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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Connection between Stinespring's factorization theorem and the spectral theorem for bounded operators

I know at least 2 versions of a Spectral theorem for operators, one of them is the following Theorem: Let H be a separable complex Hilbert space, $A\in L(H)$ self-adjoint ($L(H)$ are the bounded ...
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State of a $ C^{*} $-algebra.

Let $ (\pi,\mathcal{H}) $ be a non-degenerate $ * $-representation of a $ C^{*} $-algebra $ A $, and let $ h \in \mathcal{H} $ with $ \| h \| = 1 $. Define $ f_{h}: A \to \Bbb{C} $ by $ {f_{h}}(a) ...
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Completely positive maps

Let $B$ be a commutative C$^*$-algebra and let $M_n$ denote the algebra of $n\times n$ complex matrices. Let $f$ be a state on the tensor product of $B$ and $M_n$, $B\otimes M_n$. How can I show that ...
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Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
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What algebraic structure do self-adjoint operators form?

Consider the $\mathbb{C}$-algebra of all matrices of dimension $n$ over the complex numbers, $Mat_n\mathbb{C}$; We have here a notion of adjointness which is an involution; and thus of ...
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A necessary and sufficient criterion for an element of a multiplier $ C^{*} $-algebra to be positive.

I am trying to find a reference for the following assertion: Let $ A $ be a $ C^{*} $-algebra, and let $ M(A) $ denote its multiplier algebra. Then $ m $ is a positive element of $ M(A) $ if and ...
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38 views

$||a||\leq \sup_{||b||\leq 1} ||ab||$ in a C*-algebra

I would like to prove that, if $a$ is an element in a C*-algebra then $$\|a\|\leq \sup_{\|b\|\leq 1} \|ab\|$$ It is obvious if the algebra is unital. What if it is not?
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Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...
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positive elements in $L(H)$

At the moment I'm studying positive elements in $C^*$-algebras. Let $H$ be a complex Hilbert space, $L(H)$ the linear bounded operators $H\to H$ and $T^*=T$. Then: $$\sigma(T)\subseteq [0,\|T\| ]\; ...
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continuous functional calculus for nonunital $c^*$-algebras

In lecture we had the continuous functional calculus for unital $c^*$-algebras: Let $A$ be an unital $c^*$-algebra, $a\in A$ normal and let $$alg(a,a^*)=\overline{ \{ ...
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unitalization of the $c^*$-algebra of complex polynoms without constant term / compactness of the spectrum of elements in non-unital $c^*$-algebras

Let A be $C^*$-algebra with unit $e$ and $a\in A$ normal. We define $$alg(a,a^*)=\overline{ \{ \sum\limits_{k,l=0}^n\lambda_{k,l}a^k\overline{a}^l; \lambda_{k,l}\in\mathbb{C}, n\in\mathbb{N}\} \\}$$ ...
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Theorem about irreducible representation of $C^*$-algebra

I have been told, that there is a theorem about irreducible representation of $C^*$-algebras, but I have troubles finding it. It is also possible, that this theorem is consequence of some theorem I've ...
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positive linear maps which are involutive

Let $S$ be an operator system, $B$ a $c^*$-algebra and $\phi:S\to B$ a positive linear map. Then $\phi$ is involutive, i.e. $\phi(x^*)=\phi(x)^*$ for all $x\in S$. I want to prove this claim but I'm ...
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28 views

positive linear maps of $c^*$-algebras are bounded

Let $A, B$ be $c^*$-algebras and $\phi:A\to B$ a positive, linear map. Then $\phi$ is bounded. Proof: It is sufficient to proof boundedness of $\phi$ on the unitarization (I missunderstood that, see ...
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28 views

Unitary element in a C*-algebra

Suppose $\Bbb T$ is the unit circle and $M$ is the C*-algebra of all $2\times 2$ complex matrices. Consider the C*-algebra $A: = C(\Bbb T, M)$. Let $E$ and $F$ be the projections in $A$ given by ...
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What $\mathbb{C}I$ means?

I've come across this expression $$ \mathbb{C}I $$ while studying operators algebras. $C^*$-algebras and AF-algebras, concretely. In Kenneth R. Davidson's book $\boldsymbol{C^*}$**-algebras by ...
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Irreducible unitary representations of $ \Bbb{R}^{2} \rtimes_{\alpha} \Bbb{R} $.

