The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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Normal completely positive map in C*-algebra

Let $A$ be a C*-algebra, for a linear map $\phi: A\rightarrow M_{n}(\mathbb{C})$, we define a linear functional $\bar{\phi}$ on $M_{n}(A)$ by ...
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59 views

Approximately unitarily equivalent in C*-algebra

There is a quotation below: Definition 1.7.2. Two maps $\pi: A\rightarrow B(H)$ and $\sigma: A\rightarrow B(K)$ are called approximately unitarily equivalent if there is a sequence of unitary ...
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91 views

A question about essential representation in C*-algebra

There is a quotation of a book "C*-algebras Finite-Dimensional Approximations" below: Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero ...
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105 views

A question on simple and unital $C^\star$-algebra

There is a quotation of a book "$C^\star$-algebras Finite-Dimensional Approximations" Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero ...
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Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
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104 views

A proposition about Voiculescu's Theorem in C*-algebra

It is the quotation below: Exploiting the duality between completely positive map $A \rightarrow M_{n}(C)$ and states on $M_{n}(A)$, it is not too hard to deduce the next result from Glimm's lemma. ...
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184 views

Arveson's Extension Theorem in C*-algebra

I am reading a book C*-algebra and finite-Dimensional Approximations. There are two conclusions (in the book) below. Corollary 1.5.16. Let $E\subset A$ be an operator subsystem and $\phi: E ...
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52 views

continuous depence of the spectrum on elements

Suppose $a_n \to a$ in a unital $C^*$-algebra $A$. If $\lambda_n \in \sigma(a_n)$ converges to $\lambda \in \mathbb{C}$, then $\lambda \in \sigma(a)$. Does the converse hold? So if $\lambda \in ...
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91 views

Abelian group C*-algebras

Let G is a locally compact Abelian group $C^*$-algebra, then $C^*(G)$ is an Abelian $C^*$-algebra, so C*(G) is isomorpohism with the C$_0$(X) for some locally compact Hausdorff space X, here X is the ...
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81 views

Completely positive map on C*-algebra

There is a quotation in a book about C*-algebra. A positive linear functional $f$ on an operator system $E$ is completely positive map. Indeed, for $\xi=(\xi_{1}, \xi_{2},\ldots,\xi_{n})\in ...
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42 views

A confusion on Stinespring dilation

There is a quotation of Stinespring dilation in a book about C*-algebra. (Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there ...
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30 views

A proposition about minimal Stinespring dilation in C*-algebra

Proposition 1. Let $(\pi, \widehat{H}, V)$ be the minimal Stinespring dilation of a contractive completely positive map $\phi: A \rightarrow B$. Then, there exists a *-homomorphism $$\rho: ...
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57 views

A proposition about cyclic representation in C*-algebra

Let $A$ be a C*-algebra, if for an arbitrary cyclic representation $\pi: A \rightarrow B(H)$, we have $\pi(a) \geq 0$, $a\in A$, then can we conclude that $a \geq 0$?
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26 views

Has a *subalgebra of $L(H)$ with $1$ a trivial null space?

Like the title: is it true that a self-adjoint unital subalgebra of $L(H)$ closed in the weak operator topology (a Von Neumann algebra) has a trivial null space? Why? Thank you.
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158 views

A theorem about a tracial state in von Neumann algebra

I am reading a book about C*-algebra. There is a quotation below. Let $M$ be a von Neumann algebra with a faithful normal tracial state $\tau$ and let $1_{M}\in N\subset M$ be von Neumann subalgebra. ...
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237 views

A theorem about conditional expectation in C*-algebra

Definition 1. Let $B\subset A$ be C*-algebra. A projection from A onto B is a linear map $E: A \rightarrow B$ such that $E(b)=b$ for every $b\in B$. A conditional expectation from A onto B is a ...
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36 views

Does it hold that $\{a\}'' = \{f(a) : f$ bounded, Borel measurable$\}$?

