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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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0answers
21 views

commutant of a non unital $C^*$ algebra

If $A$ is a non-unital $C^*$ algebra,is the commutant of $A$ empty? Does there exist a theorem which states that every $C^*$ algebra has commutant(centralizer)?
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0answers
18 views

multiplication of bounded linear operators

Suppose $T_1,T_2\in B(\oplus H_n)$,each $H_n$ is an infinite dimensional Hilbert space,I want to construct $S_1,S_2\in B(\oplus H_n)$ such that $S_1T_1=T_2S_2 =(0\cdots,k I_n,\cdots,0)$,where k is ...
1
vote
1answer
31 views

Compute the norm of a bounded linear operator

Let $T$ be a nonzero bounded linear operator in $B(H)$, where $H$ is an infinite dimensional Hilbert space. If the norm of $T$ is known, how to compute the norm $\|I–T\|$,where $I$ is the identity ...
0
votes
1answer
22 views

multiplication of Hilbert Schmidt operators and bounded operators

Suppose $T_1,T_2 \in B(H)$,where $H$ is an infinite dimensional Hilbert space. $S\in \mathcal{HS}(H)$,$\mathcal{HS}(H)$ is the set of Hilbert Schmidt operators on $H$. Does there exist nonzero ...
0
votes
0answers
35 views

inverse of a bounded linear operator [on hold]

Suppose $T\in B(H)$,where $H$ is an infinite dimensional Hilbert space. If $\|I–T\|<1$,then $T$ is invertible. 1.Does there exist other methods to determine whether $T$ is invertible or not. 2....
2
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0answers
22 views

Schwarz inequality for unital positive maps on C*-algebras

I was recently studying this paper by Man-Duen Choi about inequalities for positive maps on C*-algebras. He demonstrates that Let $\phi : \mathcal{A} \to \mathcal{B}$ be a unital positive linear ...
1
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0answers
38 views

Proof that regular representation induced by faithful representation is faithful on reduced crossed product

Suppose that $A$ is a $C^*$-algebra and $G$ is a locally compact Hausdorff group acting on $A$. Gert Pedersen, in his book $C^*$-algebras and their automorphism groups, shows in theorem 7.7.5 that ...
1
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1answer
31 views

Operator norm of a family of matrices

Let $c$ be a complex number. Consider the family of $n\times n$ matrices $M_n$ which have $c$'s on one off-diagonal, $\bar{c}$'s on the other off-diagonal, and zero everywhere else. So $M_4$ looks ...
1
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2answers
35 views

state on a non-unital $C^*$ algebra

Suppose $\tau$ is a state on a non-unital $C^*$ algebra $A$.There is a well-known inequality: $$\tag{$*$}|\tau(a)|^2\leq\tau(a^*a),\ \text{ for all } a\in A.$$ Does there exist some nonzero element $...
2
votes
1answer
29 views

Distance between subalgebras and commutants in matrix algebras.

This is a question on how to relate two different distances in the matrix setting. Everywhere below, $M_n$ denotes the square matrices $n\times n$ whose entries are in $\mathbb C$. We consider the ...
1
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2answers
39 views

How to understand the spectrum of a C*-algebra?

I know that the spectrum of an element $x$ in a unital C*-algebra $A$ is defined as $$\operatorname{Sp}_{A} x=\left\{\lambda\in\mathbb{C}\mid (x-\lambda\cdot1)\ \text{is not invertible}\right\}.$$ ...
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1answer
27 views

A problem in the proof of the irrational rotational algebra has unique trace

I'm reading Davidson's book C* Algebras by Example.in the proof of the irrational rotational algebra has unique trace. he define a automorphism: $$\rho_{\lambda,\mu}(U)=\lambda U, \rho_{\lambda,\mu}...
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0answers
26 views

faithful tracial states on a $C^*$ algebra [closed]

