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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).

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1answer
67 views

7 doubts about the von Neumann algebra [on hold]

A von Neumann algebra, or $W^*$-algebra, is a $*$-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type ...
2
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1answer
42 views

Normal u.c.p extension of Schur-multiplier

I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all. Let $\Gamma$ be a discrete group and ...
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1answer
25 views

Finding the Spectrum of an element of $\ell^\infty$

I have done Q.1.2.1 already and it is quite clear.But I am not sure about the next one. How does the closure come into the picture. I have a feeling that it should be $f(S)$ only. Am I missing ...
3
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1answer
60 views

Is a Rational Rotation Algebra a Cutdown of a Matrix Algebra?

Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2\pi i \...
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1answer
41 views

Verapoulous Algebra $C(K) \mathbin{\hat\otimes} C(L)$ is a subalgebra of $C(K\times L)$?

Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. ...
2
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1answer
34 views

Almost having invariant vectors vs having almost invariant vectors?

Let $\Gamma$ be a discrete and countable Group and let $\pi:\Gamma\to \mathcal{B(H)}$ be a unitary representation. We say that $\pi$ almost has invariant vectors if for every compact (=finite) subset ...
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1answer
24 views

Self-adjoint element of $C^*$-algebra has real spectrum

I'm trying to understand the proof in $\S$3.9 on p.23 of these notes: http://strung.me/karen/CStarIntroDraft.pdf. The argument is as follows: Let $A$ be a unital $C^*$-algebra, let $a \in A$ be self-...
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1answer
30 views

C*-algebra without finite-dimensional representations is simple?

Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite ...
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1answer
13 views

C* algebra exact sequences and ideals

if you have C* algebras $A,B$ and $C$ and $\exists$ a short exact sequence as follows $0\rightarrow A\rightarrow B \rightarrow C \rightarrow 0 $ where the functions are $\phi$ and $\psi$ respectively, ...
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1answer
31 views

simple nuclear $C^*$ algebra [closed]

Does there exist an infinite dimensional simple nuclear $C^*$ algebra which admits a tracial state?
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2answers
41 views

Completely positive map is $*$-homomorphism

Suppose $A$ is a $C^*$ algebra and we have a completely positive contractive map $f \colon A\rightarrow B(H)$ such that $sup_{a,b \in A}\lVert f(ab)-f(a)f(b)\rVert =0$. Can we conclude that $f$ is a $*...
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1answer
36 views

strict topology on multiplier algebras

Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...
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1answer
61 views

Finding a Unitary to Implement the action in a Minimal Dynamical System

Let $X$ be an infinite compact Hausdorff space and let $\sigma\colon X\to X$ be a minimal homeomorphism thereof. Then $\sigma$ gives rise to an automorphism $\sigma'$ of $C(X)$ defined by $\sigma'(f):=...
4
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1answer
58 views

Equivalent condition for amenability via positive functions in $\ell^p(\Gamma)$

Let $\Gamma$ be a countable discrete group and let $C_\lambda^*(\Gamma)$ denote the corresponding reduced group $C^*$-algebra. In [Theorem 2.6.8., BO08] the authors prove the following equivalence: (...
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0answers
29 views

maximal ideal of a $C^*$ algebra

If $A$ is a non-unital $C^*$ algebra,does the maximal ideal of $A$ always exist? If $I$ is an essential ideal of $A$,can it be a maximal ideal?Is there a relashionship between maximal ideals and ...
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1answer
55 views

Weakly dense C*-algebra in a commutative von Neumann algebra and order convergence

Let $H$ be a Hilbert space and $\mathscr{A}$ a commutative norm-closed unital $*$-subalgebra of $\mathcal{B}(H)$. Let $\mathscr{M}$ be the weak operator closure of $\mathscr{A}$. Question: For given ...
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1answer
25 views

Question regarding $C^*$ algebra generated by a product

Let $V=M_{1,\infty}$,the row Hilbert space. Suppose $W$ denotes the $C^*-$ algebra generated by $V^*V=\{x^*y : x,y \in V \}$ Is it true that $W= K(l_2)$, space of compact operators on $l_2$? I can ...
2
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1answer
27 views

UHF-algebra tracial state is faithful?

It is known that a unital UHF-algebra has a unique tracial state, it is true that it is true that this trace is normal and faithful? I am particularly interested in the universal UHF-algebra, i.e. the ...
3
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1answer
37 views

Norms on Tensor Product of $C^*$- algebras

Suppose $A$ and $B$ are two $C^*$-algebras. On the algebraic tensor product $A\otimes B$ we can define the maximal and minimal tensor norms which makes $A\otimes B$ a $C^*$- algebra. what are other ...
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1answer
32 views

Pure states of $M_n(\Bbb C)$

I saw a reference book,there is a statement:the pure states of $M_n(\Bbb C)$ are the rank 1 projections of $M_n(\Bbb C)$. By definition of states,they should be the positive linear functional of norm ...
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1answer
32 views

Isn't all C$^*$-algebra have this property?