Let $ \alpha $ be the action of $ \Bbb{R} $ on the group $ \Bbb{R}^{2} $ defined by $ \alpha_{t} \! \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \exp \! \left( \begin{bmatrix} t & 0 \\ ...
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Show that $C(S^n)$ is the universal $C^*$-Algebra of selfadjoint, commutative $x_0,\ldots,x_n$ with $\sum x_i^2 = 1$

Let $x_0,\ldots,x_n$ be symbols with relations $x_i=x_i^*$, $x_i x_j = x_j x_i$ and $\sum_i x_i^2 = 1$. Then I want to show that the universal $C^*$-Algebra $A$ of these relations exists and that ...
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countable unitary element in a separable C*-algebra

How do I show that the set of unitary equivalence classes of projections is countable in a unital separable $C^*$-algebra? So I tried to show that the set of unitary elements in $C^*$-algebra is ...
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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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1answer
32 views

Projection matrix (C* algebra.. but linear algebra question) [closed]

The subject is $C^*$-algebra, but I think my question might be linear algebra related type. I have a question from the book Operator Algebras Theory of C*-Algebra by Blackadar. On page 351, in the ...
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A continuous field of C* algebra, $C(\mathbb T)\rtimes\mathbb Z_2$

Given a $C^*$-algebra, $A=${$f:[0,1]\rightarrow M_2(\mathbb C)$ where $f(0),f(1) $ are diagonal } which is isomorphic to $C(\mathbb T)\rtimes\mathbb Z_2$, How can I determine its continuous field ...
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Convergence in strong operator topology and norm topology

Let $(T_n)\subset B(H)$ be a sequence of operators such that $T_n\to 0$ in strong operator topology. Show that $\|T_nK\|\to 0$ and $\|KT_n\|\to 0$ for every compact operator $K$. Let $f,g \neq 0 ...
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41 views

Closed unit ball of positive bounded operator space and its extreme point

Let $H$ be infinite dimensional Hilbert space. Then the closed unit ball of positive bounded operator space $B(H)^+$ is not the convex hull of the projections of $B(H)$. Please help me. Thanks.
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Inclusion of representations: $\pi_1(A)^{''}\subseteq \pi_2(A)^{''} \Rightarrow \pi_1(A)\subseteq \pi_2(A)$?

Let $A$ be a $C^*$-algebra and $\pi_1 , \pi_2$ two *-representations on a same Hilbert space $\mathcal{H}$ so that the inclusions $\pi_1(A)^{''}\subseteq \pi_2(A)^{''}$ of the associated von Neumann ...
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Applications of $C^\ast$ algebras in differential topology

I was wondering if there were any useful ways $C^\ast$ algebras come into play within differential topology. I know this is a fairly broad question,so any type of input is applicable.
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question about a proof on Murphy's book about $C^*$-algebras

I'm reading Murphys book "$C^*-$algebras and operator theory" and I have a question about a proof in chapter 3. The statement is (Theorem 3.1.8): Let I be a closed ideal in a $C^*$-algebra A. Then ...
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Can $ {L^{1}}(G) $ be a $ C^{*} $-algebra?

Let $ G $ be a locally compact abelian group. Then $ {L^{1}}(G) $ is a commutative algebra when equipped with convolution. Is there an involution $ ^{*} $ on $ {L^{1}}(G) $ so that it becomes a $ ...
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Are elements of a $C^*$-Algebra strictly positive iff their spectrum is strictly positive?

Let $A$ be a $C^*$-Algebra. An element $a\in A$ is said to be positive iff $a=a^*$ and the spectrum $\sigma(a)$ is nonnegative, ie. $\sigma(a)\subset[0,\infty)$. This is equivalent to $\varphi(a)\ge ...
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Irreducible representation of $C^*(D_\infty)$

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$ Ultimately, I'm interested in finding a ...
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K-Theory of $C(X)$ for $X$ totally disconnected

I am studying K-Theory for C*-algebras by the following book: Rordam, Larsen and Laustsen. I am having a problem with the the Exercise 3.4, which is: Let $X$ be any compact Housdorff space. In the ...
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Primitive ideal space of C*(Z2*Z2)

Find the primitive ideal space, the center, a continuous field of $C^*(Z_2*Z_2)$. Here, $C^*(Z_2*Z_2)$ is the full group $C^*$-algebra. I know the definitions of all of them, but I'm having hard ...
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Universal properties of certain crossed products

So I was wondering if there are any nice universal properties that the crossed product $C^*$ algebra, $C(\mathbb{T})\times_\alpha \mathbb{Z}_2$ satisfies, where $\alpha$ is the action of conjugation. ...
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A representation of a $ C^{*} $-algebra.