Does there exist a theorem or a proposition to determine whether a $C^*$ algebra has faithful tracial state or not?
2
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1answer
20 views

why the irrational rotation algebra $A_{\theta}$ is $C(T^{2})$ when $\theta =0$

since the irrational rotation algebra $A_{\theta}$ is commutative when $\theta =0$, it has the form $C(X)$ for some space $X$ and by universal property of $A_{\theta}$, there is a homomorphism from $...
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votes
3answers
53 views

Complementary $C^*$-subalgebra

Does there exist a closed subspace of $\ell^{\infty}$ complementary to $c_0$? If $A$ is a $C^*$-algebra, $B$ is a $C^*$-subalgebra of $A$, under which condition can one ensure that there exists a $C^*...
2
votes
1answer
32 views

If $h$ is self-adjoint then $e^{ih}$ is unitary

Let $A$ be a unital C*-algebra. If $h \in A$ is self adjoint, then $e^{ih}$ is unitary. This proof is in a lecture note, which I am not able to understand. They prove it in the following way: For ...
2
votes
1answer
33 views

Nuclear $C^*$ algebra and tensor products

Suppose $A,B$ are $C^*$ algebras, $\alpha$ is some $C^*$-norm on the algebraic tensor product of $A$ and $B$. If $A\otimes_{\alpha}B$ is nuclear, can we conclude that $A$ and $B$ are nuclear? What ...
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0answers
28 views

How to imbed the irrational rotation algebra into AF algebra?

I'am reading Kenneth R.Davidson`s book C*-Algebras by Example. I feel confused by the construction of embedding irrational rotation algebra into associated AF algebra, Is there another book give a ...
2
votes
1answer
37 views

unique $*$ homomorphism of spatial tensor product

If $A$ is a nuclear $C^*$ algebra, $A^{op}$ is the opposite $C^*$ algebra, is the conclusion: "there is a unique injective $*$-homomphism $\pi \colon A\otimes A^{op}\rightarrow M(A)\otimes M(A)^{op}$" ...
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votes
2answers
54 views

Elementary tensors of tensor product of C*algebras

When $\alpha$ is a $C^*$-norm on $A \times B$, we denote the $C^*$ completion of $A \otimes B$ with respect to $\alpha$ by $A\otimes_{\alpha}B$. I feel a little confused about the elementary tensors ...
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1answer
35 views

construct a faithful tracial state throgh a tracial state on some $C^*$ algebra [closed]

Suppose $A$ is a $C^*$ algebra ,$\phi$ is a tracial state (not faithful)on $A$.Can we construct a faithful tracial state on$A$ by using $\phi$?
3
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1answer
51 views

$C^*$-algebra norm computations

I'm fairly new to $C^*$-algebras and Hilbert space. Given the algebraic relations of the $C^*$-algebra, I am having a lot of trouble computing the norm of its elements and am wondering if there are ...
2
votes
1answer
46 views

product states on the tensor product *-algebra

Let A and B be two unital $C^∗$-algebras, and $x∈A⊗B$ (the algebraic tensor product $*$-algebra), different from $0$. Is there states $ω_x∈A^∗_+$ and $φ_x∈B^∗_+$ such that, for the product state $ω_x×...
2
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1answer
34 views

Is there an example of a non-zero projection in a C$^{*}$-algebra that is infinite but not properly infinite?

For clarification: Given a projection $p$ in a C$^{*}$-algebra $A$, we say $p$ is infinite if there is a projection $q\in A$ satisfying $q\lneq p \sim q$; we say $p$ is properly infinite if there are ...
1
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1answer
35 views

Closed convex hull of pure states of non-unital $C^*$-algebras

It is known that, when $\mathcal{A}$ is a $C^{*}$-algebra with an identity element, the space $\mathcal{S}$ of states of $\mathcal{A}$ is a convex subset of the topological dual $\mathcal{A}^*$ of $\...
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0answers
17 views

tensor product of multiplier algebras

Let $A$ be a $C^*$ algebras,$A^{op}$ is the opposite $C^*$ algebra. Can we view $A \otimes_{\alpha} A^{op}$ as a $C^*$-subalgebra of $M(A)\otimes_{\alpha} M(A)^{op}$ ,where $\alpha$ is any $C^*$ norm,$...
4
votes
1answer
40 views

$GL_n^+(A)$ is open but $U_n^+(A)$ is not

Let $A$ be a C*-algebra. $\tilde{A}$ be the unitization of $A$. I checked the following lemma: If $x$ and $y$ are elements of $M_n(\tilde{A})$ such that $x$ is invertible and $\|x-y\| \leq \frac{1}{...
2
votes
1answer
49 views

Is the extension of $*$ homomorphism unique?

