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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The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
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Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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1answer
37 views

Continuity of functional calculus

Let $\mathcal{A}$ be an unital C*-Algebra. $a,b$ be normal elements in $\mathcal{A}$. $X\subset \Bbb C$ is a compact subset. $f:X\rightarrow \Bbb C$ is continuous. I need to show that for all ...
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Is a von Neumann algebra just a C*-algebra which is generated by its projections?

von Neumann algebras have the nice property that they are generated by their projections (the elements satisfying $e = e^{\ast} = e^2$) in the sense that they are the norm closure of the subspace ...
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34 views

Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
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31 views

Weak convergence of bounded operators

so let $X$ be a Banach space then we say that $A_n \in L(X)$ converges weakly to $A \in L(X)$ if for all $y \in L(X)^*: y(A_n) \rightarrow y(A).$ On the other hand, I just read that weak convergence ...
3
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1answer
76 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
23 views

Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ is not connected. Prove that $A$ contains a nontrivial idempotent $z$.

While trying to solve the exercise below, I came up with a wrong conclusion, but I can't see why it's wrong. Also I'm accepting suggestions to get the right solution. This is the problem 17 from ...
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33 views

Lebesgue measurable integration, density

Let $\mathbb{T}$ be the unit circle and $\lambda$ be the Lebesgue measure on $\mathbb{T}$. Let $A_n := e^{2\pi i[1/2^{2n},1/2^{2n+1}]}$, $n\ge 1$. Define a function $f$ on the set of all the Lebesgue ...
2
votes
1answer
13 views

Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take ...
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1answer
27 views

Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra ...
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18 views

The necessity of defining the stable equivalence in the construction of the Grothendieck group $K_0$

I am confused about the process of the construction of the Grothendieck group $K_0$ in Murphy's $C^*$-algebras and operator theory section 7.1. Let $A$ be a $*$-algebra and ...
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1answer
13 views

positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$

Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$ Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ ...
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1answer
17 views

Existence of minimal projection in a sub-algebra of compact operators

I am not sure that I explained to myself the missing details in the proof right, so please check my explanations. (The proof is taken from "$C^*$ -Algebras by Example"-Davidson) First, I don't know ...
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16 views

semifinite von Neumann algebra, spectral projection, trace

Let $\mathcal{M}$ be a semifinite von Neumann algebra and $\tau$ be a semifinite faithful normal trace on it. Let $T,P_1,P_2\in \mathcal{M}$, where $P_1,P_2$ are projections with $P_1\perp P_2$. Then, ...
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vote
2answers
32 views

positive operator, projection, Hilbert space

Let $T$ be a positive operator on a Hilbert space $H$. Let $P$ be a projection on $H$. Then, it is well-known that $PTP$ is also positive. My question is: whether $T\ge PTP$?
3
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1answer
51 views

About a relation between isometries

If we have $(T_i)_{i=1}^N$, operators on a Hilbert space, that are also isometries and satisfy the following relation: $$\sum_{i=1}^NT_iT_i^*=Id\quad (1)$$ How can you prove that they must also ...
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0answers
17 views

Show that $(Tf)(x)=\int_0^{x} {t \cdot f(t)} dt$ maps from $L^{1}(0,2)$ $\rightarrow$ $L^{1}(0,2)$

Given the mapping $T:L^1(0,2) \rightarrow L^1(0,2), (Tf)(x)=\int_0^{x} {t \cdot f(t)} dt$ Show that T actually maps from $L^{1}(0,2)$ $\rightarrow$ $L^{1}(0,2)$. I have been given the hint that i ...
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0answers
18 views

Injectivity of index map for $K_1(S^1)$

This example/problem is from Valette's notes on the Baum-Connes conjecture (p. 45). The exercise is to prove that the (trivially equivariant) $K$-homology group $K_1(S^1)$ is $\mathbb{Z}$. For this, ...
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1answer
20 views

Types of W*- algebras

Searching to find a reference for "homogeneous type $I_{{\aleph}_0}$ W*-algebra", I was not successful. Please guide me. Thanks in advance.
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1answer
331 views

Reduced Crossed Products

Given a discrete group $ G $ and a $ G $-$ C^{*} $-algebra $ A $, we can form the reduced crossed product $ A \rtimes_{\operatorname{r}} G $. I want to define it as the closure of the embedded image ...
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7 views

Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
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1answer
52 views

Hermitian Pairings from Positive Functionals

Let $A$ be $*$-algebra and $\phi:A \to {\mathbb C}$ a positive linear functional, that is, one for which $\phi(aa^*) \geq 0$, for all $a \in A$. When does it hold that a symmetric sesquilinear form, ...
3
votes
1answer
498 views

Minimal projections in a C* algebra

Let $e$ be a projection in a C* algebra $A$. Is $eAe= \mathbb{C}e$ equivalent to the nonexistence of any projection in between $e$ and $0$? I know it is true if $A$ is a Von Neumann algebra because ...
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1answer
30 views

Index of an element in C*-algebra

Suppose that $x$ is an element of abstract $C^*$-algebra $A$. For example if $x$ is normal, i.e. $x^*x=xx^*$ then if we use any representation $\pi$ of $A$ on some Hilbert space $H$ then $\pi(x)$ will ...
3
votes
1answer
94 views

Is there an example of a non von Neumann algebra with this property?