A$\ $ C$^*\!$-algebra $A$ has property $*$ if for each suquence $(x_n)_n$: $$\text{if} \ x_n\to 0(\text{weakly})\ \ \Longrightarrow \ \ \ \ yx_n\to 0 (\text{in norm}) \ \forall y\! \in \! \! A ...
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1answer
34 views

finite dimensional $C^*$ subalgebra

Given any $C^*$-algebra $A$ (not necessarily unital),can we construct a nonzero finite dimensional $C^*$-subalgebra of $A$?
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1answer
35 views

What is the definition of identity bimodule in the context of von Neumann or $\ast$-algebras?

I will just repeat the question in the title: What is the definition of identity bimodule in the context of von Neumann or $\ast$-algebras? I know what a bimodule is but I never heard up to ...
2
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1answer
29 views

About “names” of von Neuman algebra morphisms

I have actually a basic quastion about maps between von Neumann algebras. If I have a map $f:N \to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal ...
2
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1answer
35 views

Is $\|a+b\|\leq\sqrt{\|a\|^2+\|b\|^2}+2\varepsilon$ when $\|ae-a\|\leq\varepsilon$ and $\|be\|\leq\varepsilon$ in a $C^*$-algebra?

Let $\mathcal{A}$ be a $C^*$-algebra with $a,b,e\in\mathcal{A}$ such that $e\geq0$ and $\|e\|\leq1$. If $\|ae-a\|\leq\varepsilon$ and $\|be\|\leq\varepsilon$, then is $\|a+b\|\leq\sqrt{\|a\|^2+\|b\|^2}...
1
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1answer
39 views

Ideal in a $C^*$ algebra [closed]

Suppose $A$ is a non-unital $C^*$ algebra, $a\in A$, $I$ is the ideal generated by $a$. In the unital case, $a=1a1\in AaA$. But in the non-unital case, how to show that $a\in A$, can $a$ be ...
3
votes
1answer
50 views

Can we bound $\|a+b\|$ if $\|ae-a\|$ and $\|be\|$ are small, in a $C^*$-algebra?

Let $A$ be a $C^*$-algebra with $a,b,e\in A$ such that $e\geq0$ and $\|e\|\leq1$. If $\|ae-a\|\leq\varepsilon$ and $\|be\|\leq\varepsilon$, then is $\|a+b\|\leq\max\{\|a\|,\|b\|\}+2\varepsilon$? The ...
3
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1answer
51 views

Gelfand map is topologically injective

Let $A$ be a commutative Banach algebra and let $$G: A \rightarrow C_{0}(\text{Spec}(A))$$ be a topologically injective map. Recall that the map $T: X \rightarrow Y$ is called topologically ...
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0answers
29 views

What does the enveloping von Neumann algebra functor do to locally compact Hausdorff spaces?

Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is ...
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1answer
27 views

infinite dimensional representation of a $C^*$ algebra [closed]

When $H$ is separable infinite dimensional,$K(H)$ has no finite representation.Do there exist other non-unital $C^*$ algebras such that their representations cannot be finite dimensional?
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1answer
37 views

unitization of an essential ideal [closed]

Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $\tilde{I}=I\bigoplus \Bbb C$ is an essential ideal of unitization $B\bigoplus \Bbb C$...
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2answers
36 views

an element in $\prod_n M_n(\Bbb C)$

I want to find an element $x=(x_n)\in \prod_nM_n(\Bbb C)$ such that $\lim \operatorname{tr}_n(x_n)=0$ but $\lim \operatorname{tr}_n(x_n^*x_n)\not \to 0$,where $tr$ is the unique tracial state on $M_n(\...
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1answer
28 views

Inclusion of multiplier algebras [closed]

If $B$ is a $C^*$-subalgebra of $A$, i.e there exists an inclusion map $\phi\colon B \rightarrow A$, can we conclude that there exists a $*$-homomorphism beween the multiplier algebras $\bar{\phi} \...
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1answer
59 views

Type III Von Neumann algebras and spectra of the modular operators.

