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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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A sequence of points in the spectrum of a subhomogeneous C$^{*}$-algebra can converge to at most finitely many points

Let $A$ be a subhomogeneous C$^{*}$-algebra (i.e., there is a finite upper bound on the size of the irreducible representations of $A$). Let $\hat{A}$ denote its spectrum. I heard of a result that ...
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sum of ideals in $C^*$ algebra

Suppose $I_1$ is a maximal ideal in $C^*$ algebra $A$,$I_2$ is an ideal of $A$,then $I_1+I_2$ is an ideal of $A$,can we conclude that $I_2\subset I_1$? My thought:if there exists an element $x\in I_2$...
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28 views

Universal $C^*$ algebra subject to $ x^*x+y^*y=xx^*+yy^*$

What is the precise description for the universal unital $C^*$ algebra generated by two elements $x,y$ subject to relation $$ x^*x+y^*y=xx^*+yy^*$$
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positive functionals on the full group C*-algebra

It it true that every positive linear functional on the full group C*-algebra is Completely positive? I am reading Brown and Ozawa's book and they seem to use this at some point, yet I'm not sure how ...
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31 views

Does the spectrum at a point vary continuously in this case?

Let $A$ be a C$^{*}$-algebra. Let $\hat{A}$ denote the set of all irreducible representations of $A$. Suppose $\pi\in\hat{A}$ has the following property: for all $a\in A$, the map from $\hat{A}\to\...
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31 views

Maximal modular ideal

I met with some troubles with the two concepts:maximal ideal and maximal modular ideal in $C^*$ algebras. If $I$ is a maximal modular ideal in a $C^*$ algebra $A$,does this imply that for any other ...
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1answer
20 views

Is multiplication by continous functions in $B(L^2)$ norm closed?

If you consider the multiplication by a continuous function as a subset of $B(L^2(\mathbb{T}))$, is this norm closed in operator norm? i.e. if B is an operator in $B(L^2(\mathbb{T}))$ with $||B-M_{\...
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31 views

Spectrum of the difference of almost commuting elements

Suppose $A$ is a C*-algebra. Suppose we have an approximate unit $(u_\lambda)$ then one knows that $|| u_\lambda a-au_\lambda||\rightarrow 0$ in particular $u_\lambda$ almost commutes with $a$ can ...
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1answer
23 views

find a smallest nuclear $C^*$ algebra containing set S [closed]

Suppose $S$ is a set ,can we find a smallest nuclear $C^*$ algebra containing $S$
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28 views

Bounded linear functional on a $C^*$ algebra

I want to prove the following statement: Let $\mathcal{A}$ be a unital $C^*$ algebra with unit $e_\mathcal{A}$ and let $\varphi \in \mathcal{A}^*$ be a bounded linear functional in $\mathcal{A}$ ...
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1answer
30 views

stably finite $C^*$ algebras

I saw a conlusion:If $A$ is simple $C^*$ algebra,then $A\otimes \Bbb K$ contains no infinite projections? If $A$ is simple, $A\otimes \Bbb K $ contains no infinite projections,can we conclude that $A$...
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Hyperfinite II$_1$-factor

1.What is the definition of hyperfinite II$_1$-factor? Can anyone show me concrete examples? 2.If $R$ is a hyperfinite II$_1$-factor ,we can define the ultraproduct of $R^{\omega}=l^\infty(R)/I_{\...
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Quick question regarding C*-algebras/W*-algebras

I'm reading about C*-algebras and W*-algebras, and I want to know the differences between the two,in other words properties you can find in W*-algebras that aren't in C*-algebras and the other way ...
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1answer
47 views

trace on quotient of a $C^*$ algebra

Does there exist a $C^*$-algebra $A$ such that $A$ has a faithful tracial state,but the quotient of $A$ has no tracial states(there exists a nontrivial ideal $I$ of $A$ such that $A/I$ has no traces )...
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different essential ideals

Does there exist a $C^*$ algebra which has more than one essential ideal?If there exists such a $C^*$ algebra ,suppose $I,J$ are two different essential ideals,$I\subset J$,can we compare the ...
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1answer
37 views

Suspension of Cuntz algebra is traceless

I saw a conclusion in a reference book: the suspension of the Cuntz algebra $C_0((0,1))\otimes \mathcal O_2$ has no tracial states. My thought: there are many tracial states on $C_0((0,1))$. We take ...
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1answer
34 views

what's spectral axiom

I encounter a proposition in an article: For any $0\leq x\leq 1$ in a C*-algebra enjoying the spectral axiom, there are projections $(e_n)$ such that $$x=\sum_{n=1}^{\infty}\frac{1}{2^n}e_n.$$ ...
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Composition of an analytic function F with an analytic function f in a C* algebra A such that F(f) in A?

