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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4
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1answer
27 views

The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a ...
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0answers
21 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
2
votes
0answers
7 views

Extending $*$-isomorphisms between $*$-algebras to cross products.

Let $G$ be a discrete countable group and suppose I have two $G$-$C^*$-algebras $A$ and $B$ such that there exists a $G$-equivariant isometric $*$-isomorphism $\varphi \colon A \to B$. One can extend ...
1
vote
1answer
13 views

Normal states are dense in $B(H)$

Can someone explain/sketch the proof of the fact stated in the title : The set of normal states (in some $B(H)$) is weakly dense in $B(H)$ ? Or, if possible, some reference. Thank you very much.
2
votes
1answer
26 views

A question on the minimal tensor norm

Given two C*-algebras $A$ and $B$ and let $A_1$ and $B_1$ be their C*-subalgebras. Can we conclude that $A_1 \otimes_\min B_1$ is a subalgebra of $A \otimes_\min B$? I think that this is not true, ...
3
votes
0answers
34 views

Creating Bratteli diagrams for Riesz groups

The Effros-Handelman-Shen-theorem tells you that Riesz groups are the same as dimension groups -- i.e. any ordered, unperforated abelian group with the Riesz interpolation property can be realised as ...
1
vote
1answer
29 views

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 ...
3
votes
0answers
55 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 ...
0
votes
1answer
38 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?
0
votes
1answer
39 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 ...
0
votes
1answer
17 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 ...
0
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1answer
51 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 ...
2
votes
0answers
15 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 ...
0
votes
2answers
39 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 ...
0
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1answer
34 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) = ...
1
vote
0answers
12 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 ...
0
votes
1answer
40 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 ...
0
votes
1answer
36 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 ...
0
votes
1answer
24 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 $\varphi$ ...
0
votes
1answer
25 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 ...
0
votes
1answer
54 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 ...
0
votes
1answer
15 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: ...
4
votes
1answer
54 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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votes
2answers
72 views

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 ...
1
vote
1answer
29 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 ...
1
vote
1answer
33 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 $T_{1}, T_{2}\in B(H)$, ...
1
vote
1answer
39 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 ...
1
vote
1answer
63 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 $ ...
4
votes
1answer
26 views

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, $C^*(S_3)=C_c(S_3)$ ...
4
votes
1answer
56 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 $$\bigcap_{p\in{\cal P}}\{x\colon p(x)=0\}=\{0\}.$$ ...
2
votes
1answer
56 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 ...
0
votes
0answers
46 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 ...
5
votes
0answers
191 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 \ ...
2
votes
2answers
73 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 ...
1
vote
1answer
41 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: ...
3
votes
1answer
59 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 $\mathbb{R}$. ...
2
votes
1answer
35 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 $T_n \stackrel{WOT}{\rightarrow}T$ ...
2
votes
1answer
35 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 ...
0
votes
0answers
23 views

Homeomorphism from $X$ onto $\hat{\mathcal{A}}$.

STATEMENT: For each $f ∈ A$ let $f$ be the function from $\hat{\mathcal{A}}$ to $\mathbb{R}$ defined by $f(\phi) = \phi(f)$. Put on $A$ the initial topology from the collection of all the functions ...
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0answers
29 views

Prove that there is no invertible function in the kernel of a linear functional.

STATEMENT: Use the compactness of $X$ to show that if there is no point $x_0$ such that $g(x_0) = 0$ for all $g ∈ \mathcal{N}$ , then $\mathcal{N}$ contains an invertible function. Some ...
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vote
0answers
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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 ...
0
votes
1answer
24 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 ...
1
vote
1answer
32 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 ...
0
votes
1answer
30 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 ...
1
vote
1answer
31 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 ...
2
votes
1answer
22 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 ...
4
votes
0answers
47 views

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 ...
1
vote
1answer
34 views

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 ...
2
votes
1answer
34 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, ...
0
votes
2answers
89 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 ...