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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Partial Isometries: Characterization

Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: $$W^*W,WW^*$$ One derives a partial isometry: ...
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Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
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A symmetric algebra that is not a C* algebra

Recall that a commutative Banach $*$-algebra $A$ is called symmetric if the Gelfand transform replaces involution in $A$ by complex conjugation in $\mathbb{C}$. Moreover, any commutative C* algebra is ...
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Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
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States: KMS-Condition

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state $\omega$. Does it suffice to have on a dense set the KMS-condition: ...
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Commutators of Schur polynomials of Lie algebra elements

Question: Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
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Homotopy classes of $*$-morphisms and unital $*$-morphisms

Let $A$ and $B$ be C*-algebras (non necessarily unital). A homotopy between two $*$-morphisms $\phi,\psi:A \to B$ is a $*$-morphism $A \to C([0,1],B)$ such that you can recover $\phi$ and $\psi$ from ...
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Tensor Products of C*-Algebras

If A, B are C*-algebras, show that there exists a unique $*-isomorphism $ $ ‎\theta‎‎: A ‎\otimes‎_{*}‎‎ B ‎\longrightarrow‎ B ‎\otimes‎_{*}‎‎ A $ such that $\theta( a \otimes‎_{*} b) = b ...
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Question about $C_0(X)$-algebras and $C_b(X)$.

Let $X$ be a locally compact Hausdorff space. Denote by $C_0(X)$ its C*-algebra of continuous functions that vanish on infinity and by $C_b(X)$ its C*-algebra of bounded functions. Now, let $A$ be a ...
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Spectral decomposition of a an Operator on $\mathcal{L}(l^2(\mathbb{N})$

I have an exercise, where I do not really know how to solve it. Let $\{\lambda_n\}_n$ be a counting of the rational numbers in $[0,1]$ and define a bounded self-adjoint operator $T$ on ...
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Self adjoint operator has spectrum around 0 and 1

I need to prove the following statement: Let $\mathcal{A}$ be a unital $C^*-$Algebra, $A$ a self-adjoint element and $P$ a projection, so $P^2=P=P^*$. Let $\delta :=\|P-A\|$. I want to prove that ...
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Power of positive operator

Let $H$ be a complex Hilbert space and $B(H)$ be the space of all bounded linear operator on $H$. Let $T\in B(H)$ be a positive operator ($\langle Tx,x\rangle\geq0$ for all $x\in H$) and $\alpha\in ...
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States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
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Proving the unitary relation of ensemble decompositions

I apologize if this is better suited for physics.stackexchange; looking at previously asked questions it seems as if this would be a good fit here, as the arguments are mostly mathematical. In my ...
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States: Approximate Unit

Given a C*-algebra $\mathcal{A}$. Consider a state: $\omega\geq0$ Especially one has: $\sup\omega(E^2)=\|\omega\|$ Can it actually fail to be a proper limit? The problem is that the square is not ...
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Positive Elements: Square Root

Given a C*-algebra $\mathcal{A}$. Consider positive elements: $A\geq0$ Why does the square root lie within: $\sqrt{A}\in\mathcal{A}_A$
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Positive Elements: Norm (Decomposition)

Given a C*-algebra $\mathcal{A}$. Then every element decomposes into: $Z=X_+-X_-+iY_+-iY_-=\sum_{\alpha=0\ldots3}i^\alpha Z_\alpha$ Obviously, one has: $\|Z\|\leq\sum_{\alpha=0\ldots3}\|Z_\alpha\|$ ...
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States: Positivity vs. Continuity

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$. Consider a ...
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States: Extension

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$ and adjoin a unit ...
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An elementary perturbation result in C*-algebra

The following question was raised when I read a papaer "MF actions and K-theoretic dynamics". In one of the proofs in that paper, the author utilized a so called "elementary perturbation result in ...
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On the exponential form of a unitary matrix

A unitary matrix $U \in \large C_{n,n}$ can always be written in the exponential form: $U = e ^{iA}$ (1) where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary ...
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Approximate Identity: Projections

Problem Given a C*-algebra $\mathcal{A}$. Denote the positive open unit ball by: $\mathcal{B}^+$ Then it has an approximate identity: ...
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Positive Elements: Decomposition?

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Then every selfadjoint element decomposes into positives ones: $$A=A^*:\quad A=\frac12(|A|-A)-\frac12(|A|-A)\quad\left(\frac12(|A|\pm ...
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Dynamics: EQ-States vs. NESS-States

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state that relaxes towards equilibrium: $$\omega_T(A):=\omega\circ\tau^T(A)\stackrel{T\to\infty}{\to}\omega_\infty(A)$$ Then it ...
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Questions about definitions of crossed product $C^\ast$-algebras and group $C^\ast$-algebras

In Dana P. Williams's book Crossed Products of $C^\ast$-Algebras, the author defined the crossed product of a $C^\ast$-algebra $A$ by a local compact group $G$ as the completion of $C_c(G,A)$ with ...
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A sufficient condition for a strictly positive linear map

Let $\mathcal{A}$ and $\mathcal{B}$ be ${\rm C}^*$-algebras, $\psi: \mathcal{A} \to \mathcal{B}$ a linear positive map (i.e. $a \ge 0 \Rightarrow \psi(a) \ge 0$) and $p \in A$ projection such that ...
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A question about the extension of homomorphism

Here is a quotation of a book "C*-algebras and Finite-Dimensional Approximations" P275 Let $A$ be a C*-algebra. Suppose $I\triangleleft A$ be a closed ideal. If the representation ...
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An exercise about nondegenerate representation

Suppose $I\subset J$ is an inclusion of nonunital C*-algebras such that some approximate unit $\{e_{n}\}\subset I$ is also an approximate unit of $J$. If $J\subset B(H)$ be a nondegenerate ...
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trace identities for the functional calculus

I'm sorry if this is a trivial question, but I cannot convince myself of why $\text{Tr}\,f(EFE)=\text{Tr}\,f(FEF)$ for projections $E$ and $F$ on a Hilbert space and a (say, continuous or Borel) ...
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about ultrapowers

Let $\mathcal{R}$ denote the hyperfinite type $II_{1}$ factor, with $\mathcal{R}^{\omega}$ the ultrapower of $\mathcal{R}$, in respect to some ultrafilter $\omega$ on $\mathbb{N}$. I'm reading a paper ...
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A question about the definition of group $C^\ast$-algebra

Let $G$ be a local compact group, then group $C^\ast$-algebra of $G$ is defined as the completion of $C_c(G)$ with respect to some norm. By now, I have seen three norms. $\|f\|=\sup\|\pi(f)\|$, ...
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Prove a factor is III$_{\lambda}$ type

This question is from the Sunder's book An Invitation to von Neumann Algebras Ex 4.2.13 Let M be a semifinite factor with fns trace ${\tau}$. Let ${\theta}$ be an automorphism of M such that ...
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A question about essential ideal

Let $I$ be a nonunital C*-algebra and $I\subset B(H)$ be any nondegenerate representation and define $$M(I)=\{T\in B(H): Tx\in I~and ~xT\in I, ~for ~all~ x\in I\}.$$ Then, how to prove $I$ ...
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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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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 ...
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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 ...
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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.
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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, ...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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 ...