Let $H$ be an Hilbertspace. Let $a\in\mathcal{B}(H)$ be a self-adjoint operator. Does it hold that $\{a\}^{\prime\prime}=\{f(a):f\in L^{\infty}(\sigma(a))~\text{Borel measureable}\}$?
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freely generating elements in an algebra

Let $(\mathfrak{M}, \tau)$ be a W${}^{\ast}$-Algebra with (finite, normal, etc., whichever nice conditions one may find need for) tracial state. Elements $(a_{i})_{i\in I}\subset\mathfrak{M}$ shall be ...
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65 views

The proof of Stinespring dilation

(Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there exist a Hilbert space $H_{1}$, and a *-representation $\pi: A \rightarrow ...
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54 views

The comprehension of Stinespring dilation in C*-algebra

(Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there exist a Hilbert space $H_{1}$, and a *-representation $\pi: A \rightarrow ...
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111 views

A simple question about positive element in C*-algebra

I am reading a book about C*-algebra. There is a quotation below. An $operator~system$ $E$ is a closed self-adjoint subspace of a unital C*-algebra $A$ such that $1_{A}\in E$. The $n \times n$ ...
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43 views

An exercise in operator theory

Let $H$ be a Hilbert space and $P$ be a projection to a finite dimensional subspace $K$ of $H$, for a $T\in B(H)$, if $||PTP||=1$, then, for arbitrary $\epsilon>0$, there exists a vector $\alpha$ ...
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Powers of a closed range operator

suppose that $S$ and $S^2$ are operators with closed range. Does it follow that $S^n$ is an operator with closed range for all natural numbers $n$?
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30 views

Concrete representation of the annihilating algebra

Suppose $\mathfrak{M} = A^{\prime\prime}$, where $A$ is a concretely described subalgebra of $\mathcal{B}(\ell^{2}(\mathbb{N}))$. In some instances, it is possible to provide a concrete description of ...
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38 views

Classification of Banach Algebras?

Is there a classification theorem for Banach algebras, or even for Banach *algebras, similar to the GNS representation theorem for $C^*$-algebras? If yes, please provide a reference where I can read ...
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42 views

A proposition in C*-algebra

Problem: Let $A$ and $B$ be C*-algebra and $\varphi:A \rightarrow B$ be a contractive completely positive map. $A_{\varphi}=\{a\in A: \varphi(a^{\ast}a)=\varphi(a)^{\ast}\varphi(a)$ and ...
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82 views

The excision theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". I can not understand the proof of the excision theorem in the fundamental facts of the book. Theorem 1.4.10(Excision) Let ...
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77 views

An exercise in C*-algebra

Let $A$ be a C*-algebra, $\phi$ be a pure state and $L=\{a\in A:\phi(a^{\ast}a)=0\}$, how to prove that $L+L^*\subseteq ker\phi$. ($L^*=\{a^{\ast}: a\in L\}$) I think it is an easy exercise, ...
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42 views

A question about $C^\ast$-algebra

In Kadinson's book Fundamentals of The Theory of Operator Algebra, when the author proved the Theorem 7.2.1, he let $V$ be an extreme point of the unit ball of $C^\ast$-algebra $\cal{U}$, $h$ be a ...
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100 views

Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”

I'm going over the proof of this theorem about strong Morita equivalences on page 253 of "On Morita equivalence of nuclear $C^*$-algebras" by Walter Beer (http://bit.ly/1fOZiOw), I want to make sure I ...
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Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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65 views

How to explain a theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem: Let $A\in B(H)$ be a nondegenerate C*-algebra ...
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147 views

A theorem about approximate units in C*-algebra

I am reading a book about C*-algebra. I encounter a theorem without proof. Could someone help me to complete its proof or give me some hints. Theorem 2.1. Let $I$ be an ideal of C*-algebra $A$. Then ...
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The unitary implementation of $*$-isomorphism of $B(H)$