If $\phi:A\to B(H)$ is a $*$ homomorphism, do there exist two different $*$ homomorphisms $\phi_1,\phi_2:M(A)\to B(H)$ which extend $\phi$, where $M(A)$ is the multiplier algebra of $A$?
2
votes
0answers
72 views

Sufficient condition for element to be close to an invertible in Rordam's Simple C$^{*}$-algebras paper

I am reading Rordam's paper "On the Structure of Simple C$^{*}$-Algebras Tensored with a UHF-Algebra." I came across the following passage: I am having trouble understanding the $(\impliedby)$ ...
2
votes
1answer
22 views

Isomorphism between $\mathcal{L}_{M_n(A)}(E^n)$ and $\mathcal{L}_{A}(E^n)$

The following doubt came after reading the book "Hilbert C*-modules" by E.C. Lance. Let $A$ be a C*-algebra and $E$ a Hilbert $A$-module, there's a natural structure of Hilbert $A$-module on $E^n$ ...
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1answer
30 views

Bases of the tracial cone and full elements

Say $A$ is an exact C*-algebra and let $T(A)$ be the cone of densely defined lower semicontinous traces. It is known that if $a \in \mathrm{Ped}(A)$ is full, then $T_{a\to 1} := \{\tau \in T(A) \mid \...
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1answer
47 views

Dependence of operator topologies in a $C^*$ algebra on the representation

Let $A$ be a $C^*$ algebra. Given a faithful representation $\pi:A\to \mathcal{B}(H)$, we can define the weak operator topology with respect to $\pi$ as initial with respect to the maps $a\mapsto \...
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1answer
36 views

Sufficient conditions for a C* algebra to be separable

Do you know of any (necessary and) sufficient conditions for a C* algebra to be separable? Reference to bibliography is welcome.
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1answer
33 views

Some concepts of $C^*$-algebra generalized from linear algebra. Can anyone help me to check if they are correct, and give some examples?

A Banach algebra is just a Banach space equipped with an operation of multiplication defined such that $\|a b\| \le \|a\|\|b\|$ for all $a,b$ in it. If, in addition, there exists an identity, then it ...
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1answer
23 views

Conditional expectation that preserve involution

Consider $B$ a C*-algebra and $A$ a C*-subalgebra such that $1_A=1_B$. If $E:B\rightarrow A$ is a faithful conditional expectation (that is a projection of norm 1, by Tomiyama's theorem) then is it ...
2
votes
1answer
27 views

the commutant of $\mathcal{HS(H)}$

Let $\mathcal{HS}(H)$ be the set of Hilber Schmidt operators on a Hilbert space,it is a $C^*$ algebra.I wonder whether we have an explicit description of the commutant of $\mathcal{HS(H)}$.Is the ...
0
votes
1answer
27 views

construct a closed subspace of $\mathcal{HS}(H)$ such that all elements in the subspace are commutative.

Suppose $K=\mathcal{HS}(H)$,where$\mathcal{HS}(H)$ is the set of all Hilbert Schmidt operators on the Hilbert space $H$.I have two questions. 1.Can we construct a closed subspace $K_1$ of $K$ such ...
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0answers
61 views

Prove a mapping $C(X,Y)\to \operatorname{Hom}(C(Y), C(X))$ is surjective [duplicate]

Let $X$ and $Y$ be compact Hausdorff spaces, and let $F$ be a continuous function from $X$ to $Y$. Define a function $\Phi_F$ from $C(Y)$ to $C(X)$ by $\Phi_F(f)=f\circ F.$ I have shown $\Phi_F$ is ...
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1answer
60 views