I´m studying the paper of Fredenhagen. There he said that he would prove that the algebra of local observables under certain conditions is of type III, by showing that all modular operators satisfy $...
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1answer
34 views

Essential ideals in sums of matrix algebras [closed]

Given the $C^*$-algebra $A=\prod_{n}M_n(\Bbb C)$, how many essential ideals has $A$? Is there a unique one ?
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2answers
66 views

Example of norm separable c-star algebras [closed]

I want to know enough examples of norm separable $C^{*}$-algebras which are neither finite dimensional nor commutative.
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1answer
33 views

examples of non-unital commutative $C^*$-algebras

I know that all the non-unital commutative $C^*$ algebras are isomorphic to $C_0(\Omega)$,where $\Omega$ is a locally compact space. Can anyone show me some common non-unital commutative examples.I ...
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1answer
36 views

non isomorphic finite dimensional $C*$ algebras

How many non isomorphic finite dimensional $C^*$ algebras if the dimensions without a bound? Is it countable or uncountable?
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1answer
121 views

If $\ $ $yx_n\to 0 $ for all $y$ in the C$^*$-algebra A, Is it true that $x_n$ is weakly convergent to $0$?

$A$ is a C$^*\! $-algebra and $(x_n)_{n\in \mathbb{N}} \subseteq A $. If $\ $ $yx_n\to 0 $ for all $y\in A$, Is it true that $x_n$ is weakly convergent to $0$ ? For unitals this is trivial ...
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1answer
67 views

Rigorous Computation of the Spectrum of a Certain C$^{*}$-algebra

Let $$ A=\left\{f\in C([0,1],M_{2}(\mathbb{C})):f(1)=\begin{pmatrix}\xi & 0 \\ 0 & \lambda\end{pmatrix} \text{ for some }\xi,\lambda \text{ in } \mathbb{C}\right\}. $$ In his Operator ...
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1answer
26 views

spectrum of $C^*$ algebras

When $A=\bigoplus B(\Bbb C^n)$ ($c_0$ direct sum),how to compute the spectrum of $A$ ?What about the conclusion If we replace the $\ell ^\infty $ direct sum with $c_0$ direct sum?
3
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2answers
37 views

Can distinct elements of a $C^*$-algebra be separated by a maximal left ideal?

Let $A$ be a $C^*$-algebra, and let $f\neq g\in A$. Does there exist a maximal ideal $J\trianglelefteq A$ with $f+J\neq g+J$? I'm particularly interested in the case of $A=B(\mathcal H)$, and why ...
0
votes
1answer
27 views

approximate identity element

Let $I\subset \prod_n B(H_{m_n})$ be a separable $C^*$ algebra,where $\prod B(H_{m_n})$ denotes the $\ell ^{\infty}$direct sum of $B(H_n)$ and $dim(H_{m_n})<\infty$. We suppose that there exits a ...
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1answer
25 views

image of some ideal under the quotient map

! I am still confused about the range of $\sigma(I_{\omega})$.Since $\|x_n\|_2\to 0,$we have $\|x_n\|\to 0$,$\sigma(I_{\omega})=0,$then $J=0$,it is trivial. Is my understanding correct?
1
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1answer
26 views

identify GNS construction as asubalgebra of $R^{\omega}$

I have two questions In lemma6.5.5. 1.why can we identify the GNS representation of $\prod M_{k(n)}(\Bbb C)/\bigoplus M_{k(n)}(\Bbb C)$ with respect to $\tau _{\omega}$ with a subalgebra of $R^{\...
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1answer
32 views

intersection of sets corresponding to free ultrafiler

If $\omega \in \beta \Bbb N\setminus \Bbb N$,we define $S_{\omega}=\{(x_n) \in \prod M_n(\Bbb C):lim_{n \to \omega}tr_n(x_n)=0\}$ Is the intersection $\cap_{\omega \in \beta \Bbb N \setminus \Bbb N}...
0
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1answer
21 views

strictly positive element vs positive definite matrix

If $A=\prod_{n=1}^{\infty}M_{k(n)}(\mathbb{C})$,$x=(x_1,\cdots,x_n,\cdots)$ is strictly positive in $A$,does this mean that each $x_n\in M_{k(n)}\mathbb{C}$ is a positive definite matrix?
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1answer
22 views

tracial state of a orthogonal projection

Suppose $A\in M_n(\mathbb{C})$,$A$ has eigenvalues$\lambda_1,\cdots,\lambda_n$,$P$ is the orthogonal projection from $\mathbb{C}^n$ onto the span of eigenvectors associated with $\lambda_1,\cdots,\...
0
votes
1answer
13 views

central projections

Suppose $A$ is a separable $C^*$ algebra(not necessarily unital),and let $I$ be the ideal generated by central projections in $A$,does there exist nonzero pairwise projections in the quotient $A/I$?
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1answer
17 views

strictly positive elements under a nonzeo $*$ homomorphism

Suppose $A$ is a separable $C^*$ algebra,x is a strictly positive element in $A$,$\phi:A\rightarrow B$ is a nonzero $*$ homomorphism,is $\phi(x)$ also strictly positive in$B$?