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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embedding of $\prod_{n\in\mathbb{N}}M_{n}(\mathbb{C})$ in a type $II_{1}$ factor

Suppose $M$ is a type $II_{1}$ factor with trace $\tau$. Let $\lbrace p_{n}\rbrace_{n\in\mathbb{N}}$ be an increasing sequence of projections such that $\tau(p_{n})\rightarrow 1$. Now, let's consider ...
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75 views

Maximal abelian subalgebras of SAW*-algebras

Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras: A C*-algebra $A$ is an SAW*-algebra if for each pair of orthogonal, positive elements $x,y\in A$, there exists a ...
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57 views

The cone over separable C*-algebra is also separable?

For a C*-algebra $A$, the cone over $A$ is $CA=C_{0}(0,1]\otimes A$ , My question: If $A$ is separable, $CA$ is also separable?
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50 views

Two questions about orthogonal projections on Hilbert space

Let $l_{k}^{2}$ denote the k-dimensional Hilbert space and $\oplus_{1}^{\infty} l_{k}^{2}$ be the infinite direct sum of $l_{k}^{2}$. Let $P_{M}\in ...
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24 views

Types of von neumann algebras

A von Neumann algebra M is said to be finite, infinite, properly infinite, or purely infinite according to the property of the identity projection 1. I think that this classification of types are ...
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71 views

Soft questions: $C^\ast$-dynamical systems

I have read some papers about $C^\ast$-dynamical systems. But there are still some questions in my mind which I can not answer. When is the $C^\ast$-dynamical system introduced? Why is the ...
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21 views

Relation between diferent definitions of Quasicontinuous functions

Defining the class of quasicontinuous functions by \begin{equation} QC=(H^{\infty }+C(\dot{{\mathbb R}}))\cap (\overline{H^{\infty }}+C(\dot{{\mathbb R}})). \end{equation} Where $H^{\infty }$ denotes ...
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60 views

construction of an injective representation of $C_0(X)$

Let X be a locally compact noncompact Hausdorff space and consider the C$^*$-Algebra $C_0(X)$ of continuous functions vanishing at infinity. I want to construct an injective *-represenatation of ...
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40 views

Unital maps taking values in abelian C*-algebras

It is known that a bounded linear functional $f$ on a unital C*-algebra $A$ is positive if and only if $f(I)\geqslant 0$. Is the same true for bounded linear operators $T\colon A\to C(X)$ with $T(I) = ...
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25 views

Wiener's lemma and Hulanicki's lemma

Let $\mathcal{A}(\mathbf{T})$ be the Banach algebra of continuous complex-valued functions on the unit circle with absolutely convergent Fourier series. Then Wiener's lemma states that if $f \in ...
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69 views

Important and simple example of application for functional calculus?

I reently proved the theorem for unital $C^\ast$-algebras that for $a\in A$ normal there exists a unique unital isometric $\ast$-homomorphism $\varphi : C(\sigma(a))\to A$ with $\varphi(i) = a$ where ...
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61 views

positive elements in c*algebras and states

I have problems to prove that an element $a $ is a $C^*$-algebra is positive if and only if $f(a) \geq 0$ for all states $f$. The definitions I use: -$f:A\to\mathbb{C}$ linear functional on a ...
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27 views

limit of state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
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37 views

strictly positive element iff A contains a countable approximative unit

I search a proof of: Let A be a c$^*$-algebra and let $(u_n)_{n\in\mathbb{N}}$ an approximative unit in A. Then $a=\sum\limits_{n=1}^{\infty}\frac{u_n}{2^n}$ is strictly positive. Could anybody tell ...
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64 views

Positive Elements in a C*algebra

Let A be a C$^*$-Algebra, $a\in A$. Why is $a\ge 0$ (a is called "positive") iff $\forall \varphi\in S(A): \varphi\ge0$? S(A) is the set of linear positive functional $\eta:A\to\mathbb{C}$ with ...
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19 views

Spectrum of $C^\ast$ subalgebra

Let $A$ be a unital $C^\ast$ algebra. It is stated in this book that for any $C^\ast$ subalgebra we have $\sigma_B(b)\cup\{0\} = \sigma_A(b)\cup\{0\}$. The reasoning why this should be true is this: ...
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130 views

$\sigma$-weak topology versus weak operator topology

The reference text for this question is: Pedersen, Analysis Now, GTM 118. The $\sigma$-weak topology on $B(H)$ (the bounded linear operators on a Hilbert space $H$) is the weak$^*$-topology on ...
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C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
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1answer
39 views