Is it possible to construct $*$-isomorphism of (factor von Neumann) algebra $B(H)$ which is not unitary implementable?
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Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
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62 views

Automorphism of $W^*$ algebra

Let $\mathfrak{A}$ be von Neumann algebra. It is in particular $C^*$ algebra. Is it true that every $*$-isomorphism of $\mathfrak{A}$ is also $W^*-$isomorphism? (Note that every $*$-isomorphism of ...
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25 views

A separating set which is not cyclic

Let $H=L^2[0,1]$ , $T_g$ be the multiplication operator on $H$, i.e. $f\to fg$ . Let $A$ be the set of the $T_g$ as $g$ runs through the set of polynomials with complex coefficients. Let $h$ be te ...
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Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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Examples of operator theory on Hilbert space

$(1)$ If $T \in B(H)$ is self-adjoint and $T \neq 0$ then $T^n \neq 0$ $(a)$for $n=2,4,8,16,... (b)$ for every $n$ $(2)$ Show that any $T \in B(H)$ can be uniquely expressed as $T=T_1+iT_2$ ...
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37 views

How to prove that $A B A^* \leq \|B\| A A^*$ for operators A,B?

Let $A$, $B$ bounded operators on a Hilbert space $H$. Further let $B$ be self-adjoint. Then we have that $A B A^* \leq \|B\| A A^*$. I wanted to ask how to prove this inequality or where I can find ...
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64 views

Closed graph theorem question?

Let $H$ be a Hilbert space. Let $A:\operatorname{dom}A\to H$ has a closed graph, where $\operatorname{dom}A$ is dense in $H$. Let $S\subseteq \operatorname{dom}A$ be dense. Is it true $A_{|S}$ has a ...
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A funny way of expressing the identity operator

I have encountered the following trick that people in C*-algebra use; but frankly I don't understand why really this is true. Let $A$ be a unital C*-algebra acting non-degeneratly on a Hilbert space ...
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38 views

An inequality for completely positive maps.

Let $f\colon A\to B$ be a contractive completely positive, ${}^*$-preserving map between C*-algebras and take $a\in A$. How one can prove that $$0\leqslant f(a)f(a^*)\leqslant f(aa^*)?$$ Some authors ...
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Proof: $C(X×Y)=C(X)⊗C(Y)$

Where I can find the proof of the following theorem: Let $X$ and $Y$ be compact Hausdorff spaces, $C(X)$ and $C(Y)$ the space of continuous functions on $X$ and $Y$ respectively, then we have ...
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159 views

Positive elements of a $C^*$ (MURPHY, ex 2-2).

I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise: Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra. (a) If $a,b$ are positive ...
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26 views

Question about equivalent representations of von Neumann algebras in Kadison's book

When I read Kadison's book Fundamentals of The Theory of Operator Algebras, I meat a qeustion on page 460, at line 14, I can not understand that $A\mapsto AE'$ and $A\mapsto \Phi_n(A)F'$ are unitarily ...
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41 views

A conjugation of an operator, which commutes with all permutations, still commutes with all permutations

Assume $v:H^{\otimes m}\to H^{\otimes m}$ is a linear operator on the $n^{\text{th}}$ tensor power of a vector space $H$. For each permutation $p$ on the $m$-element set define the linear operator ...
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Show this representation is irreducible and faithful

Let A be a prime $C^{*}$ algebra and $e= \alpha^{-1} c^{*}c$ for some $c \in A$ which satisfies $(c^{*}c)^{2}=\alpha c^{*}c$. Then $e^{2}=e=e^{*}$ and $eAe=\mathbb Ce$ so $eAe$ may be identified with ...
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Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
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64 views

Cross Product Algebras references

Can someone give some references to introductory books or online notes about group algebras and cross-product algebras ? I've already searched on Google (but only for some online notes). The purpose ...