I have a C* algebra A, a function $f(x)\in A$ and an analytic function $F:\mathbb{C}\rightarrow\mathbb{C}$. I would like to know what condition must have $F$ such that $F(f)\in A$. The idea is the ...
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34 views

center of a $C^*$ algebra

Suppose the center of $C^*$ algebra $A$ is 0,does this $C^*$ algebra constructed by simple non-unital $C^*$ algebras.To be more precise,$A$ can be only simple or direct sum of simple non-unital $C^*$ ...
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a question on GNS space

If a $C^*$ algebra have a faithful tracial state ,we can construct a representation $(\pi_{\tau},H_{\tau})$ of $A$.$H_{\tau}$ can be obtained as following: Let $N=\{a\in A,\tau(a^*a)=0\}$,then $H_{\...
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construct a representation of a $C^*$ algebra

If $A$ has a tracial state,we can construct a representation $(\pi,H)$ by the GNS theorem. My question is: If $A$ has no tracial states,how can we construct a representation of $A$?Do there exist ...
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32 views

Von Neumann algebra generated by a set

Suppose $A$ is a unital $C^*$ algebra,$\pi:A\to B(H)$ is a representation of $A$.Then the von Neumann algebra generated by $\pi(A)$ is equal to $\pi(A)^{"}$. Is the weak$*$ closure of $\pi(A)$ equal ...
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1answer
14 views

invariant subspace of a representation

Suppose $\pi:A \to B(H)$ is a representation of $A$ such that $\pi(A)K_1\subset K_,\pi(A)K_2\subset K_2$,where $H=K_1\oplus K_2$,can we conclude that there exist a projection $p\in \pi(A)^{'}$ such ...
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25 views

Decomposition of self adjoint elements by positive elements

Let $a \in A$ be a self adjoint element of a $C^*$ algebra. There exists positive elements $a_+, a_-$, such that $$a=a_+ - a_{-} $$ $$a_+a_-=a_-a_+=0$$ Is the statement true? This is ...
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Showing isomorphism of two $C^*$ algebras

It seems that quite a standard trick of showing two $C^*$ algebras are as follows: Let $A$ be a $C^*$ algebra $B$ another $C^*$ algebra. $A' \subseteq A$ be a subalgebra that is closed under $*$. (...
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1answer
20 views

Maximal abelian subalgebra in generated von Neumann algebra

Let $D \subseteq A$ be an abelian C*-subalgebra of the C*-algebra $A$ where $A \subseteq B(H)$ for some separable Hilbert space $H$. Assume that the von Neumann Algebra generated by $D$ is a maximal ...
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1answer
23 views

double commutant of the left representation of $C^*$ algebra

If $A$ is a unital $C^*$ algebra with a trace $\tau$,$\lambda:A\to B(L^2(A,\tau))$ is the left representation.where $L^2(A,\tau)$ is the GNS space.Let $\lambda(A)^{"}$ be the double commutant of $\...
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1answer
35 views

Injective $*$-homomorphism is isometric

I am aware there are other proofs of line of this statement. But I am interested in the argument outlined here on page 62-63 Corollary II.2.2.9 Let $A$ and $B$ be $C^*$ algebras, $\phi:A \...
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Universal enveloping von Neumann algebra of a separable $C^*$ algebra

Let $A$ be a separable $\mathrm{C}^*$-algebra and let $\pi_U$ be its universal representation. Denote by $M=\pi_U(A)''$ the universal enveloping von Neumann algebra of $A$ (which is isomorphic to $A^{*...
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Projection on the GNS subspace

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. If $\omega$ is a positive linear functional on $\mathcal{A}$, then we may perform the so-called GNS construction in order to obtain ha Hilbert space $\...
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1answer
64 views

Showing a C* algebra with certain properties has a minimal projection

I am trying to show the following which is stated in Exercise 10.11.10 of Blackadars book on K-theory for operator algebras. A unital, simple, nuclear, stably finite, infinite dimensional C*-algebra ...
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Hilbert C*-Modules: Inner *-Isomorphisms

I have got a very basic question, but it would simplify some things, so I hope this resolves in either the affirmative or maybe someone can provide an I guess simple non-example. Given pairs of ...
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Maximal tensor product of quotient C*-algebras

This question is from the book: C*-algebras and Finite-Dimensional Approximations by N.P.Brown and N. Ozawa Ex 13.3.5. Let C$_i$ (i=1,2) be C*-algebras with the LLP and J$_i$ be a closed two-...
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$f+g$ $*$-homomorphsim if and only if $im \, f \cdot im\, g = 0$

Let $f,g:A \rightarrow B$ be $*$-homomorphisms of $C^*$ algebras. Then $f+g$ is a $*$-homomorphism if and only if $im \, f \cdot im \, g =0$. How does this hold? My thoughts: We know that $f+g$ is ...
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1answer
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Inductive limit of unitization is unitization of inductive limit?

