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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Gelfand Topology and C*-algebras

Before we start here some notations to have no confusion: Suppose $A$ is a commutative $C^*$-algebra with unit. $\Sigma(A)$ is the Gelfand spectrum, given by all linear maps ...
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When is a $*$-homomorphism between multiplier algebras strictly continuous?

The strict topology on the multiplier algebra $M(A)$ of a C*-algebra $A$ is that generated by the seminorms $$ x\mapsto \| ax \|\qquad x\mapsto\| xa \| \qquad (x\in M(A), a\in A) $$ Whereas a ...
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98 views

Spectrum in Banach Algebra

Let $A$ be a unital Banach algebra and $a\in A$. Let $U$ be an open subset of $\mathbb C$ containing $\sigma (a)$. Prove that there is $\delta>0$ such that for every $b\in A$, if ...
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602 views

Trace of an operator

Suppose $\rho$ is a positve trace-class operator and $x$ is positve bounded operator on a Hilbert space $\mathcal{H}$, I am unable to prove that trace of $\rho x$ is positive, where trace($x$):= $\sum ...
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maps from a convex set to itself

Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ ...
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Double centralizers in the Murphy book

I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake: ...
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78 views

Trace-preserving $\Rightarrow$ Norm-preserving?

Let $\theta\in\text{Aut}(\frak{M},\tau)$, where $\frak{M}$ a von-Neumann algebra and $\tau$ a faith, finite, normal trace. Does $\theta$ preserve the norm structure? (Here assumed, that $\frak{M}$ ...
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78 views

A reference request for sums of $C^*$-algebras

Does anyone know where I can find a reference for the following well-known fact: Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s. Then ...
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207 views

Composition of Fredholm Operators

If $ST$ is a Fredholm operator, then show that $T$ is Fredholm if and only if $S$ is Fredholm.
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59 views

Given $\theta(p)\neq p$ does there exist $q\leq p$, so that $\theta(q)q=0$?

Let $\mathfrak{M}$ be a vN-Alg. Let $\theta\in \text{Aut}(\mathfrak{M})$. Let $p\in\mathfrak{M}$ be a projection, so that $\theta(p)\neq p$. Is there a projection $q\in\mathfrak{M}$ with $0\neq q\leq ...
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75 views

What's the application of C*-algebra in topology?

C*-algebras are thought be be non-commutative topological spaces because of Gelfand's theorem that any commutative C*-algebra are isomorphic to C(X) for some locally compact Hausdorff space X. I've ...
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45 views

Kernel inclusion implies factorization

I have a question whether a certain fact is true for arbitrary operators on a Hilbert space. Namely, consider Hilbert spaces $H,K$, an operator $A\in B(H)$ and another $B\in B(H,K)$. Moreover, assume ...
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Polar decompostion for the operator algebras

I find that most of books discussing the polar decompostion at the W*-algebras, but not C*-algebras. I guess the rough reason is that the element of W*algebras has the well supported set, but I want ...
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68 views

Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?

Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
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172 views

Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
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56 views

basic question on normed algebra

Prove the multiplication map $m:A\times A\rightarrow A$, sending $(x,y)\rightarrow x*y$ is jointly continuous in a normed algebra. i can't understand what's jointly cont.? this is a problem from ...
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41 views

Skew-symmetric unitary on $\mathcal{B(H)}$

We know that there exists skew-symmetric unitary on $\mathcal{B(H)}$ when $\mathcal{H}$ is of even dimensions. In particular for $\mathcal{H}=\mathbb{C}^2$, any such matrix is scalar multiple of Pauli ...
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Spectral measures

Let $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a spectral measure on the Borel $\sigma$-algebra $\Sigma$ of $\mathbb{C}$. Assume also that $E$ is compactly supported in the sense that ...
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42 views

Properties of a vector operator

Suppose I have a vector operator which angle dependence is given by $$\hat O(\theta)=A\sin\theta+B\cos\theta+C$$ What can I say about $\hat O$? Sorry, I do realize that it is a bit vague. Assume $\hat ...
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152 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
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1answer
81 views

Is the cone of squares in a Jordan algebra a cone?

