarily unital) complex $*$-algebra, i.e. an algebra over $\mathbb{C}$ together with an involution $*: A \to A$. There exists at most one norm on $A$ turning $A$ into a ...
2
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1answer
36 views

Matrices over the Cuntz algebra

Consider the Cuntz algebra $O_2$. Is it true that $M_2(O_2)$ is isomorphic to $O_2$? I was trying to show that is impossible but now I am not sure.
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1answer
33 views

prove that a map is well-defined and closed ideals in $C_0(X)$

Let $X$ a locally compact Hausdorff space, $Y\subseteq X$ a closed subset and $I_Y:=\{f\in C_0(X): f_{|Y}=0\}$. a)Let $U=X\setminus Y$. Prove that $I_Y\cong C_0(U)$. b)For every closed ideal $I$ in ...
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0answers
29 views

Unital amenable Banach algebras which is a proper two sided ideal in its second dual

I need some examples of "unital amenable Banach algebras which is a proper two sided ideal in its second dual".
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1answer
41 views

Partial isometries

Given a unital $C^*$-algebra A and partial isometries $w_1, \cdots, w_n$ such that $\sum_{i = 1}^{n}w_iw_i^* = 1$ and $w_i^*w_j = 0$ if $i \neq j$ then is it true that $w_i$ is an isometry?
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1answer
42 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
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1answer
29 views

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 ...
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40 views

approximate unit of $K(H)$- ordering on $K(H)$ and finite rank operators

Let $H$ be a complex Hilbert space with orthonormal basis $\{e_i:i\in I\}$ . Consider the $C^\ast$-algebra of the compact operators on $H$, $K(H)$. For a finite subset $F\subseteq I$, let $P_F$ be the ...
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1answer
33 views

Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
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1answer
79 views

Why is the image of a C*-Algebra complete?

I am currently working through the book by Bratteli and Robinson on C* and W* algebras, there is one point at the beginning of chapter 2.3 that is frustrating me. If we take *-morphism to be a ...
2
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1answer
99 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
3
votes
1answer
89 views

concept of the classification of $C^\ast$-algebras, introduction/overview

I don't have a specific mathematical problem at the moment but nevertheless I hope, my question is suitable for math.stackexchange. I'm interested in $C^\ast$-algebras and I would like to begin with ...
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1answer
29 views

Ordering: Normalized

Given a Hilbert space $1\in\mathcal{A}$. Denote selfadjoints: $$\mathcal{S}:=\{A\in\mathcal{A}:A=A^*\}$$ Introduce an order: $$A\leq A':\ \ \sigma(A'-A)\geq0$$ Regard a projection: $$P\neq0:\quad ...
2
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1answer
69 views

From Hopf Algebras to quantum groups

I start with self study about quantum groups. Until now I covered Hopf $*$-algebras and there representations (for example by the book of Klymik and Schmüdgen). Now I want to understand the step from ...
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1answer
42 views

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 $A$ ...
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2answers
31 views

How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
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1answer
53 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
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1answer
27 views

Why is $ab=ba=a^\ast b=ab^\ast=0$ (orthogonal elements in a $C^\ast$-algebra)?

Let $a,b$ be elements in a $C^\ast$-algebra $A$, such that $$a^\ast ab^\ast b=b^\ast ba^\ast a=0$$ $$a^\ast abb^\ast=bb^\ast a^\ast a=0$$ $$aa^\ast b^\ast b=b^\ast b aa^\ast =0$$ $$aa^\ast ...
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1answer
36 views

composition and strong limits of completely positive maps is completely positive

I have two claims about completely positive maps. Let $X$, $Y$, $Z$ be $C^\ast$-algebras. 1) Let $f:X\to Y$ and $g:Y\to Z$ be completely positive maps. I want to know, why $g\circ f$ is completely ...
2
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1answer
45 views

the $C^\ast$-algebra $M_n(A)$, understanding the $C^\ast$-norm on $M_n(A)$

Let $A$ be a $C^\ast$-algebra. I want to understand $M_n(A)$, the vector space of $n\times n$-matrices with entries in $A$, as a $C^\ast$-algebra. On $M_n(A)$ you can define an involution ...
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1answer
126 views

GNS construction and representations

I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ). In the proof of GNS construction theorem 1.12.4 page 50 there is something I ...
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1answer
33 views

C*-Algebra: Cyclic Elements

Given a locally compact Hausdorff space $\Omega$. Consider the C*-algebra: ...
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1answer
27 views

Is the identity in unital, simple, purely infinite $C^*$-algebra always infinite?

I'd like to prove that the identity, $I$, of a unital, simple, purely infinite $C^*$-algebra is always an infinite projection. What I'm hoping is that the following is true: If $p$ in $\mathfrak{A}$ ...
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1answer
63 views

Square root of a compact normal operator

Halmos expresses below problem in his book; Problem: If $A$ is a normal operator and if $A^n$ is compact for some positive integer $n$, then $A$ is compact. I have an example in my mind which I ...
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1answer
58 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
3
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0answers
28 views

Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
2
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2answers
53 views

Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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0answers
38 views

Uniqueness of c.p.c. order zero extensions

I have a question about a passage in the proof (on page 316) of proposition 3.2 in this paper http://wwwmath.uni-muenster.de/42/fileadmin/Einrichtungen/mjm/vol_2/mjm_vol_2_14.pdf. My question is ...
2
votes
3answers
44 views

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 ...
2
votes
2answers
56 views

Question about $C_0(X)$ is unital iff X is compact

I am sorry if this question is trivial I don't know much about $C^*-algebra$ and learning introduction about them from the following link notes about C* algebras. According to the notes given above ...
4
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1answer
86 views

Understanding the bidual of a $C^*$-algebra as a $C^*$-algebra

I have a lot of problems trying to understand the double dual of a $C^*$-algebra. Let $A$ be a $C^*$-algebra, I read that if you endow the bidual Banach space $A^{**}$ of $A$ with the weak-*topology, ...