# Tagged Questions

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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### Ordering: Definition

This was a real question! Given a unital C*-algebra $1\in\mathcal{A}$. For $A\in\mathcal{A}$ denote its spectrum by $\sigma(A)$. Consider the selfadjoints: ...
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### An strange trace class operator

Assume $H$ is a non separable Hilbert space. 1- Let $\{a_n\}$ be a sequence of bounded linear operators on $H$. Any operator $a_n$ induces the bounded linear functional $x\to tr(xa_n)$ on $L^1(H)$, ...
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### Gelfand-Naimark for $C^*$-categories

What is a reference for the following Theorem? If $A$ is a small $C^*$-category, then there is a faithful $C^*$-functor $A \to \mathsf{Hilb}$. $C^*$-categories with exactly one object are just ...
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### Unit of a purely infinite, simple C*-algebra

Suppose that we have a purely infinite, simple C*-algebra with unit $1$. Can we find two projections $p,q$ both equivalent to the identity such that $1=p+q$ and $pq=0$? Well, there are two ...
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### A question about corners of $C^\ast$-algebras

Let $\mathcal{A}$ be a $C^\ast$-algebra, $p\in M_n(\mathcal{A})$ a projection, is there a $k\in\mathbb{N}$ such that $pM_n(\mathcal{A})p\cong M_k(\mathcal{A})$ ? Thanks a lot!
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### States on a non-unital $C^*$-algebra

Let $\mathcal{A}$ be a unital $C^*$-subalgebra of $B(H)$. Then the definition $\phi(a):=\langle ah,h \rangle$ for a fixed $h\in H, \|h\|=1$ and for all $a\in\mathcal{A}$ defines a state on ...
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### When can $*$-algebras be turned into $C^*$-algebas?

Let $A$ be a (not necessarily unital) complex $*$-algebra, i.e. an algebra over $\mathbb{C}$ together with an involution $*: A \to A$. There exists at most one norm on $A$ turning $A$ into a ...
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### Matrices over the Cuntz algebra

Consider the Cuntz algebra $O_2$. Is it true that $M_2(O_2)$ is isomorphic to $O_2$? I was trying to show that is impossible but now I am not sure.
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### What, and how can, topological invariants can be computed from a space's algebra of functions?

The Gelfrand duality says that the category of locally compact Hausdorff spaces (with proper continuous functions) is equivalent to the category of commutative $C^*$ algebras (with proper ...
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### Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
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### Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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### Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
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### composition and strong limits of completely positive maps is completely positive

I have two claims about completely positive maps. Let $X$, $Y$, $Z$ be $C^\ast$-algebras. 1) Let $f:X\to Y$ and $g:Y\to Z$ be completely positive maps. I want to know, why $g\circ f$ is completely ...
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### the $C^\ast$-algebra $M_n(A)$, understanding the $C^\ast$-norm on $M_n(A)$

Let $A$ be a $C^\ast$-algebra. I want to understand $M_n(A)$, the vector space of $n\times n$-matrices with entries in $A$, as a $C^\ast$-algebra. On $M_n(A)$ you can define an involution ...
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### The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
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### Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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