I have a quick question about the representation theory of $ C^{*} $-algebras. A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} ...
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Determining primitive ideal space of C* algebra

What is the general way of determining the space of primitive ideals of the C* algebra if there is any? Thanks.
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A Lemma about the operator space

The following lemma comes from the book "C*-algebras Finite-Dimensional Approximations" by N.P. Brown and N. Ozawa P379 Lemma 13.2.3 Let $X_{i}\in B(H_{i})$ (i=1,2) be unital operator subspaces ...
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why is $f(xx^*)$ is defined and $xf(x^*x)=f(xx^*)x$?

Let $A$ be a $c^*$algebra, $x\in A$ and $f:\sigma(x^*x)\to\mathbb{C}$ continuous and $f(0)=0$ ($\sigma(x^*x)$ is the spectrum of $x^*x$). Why is $f(xx^*)$ is defined and $xf(x^*x)=f(xx^*)x$? It is ...
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1answer
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Arveson spectrum for a unitary representation of a group on a Hilbert space

Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a self-adjoint operator $H$ (for which there is a resolution of the identity P(p), by the spectral theorem) ...
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Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
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Strictly positive element in a C*-algebra

Searching about strictly positive elements, I found this exercise. I tried to solve it, and the following is my attempt. Please check my proof. Is it correct? Suppose $a$ is strictly positive. By ...
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A question about $n$-dimensional operator space

Let $F_{n-1}$ be the free group of rank $n-1$ and $C^{*}(F_{n-1})$ be the universal group C*-algebra of $F_{n-1}$. And if $E_{n}$ is the $n$-dimensional operator space in $C^{*}(F_{n-1})$ spanned by ...
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Characterization of positive elements of a sub-C*-algebra of $B(H)$

Let $A$ be a non-unital sub-C*-algebra of $B(H)$. I want to show that if $T\in A$ and $\lambda \in \mathbb{C}$ are such that $T+\lambda I_H\geq 0$ then $T$ is self-adjoint and $\lambda \geq 0$. Let ...
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1answer
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Operator matrix is invertible if and only if its determinant is invertible

Let $A,B,C,D$ are pairwise commutative operators on a Hilbert space $H$, then a necessary and sufficient condition that the operator matrix $$\begin{pmatrix} A&B\\C&D\end{pmatrix}$$ be ...
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1answer
34 views

Natural Ordering of the Class of Hermitian Preserving Maps

I am trying to understand Man-Duen Choi's remark 3 in his paper Completely Positive Linear Maps on Complex Matrices: For a linear map $\Phi : \mathcal{M}_{n} \to \mathcal{M}_{m}$, it is obvious that ...
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realizing/ understanding $C^*(\phi_g(C([0,1])))$ and "support projection of an element of a $C^*$-algebra

someone asked me the following question but I don't know the answer, but and I'm interested in it too. Let $C([0,1])=\{f:[0,1]\to\mathbb{C}; \text{f is continuous}\}$, $g\in C([0,1])$ be a positive ...
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Does every Hausdorff Compact space admits a (Radon) measure with full support?

The title says it all. I know a bit of C*-algebra theory, so using Gelfand and Riesz theorems, this is equivalent to Does every unital abelian C*-algebra admit a faithful state? I believe this ...
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Why the self-adjointness condition for positivity of an element of a C*-algebra?

A positive element x of a C*-algebra A is a self-adjoint element whose spectrum is contained in the non-negative reals. If there's a faithful finite-dimensional representation of A where the ...
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understanding a theorem in c*algebra

I want to understand the following theorem: Theorem: Let A and B $C^*$-algebras with A unital, and let $\varphi:A\to B$ a bounded linear selfadjoint map such that: for every self-adjoint elements ...
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Questions about the Kapalansky density theorem

I'm studying Takesaki's Theory of operator algebras book by myself. The following is a theorem from that book: I have several questions about this proof: 1- He claims, in the first line of ...
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1answer
54 views

Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...