Let $A=\mathbb C[x] $ prove there is no norm on $A$ in which it is a C* algebra

Let $A=\mathbb C[x] $ prove there is no norm on A in which it makes a C* algebra. i think this is true because the spec(a) is infinity for any $a\in A$ ? but im not sure how to prove it. I did try ...
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votes
1answer
37 views

multiplication of Hilbert Schmidt operators

Suppose $H$ is an infinite dimensional Hilbert space,$B(H)$ is the set of bounded operators on $H$,$\mathcal{HS}(H)$ is the set of Hilbert-Schmidt operators on $H$. I have two questions: 1.If $T$ is ...
1
vote
0answers
20 views

bimodule over a non-unital $C^*$ algebra

Let $A$ be a $C*$ algebra,suppose $\mathcal{H}$ is a $M(A)–M(A)$ bimodule,where $M(A)$ is a multiplier algebra of $A$.Can we deduce that $\mathcal{H}$ is a $A-A$ bimodule? My thought:since $\mathcal{...
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vote
0answers
39 views

Goldstine's theorem for positive linear functionals

I am trying to show that the statespace $S(A)$ of a $C^*$-algebra $A$ lies weak-* dense in the statespace $S(A'')$ of the enveloping von Neumann Algebra $A''$. I came across Goldstine's theorem and ...
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votes
1answer
28 views

Bicommutant of self-adjoint subset of an involutive algebra

1.1.9. Let $A$ be an involutive algebra and $M$ a self-adjoint subset of $A$... If the elements of $M$ commute pairwise, then $M\subset M'$, so that $M'\supset M''$ and $M''$ is commutative... ...
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1answer
26 views

tracial states on $C^*$ subalgebra

Let $A$ be a $C^*$ algebra,$A=A_1\oplus A_2$.If $A$ has tracial state $\tau$,I want to show $A_1$ also has tracial state,say $\tau_1$. My thought: let $\tau_1(a_1)=\tau(a_1,0)$,where $a_1 \in A_1$,...
2
votes
1answer
71 views

Homomorphism between compact Hausdorff spaces.

Suppose that X and Y are compact Hausdorff spaces and $p:C(X)\to C(Y)$ is a unital * homomorphism. Prove that there exists a continuous function $h: Y \to X $ such that $p(f)=f\circ h $ for all f in $...
2
votes
0answers
59 views

Proving a Certain Inequality Without Utilizing the Full Theory of $ C^{\ast} $-Tensor Products

Suppose that we have the following objects: $ X $ — a locally compact Hausdorff space. $ A $ — a $ C^{\ast} $-algebra. $ \pi $ and $ \rho $ — commuting representations of, respectively, $ {C_{0}}(X) $...
2
votes
1answer
16 views

Show that a specific linear map from a Hilbert module into a C*-algebra satisfies a certain equation

Let $A$ be a C*-algebra, $\mathcal{H}$ be a Hilbert $A$-module and $B$ be a C*-algebra. Let $\pi: A \to B$ be a $*$-homomorphism and $\tau: \mathcal{H}\to B$ be a linear map such that $\tau(\xi)^*\tau(...
1
vote
1answer
61 views

tracial states on corona algebra

Let $A$ be the $c_0$ direct sum of $M_{n}(\mathbb{C})$,I know the fact that the multiplier algebra of $A$ ,M($A$) is $\prod M_n(\mathbb{C})$. Does the corona algebra $M(A)/A$ have uncountable tracial ...
0
votes
1answer
21 views

existence of finite irreducible reprentation of a nonunital $C^*$ algebra

Suppose $A$ is a non-unital $C^*$ algebra,can we conclude that there must exist a nonzero finite irreducible representation of $A$.
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0answers
36 views

GNS representation and states of $C^*$-Algebras

Let $\varphi$ be a state of the $C^*$-algebra $A$, $B\subset A$ a hereditary subalgebra and $K_\varphi:=\{x\in B : 0\le x \le 1, \varphi(x)=1\}$. Let $\pi_\varphi:A\rightarrow \mathcal{B}(H_\varphi)$ ...