What is an example of a $C^{*}$ subalgebra $A$ of $B(H)$ such that $A$ contains the identity $I_{H}$ and satisfies the following properties: 1) For every $T\in A$, The orthogonal projection ...
2
votes
1answer
39 views

What is the motivation for Murray-von Neumann equivalence

Definition: If $p,q$ are projections in a $C^*$-algebra $A$, we say that they are Murray-von Neumann equivalent, and we write $p\sim q$, if there exist $u\in A$ such that $p=u^*u$ and $q=uu^*$. I ...
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1answer
74 views

Most natural equivalence between $C^*$-algebras

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
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1answer
47 views

reduced density matrix for the given composite system

Given the composite system of two qubits $$ |\psi^{AB}\rangle=\frac{1}{\sqrt{2}}(|0^{A}\rangle \otimes|0^{B}\rangle+|1^{A}\rangle\otimes|1^{B}\rangle) $$ with the density matrix of the composite ...
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votes
2answers
99 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
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1answer
41 views

Singular value decomposition of sum of single particle operators

here is my question. Suppose to have an operator $L$ in a composite Hilbert space $A\otimes B$ which can be written as sum of single particle operator as: \begin{equation} L = (L_0\otimes \mathbb{I} + ...
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Does there exist a countable basis for the space of all continuous meta operators?

Consider operators $O: \left( f: \mathbb{R} \rightarrow \mathbb{R} \right) \rightarrow (f: \mathbb{R} \rightarrow \mathbb{R})$ We define a continuous operator $O$ over an interval $[a,b]$ if: $$ ...
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21 views

Second dual of Calkin algebra

1- I am looking for any information concerning the second dual of Calkin algebra $\frac{B(H)}{K(H)}$. 2- What about the non-degenerate representations of the Calkin algebra?
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1answer
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Is $\mathbb{C}^n$ a C*-algebra?

Hi I am new in learning C*-algebra. I know that under the usual $\mathbb{C}^n$-norm, $\mathbb{C}^n$ is a Banach Space. However, what will be an intuitive multiplication on $\mathbb{C}^n$ so that the ...
2
votes
1answer
29 views

How can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\mathrm{End}V)[[x,x^{-1}]]$?

In the context of vertex operator algebras, if $V$ is a vector space, how can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\operatorname{End}V)[[x,x^{-1}]]$? The notation $V((x))$ is the ...
2
votes
0answers
47 views

An strongly open set which is not measurable in the weak operator topology.

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a proper subset in $I$. Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: j\in J\}$$ ...
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24 views

Eigenvalues of an exponential operator, quick question

\begin{equation} A= e^{Q^*Q} \end{equation} Q is diagonalizable, with eigenvalues q. Show that the eigenvalues of A have a lower bound. What's the minimal one? Thanks!
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vote
1answer
23 views

Two identities concerning reduced crossed products

I am currently writing a project wherein I need two identifications. Let's fix some notation. Let $\Gamma$ be a discrete group acting on a unital C*-algebra A by $\ast$-automorphisms $\alpha: A ...
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vote
1answer
32 views

Positive Map: Reduction

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. (Both possibly nonunital!) Linear Map: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\varphi\in\mathcal{L}$$ Implication: ...
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1answer
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Is this a characterization of commutative $C^{*}$-algebras

Assume that $A$ is a $C^{*}$-algebra such that $\forall a,b \in A, ab=0 \iff ba=0$. Is $A$ necessarily a commutative algebra? In particular does "$\forall a,b \in A, ab=0 \iff ba=0$" ...
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Why is every positive linear map between $C^*$-algebras bounded?

We know that every positive linear functional on a $C^*$-algebra is bounded. How can we prove every positive linear map between $C^*$-algebras is bounded?
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Classification of vector bundles over the torus

In M. Rieffel's paper "The Cancellation Theorem for projective modules over irrational rotation $C^*$-algebras", he classifies finitely generated projective modules over the $C^*$-algebra ...
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1answer
86 views

Normal Operators: “Zorn”

Problem Given a Hilbert space $\mathcal{H}$. Normal Operators: $$\mathcal{N}(\mathcal{H}):=\{N:N^*N=NN^*\}$$ (Possibly unbounded!) Borel Calculus: ...
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1answer
19 views

What are some specific operations for matrices, like is dot product for vectors?

I read this question and the associated answer (What is the dot product between a vector of matrices?), and it wasn't what I am looking for. I want to know if there are some special operations ...
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15 views

about a definition in the representation of $L^1(G)$, where $G$ is locally compact

I try to learn group C*-algebra by reading the Chapter 7 in the book C*-algebras by example. It says that if $\pi$ is a unitary representation of $G$, then it induces a representation of $L^1(G)$ by ...
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1answer
24 views

When set of projections is closed under multiplication

Let $\mathcal A\subset B(H)$ be an unital $C^*$ algebra of operators on a Hilbert space $H$. Let's denote by $\mathcal P$ the set of projections in $\mathcal A$, that is $\mathcal P:=\{a\in \mathcal A ...
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Relation of fundamental group $\pi_1(X,x_0)$ and properties of C(X)

As stated by commutative Gelfand Naimark theorem, every unital C* algebra is of the form C(X) for some compact Hausdorff space X. Moreover two such algebras are isomorphic iff the corresponding ...
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1answer
87 views

Proving an isometric dilation of a non unitary operator on Hilbert space implies infinite dimensional space involving matrices

I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows: Let H be a Hilbert space. We are to prove, in two distinct ways, that ...
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1answer
20 views

irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book "A short course on spectral theory", William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be ...
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1answer
35 views

Uniqueness of positive square root of postive element in C* algebra

If a is a positive element then it has a unique positive square root, i.e. a unique b positive such that b^2=a. I understand the existence part of the proof. If ...