Existence of countable approximative unit in $C_0(X) \iff X$ is $\sigma$-compact

Let $X$ be a locally-compact Hausdorff space. The following are equivalent: There is a countable approximate unit $(u_n)_{n\in\mathbb{N}}$ for $C_0(X)$ such that $\|u_n\|\le 1$ for all ...
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1answer
37 views

A convergence in norm topology

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then, for any $T_{1}, T_{2}\in B(H)$, ...
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1answer
41 views

Isomorphy of $C_0(U)$ and an ideal

Let $X$ be a topological space, $Y\subseteq X$ closed and $U:=X\backslash Y$. Then $I_Y:=\{f \in C_o(X)\mid f_{| Y}=0\} \subseteq C_0(X)$ is a closed Ideal. I want to show that $I_Y \cong C_0(U)$. I'm ...
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79 views

Partial Isometries: Introduction

Attention This question has been modified drastically. It is done so the answer below is still correct. It is done so to allow more specialized threads. Problem How do I deal with partial ...
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83 views

Example of a wot convergent net but not $\sigma -$ weak convergent

Let $B(H)$ be the space of bounded linear operators. Define the $\sigma-$ weak topology on it by seminorms $p_{h,k} (x)=|\sum_{n\geq 1}(xh_n,k_n)$ where $h=\{h_n\}\subset H ,~~ k=\{k_n\}\subset H $ ...
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C*-algebra generated by the symmetric on 3 elements

I want compute $C^*(S_3)$ where $S_3$ is the symmetric group on $\{1,2,3\}$ and $C^*(S_3)$ is the (full) C*-algebra generated by $S_3$. My attempt: Since $S_3$ is a finite group, $C^*(S_3)=C_c(S_3)$ ...
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66 views

A question about local convexity of the weak operator topology

By definition, I know a locally convex space is a topological vector space whose topology is defined by a family of seminorms $\cal P$ such that $$\bigcap_{p\in{\cal P}}\{x\colon p(x)=0\}=\{0\}.$$ ...
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67 views

A question about maximal and minimal tensor product

Let $A, B$ be two C*-algebras and $\pi: A\otimes_{\max} B\rightarrow M_{n}(\mathbb{C})$ be a representations, then this $\pi$ can factor through the minimal tensor product $A\otimes_{\min} B$ ? (That ...
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61 views

A isomorphism between full group C*-algebras of free group

Fix $n\in \mathbb{N}$ and let $\mathbb{F}_{n}$ be the rank-$n$ free group, can we use the universal property to illustrate the following isomorphism: $$C^{*}(\mathbb{F}_{n}\times \mathbb{F}_{n})\cong ...
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216 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
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77 views

Positive elements of a $C^*$-algbera form a poset

My knowledge of $C^*$-algebras is very little. We call an element positive if $a=b^*b$ for some $b$ and make a relation on all positive elements by saying $$ b \geqslant a \iff b-a \text{ is ...
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45 views

I need help with a proof: invertibility of $b-\lambda$ in $B$ iff $b-\lambda $ invertible in $A$

Let $A$ be a unital $C^\ast$ algebra and let $B$ be a $\ast$ subalgebra such that $B \oplus \mathbb C = A$ and such that the unit in $B$, $1_B$, is not equal to the unit in $A$. I am trying to show: ...
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1answer
66 views

Show that a space X is homeomorphic to the space of multiplicative linear functionals

Let $\mathcal{A}=C(X,\mathbb{R})$ where $X$ is a compact Hausdorff space. Let $\hat{\mathcal{A}}$ be equal to the set of multiplicative linear functionals from $\mathcal{A}$ to $\mathbb{R}$. ...
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1answer
47 views

WOT convergence to SOT convergence

Let $H$ be a Hilbertspace and $T_n \in B(H)$ a sequence of operators with $T_{n+1} \geq T_{n}$. I want to to show that if there is a self-adjoint $T\in B(H)$ with $T_n \stackrel{WOT}{\rightarrow}T$ ...
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1answer
51 views

continuous functional calculus; spectrum of an self adjoint element in a c*algebra

Let A be a C$^*$-Algebra, $a\in A$ selfadjoint and $\|a^2-a\|<\frac{1}{4}$. The claim is: $\sigma(a)\subseteq (-\frac{1}{2},\frac{1}{2}) \cup (\frac{1}{2},\frac{3}{2})$ and there is a projection ...
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How to prove the following isomorphism?