Let $A$ be a finite-dimensional Jordan algebra over $\mathbb{R}$, i.e. a finite-dimensional real vector space with a commutative bilinear product $\circ: A \times A \rightarrow A$ satisfying $(a^2 ...
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155 views

Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
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152 views

Inequality - tensor product, Hilbert spaces

Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{K}$ be a Hilbert space with an orthonormal basis $\{ e_i \}_{i \in I}$. Let $A$ be bounded linear operator from $\mathcal{H} \otimes \mathcal{K}$ ...
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K-theory for non-separable C*-algebras

Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...
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63 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
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111 views

Seek results in group theory obtained by applping algebraic topology tools

After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have ...
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64 views

Positive maps on $\mathcal{B}(\mathcal{H})$ to itself

Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
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100 views

Time-derivative of an operator

Would I be right in thinking that the operator $$\hat O'(t)$$ is different from the operator $$D\hat O(t)$$ where $D={d\over dt}$, since when acting on a function $f$, the second corresponds to $$\hat ...
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385 views

Dot products in commutators

Suppose $\hat r$ is an position operator, $\hat p$ is a momentum operator and $\vec c$ is a constant vector. What does the commutator $[\hat p, \vec c\cdot\hat r]$ mean? I see that you can expand ...
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156 views

Abelian von Neumann Algebras on non-separable Hilbert spaces

Is there a classification of Abelian von Neumann algebras on non-separable Hilbert spaces? For a classification of Abelian von Neumann algebras on separable Hilbert spaces, see this link.
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Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
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Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ ...
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Definition by commutation property on structures : continuity and where?

(This is very vague, so sorry if there are approximations) I remember that one can define continuity as a commutation property of a function with the limit operation. Structurally, i think it maps a ...
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105 views

Characterizing positive semi-definite operators in $\mathcal{B}(L^2)$

I am asking perhaps a stupid question. How can I characterize all positive semi-definite operators in $\mathcal{B}(L^2(X,\lambda))$, where $\lambda$ is the Lebesgue measure. For a start, let us ...
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$\mathcal{K}(L^2(\mathbb{R}^m \times \mathbb{R}^n)) = \mathcal{K}(L^2(\mathbb{R}^m)) \otimes \mathcal{K}(L^2(\mathbb{R}^n))$?

QUESTION: Is it true that for the algebra of compact operators: $\mathcal{K}(L^2(\mathbb{R}^m \times \mathbb{R}^n))$ is as a $C^{\ast}$-algebra isomorphic to $\mathcal{K}(L^2(\mathbb{R}^m)) \otimes ...
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Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
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1answer
186 views

Extension of Choi's theorem on extreme completely positive maps

In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below. Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is ...
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Strong continuity of the Borel functional calculus

I have sometimes heard that the Borel functional calculus maps bounded pointwise convergent sequences of Borel functions to strongly convergent sequences of operators. I gather "sequence" is ...
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A looser condition for these operators?

Let $q_i,p_j$ be canonically conjugate operators for $i,j=1,2,\ldots,n$ that satisfy the relation $[p_i,q_j]=c\delta_{ij}$ where $c$ is constant and $[\cdot,\cdot]$ is a commutator. What sort of ...
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1answer
185 views

How to make a topological subspace Hausdorff

If $X$ is a compact topological space then $C(X)$ is a C* algebra. I'm not going to attempt to discuss the locally compact case with $C_0(X)$ because usually the definition of that space requires the ...
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3answers
126 views

What if every function in $C(X)$ has finite spectrum?

Suppose that $X$ is a compact Hausdorf space, and that every continuous function on $X$ has finite range. How do I conclude that $X$ is a finite set, hence with discrete topology? So far, I have ...
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204 views

Transpose of an operator

If $\hat O$ is an operator and $c\in \mathbb C$ Then can I safely say that $[\exp(c\hat O)]^\dagger=\exp(c^*\hat O^\dagger)$? My reasoning is by taking the transpose term by term in the power series ...
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473 views

An application of Artin Wedderburn and Schur's Lemma

In an operator algebras class, the professor said the following. Let $G$ be a finite group, and consider the complex group algebra it generates. Using the following 3 facts, we're supposed to see ...
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Question about Notes on Operator Algebra

This question is concerned with the last line of these notes by M. Fowler titled "Coherent States of the Simple Harmonic Oscillator." I understand the things before that but I don't see how the last ...
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238 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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Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?

Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$. An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space. What are the necessary and ...
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Extreme points in the set of positive linear functionals of norm $\leq 1$

Let $A$ be a C* algebra, and $S$ the set of positive linear functionals on $A$ in the unit ball of $A^*$ (Which has the weak-* topology.) I am having difficulty seeing that all nonzero extreme points ...
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Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $ \geq 0$) that is not ...
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Composition of positive maps

Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map ...
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Abstract Von Neumann Algebras

I have just read this question Is a von Neumann algebra just a C*-algebra which is generated by its projections? and am wondering about Robert Israel's answer when he says that a subalgebra of $C(X)$ ...