let $(V_n,f_n)$ be an inductive sequence with inductive limit $V$ in the category of non Unital $ C^* -$ algebras. Let $V_n^*$ denotes the unitization of $V_n$. Is it true that inductive limit of $V_n^...
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Inverse Limit in the category of $C^{\ast}$-algebras or operator spaces [duplicate]

Does the inverse limits (Projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces? I tried to search but could not find a proper reference. Any reference or comments about ...
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Behaviour of Maximal tenor product with inductive limits

I am trying to prove the following result: Let $(A_{i})_{i\in I}$ be an inductive system of non unital $C^{\ast}$-algebras with connecting homomorphisms $f_{ij}: A_{i} \to A_{j}$ and let's denote ...
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Any isometry in $\mathcal{L}(G)$ must be a unitary

Let $G$ be a countable group with neutral element $e$. Consider the Hilbert space $$\ell^2(G):=\left \{ x:G\to \mathbb{C}\mid \sum_{t\in G}|x(t)|^2<\infty \right \}$$ with inner product $\left \...
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1answer
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multiplier algebra of a non-unital $C^*$ algebra

Given any infinite dimensional unital $C^*$ algebra $A$,does there must exist a non-unital $C^*$ algebra $B$ such that the multiplier algebra $M(B)$ of $B$ is $A$?
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Is the set of normal, positive, faithful, linear functionals on a W*-algebra open?

Let $\mathcal{A}$ be an infinite-dimensional W*-algebra, that is, an infinite-dimensional $C^{*}$-algebra which is the Banach dual of a Banach space $\mathcal{A}_{*}$ (equivalently, $\mathcal{A}$ is ...
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1answer
27 views

Unitarily equivariant, linear, Hermitian maps on matrix algebras

I realized my question was not well-posed, hence I proceeded to rewrite it from scratch. Denote by $\mathcal{M}_{n}(\mathbb{C})$ the $C^{*}$-algebra of complex matrices. Let $L\colon\mathcal{M}_{n}(\...
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Upper semicontinuity on compact Hausdorff $K$ implies upper semicontinuity on state space of $C(K)$.

This is a question about Proposition 9.1 in an article by Cormac Walsh: https://arxiv.org/pdf/1610.07508.pdf. Let $K$ be a compact Haussdorf space, let $(C(K),\|\cdot\|_{\infty})$ be the space of ...
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$\ell^2$-convergence of convolution square root implies uniform convergence

I have problems understanding parts of a proof in [Proposition 18.3.5., Diximier, C*-algebras] for the special case of discrete groups. Let $G$ be a discrete group and let $\phi\in \ell^2(G)$. By 13....
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1answer
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When is the set of positive elements in a C* algebra a totally ordered set

As the title suggests I was wondering if there are any properties that ensure a C*-algebra has all positive elements comparable to each other. ( Recall $a\leq b$ if $b-a$ is a positive element in the ...
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2answers
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a question on the definition of direct sum of $C^*$ algebras

According to the definition in the Olsen's book,if $A=B\bigoplus C$,the intersection of $B$ and $C$ should be zero.But in other reference books ,when talking about direct sum of matrix algebras,there ...
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1answer
21 views

GNS representation of a nuclear $C^*$-algebra

Suppose $A$ is a nuclear $C^*$-algebra with a tracial state $\psi$, $(\pi_{\psi},H_{\psi})$ is the GNS reprsentation with respect to $\psi$. My question: Does there exist $A$ which satisfy the above ...
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1answer
12 views

find all ideals of direct sum of simple $C^*$ algebras

Suppose $A=\bigoplus A_n$ where each $A_n$ is a simple $C^*$ algebra(The direct sum is $c_0$ direct sum).I guess all the ideals of $A$ are precisely the direct sum of ideals of $A_n$ .It is easy that ...
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1answer
24 views

GNS representation

Suppose $A$ has a tracial state $\psi$, I want to prove $A/ker(\pi_{\psi})$ has a faithful tracial state,where $\pi_{\psi}$ is the $GNS$ respresentation with respect to $\psi$. My thought: define $\...
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2answers
54 views

Automorphisms of a particular abelian group

I am trying to solve Exercise 7.9 of An Introduction to K-theory of C*-algebras, It remains for me to show that the Automorphisms of $\mathbb{Q}\oplus\mathbb{Q}$ which send $ \{(x,y) \in \mathbb{Q}\...
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1answer
44 views

Spectrum invariance under the passage to a sub Banach Algebra

Let $\mathcal{B}$ be a unital Banach Algebra, fix $A \in \mathcal{B}$. $\sigma_\mathcal{B}(A) = \{ \lambda \vert \lambda I - A \, not \,invertible \, in \, \mathcal{B}\}$ the specturm of $A$ in $\...