Let $A, B$ be two C*-algebras, $\pi:B\rightarrow A$ and $\sigma: A\rightarrow B$ be *-homomorphisms such that $\sigma\circ\pi$ is homotopic to $1_{B}$. Define a *-homomorphism $\delta: B\rightarrow ...
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29 views

A question about finite-rank projection

Let $B, C$ be two C*-algebras and $\sigma_{0}: B\rightarrow C$ be *-homomorphism such that $\sigma_{0}$ is injective. Then, for a finite set $F\subset B$ of the unit ball and $\varepsilon>0$, Can ...
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1answer
56 views

p is a projection iff p is normal the spectrum of p is contained in {0,1}

I want to know why the following claim is true: Let A be a C$^*$-Algebra. $p\in A$ is a projection (that means $p^2=p^*=p$) iff p is normal and $\sigma (p)\subseteq \{0,1\}$. "=>" why p normal, it is ...
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52 views

A question about finite-rank projection on Hilbert space

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then, Can we verify that ...
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1answer
43 views

A lemma about quasicentral-approximate-unit

Here is a lemma about quasicentral-approximate-unit: Lemma 7.3.1Let $J\triangleleft A$ be a separable ideal. Then there exists a quasi-central approximate unit $\{e_{j}\}\subset J$ such that ...
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1answer
28 views

QD C*-algebra's representation theorem

Here is a question from the proof of the "QD C*-algebra's representation theorem" in P245 of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. For a separable unital ...
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Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

Let $\mathcal{H}=L^{2}[0,2\pi]$, and let $L=-\frac{d^{2}}{dx^{2}}$ on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions $f$ on $[0,2\pi]$ with $f''\in\mathcal{H}$ and ...
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1answer
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Isomorphism between compact operators and compact operators tensor matrices ($\mathbb{K}\otimes M_n(\mathbb{C})\cong \mathbb{K}$)

Let $\mathbb{K}$ be the compact operators and $M_n(\mathbb{C})$ the complex valued matrices. I have read the algebra $\mathbb{K}\otimes M_n(\mathbb{C})$ is isomorphic to $\mathbb{K}$. Could you tell ...
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51 views

Infinite projections in the Cuntz algebra

I am studying the Cuntz algebra $\mathcal{O}_n$, $(n \ge 2)$ with generators $S_1, S_2, \ldots, S_n$ and in my class notes there is a statement about the projections $S_1S_1^*, S_2S_2^*, \ldots, ...
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96 views

a question about contractions on Hilbert spaces

Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$, $\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$. $T_1T^\ast_1+T_2T^\ast_2\leq I$. Then can we verify that 1 ...
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Getting U.C.P map on group operator algebras using Fell's absorbtion principle.

I'm struggling a bit with this theorem: Let $\Gamma$ be a discrete group and $\mathbb{C}\Gamma$ be the group ring of $\Gamma$ i.e. the set of formal sums $\sum_{t \in \Gamma} \alpha_t t$. Furthermore ...
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1answer
88 views

The spectrum of $L:=-\Delta+V(x)$ on complex $L^2(\mathbb{R}^N)$ and real $L^2(\mathbb{R}^N)$

In general, when one talks about the spectrum of an self-adjoint operator, it is naturally considered in a complex Hilbert space (say $L^2(\mathbb{R}^N,\mathbb{C})$). Moreover, the spectral ...
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1answer
46 views

Smooth Bump Functions are Square Integrable

I am currently trying to prove that the smooth functions with compact support on $R^{n}$ (i.e. smooth bump functions) $C^{\infty}_{0}(\mathbb{R}^n)$ are a subspace of $L^{2}(\mathbb{R^n})$, i.e., the ...
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1answer
69 views

An exercise (about positive elements) in C*-algebra

Let $A$ be a C*-algebra, $a\in A$ be a positive element and $b\in A$ be an arbitary element in $A$. Can we verify that $$b^{*}ab\leq \|b\|^{2}a~~?$$
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38 views

Spectral measure and commutativity.

I want to prove that if $A\in B(H)$ and $N\in B(H)$ is a normal operator, and $AE(\Delta)=E(\Delta)A$, where $E$ is the spectral measure given by $N$ and $\Delta$ is a Borel subset of $\sigma(N)$, ...
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2answers
37 views

Some doubts concerning spectral theory.

Probably I'm saying something wrong (that's why the conclusions are strange) so please correct me! There is the continuous functional calculus for a normal element $N$ of a C*-Algebra. This means ...
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2answers
158 views

Spectrum bilateral shift

Let $U \in \mathbb{B}(\ell^2(\mathbb{Z}))$ be the bilateral shift. I want to shoow that $\sigma(U)=\mathbb{T}$. Using functional Calculus I have shown that $\sigma(U)\subseteq\mathbb{T}$. In order to ...