Questions tagged [operator-algebras]

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 of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

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Are there free ternary ring of operators?

I am interested in separable ternary rings of operators. For separable $C^*$-algebras we have the maximal group C*-algebra of the free group on countably many generators that quotients onto every ...
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Find a partial isometry in a von Neumann algebra [closed]

Let $M$ be any factor and $\rho$ be a faithful normal state on $M$ such that $M_\rho$ is a type II$_1$ factor. Suppose that $c$ lies in the point spectrum of the modular operator $\Delta_\rho$. Can we ...
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Appendix to Finite von Neumann Algebras and Masas

In Section A.3 of Finite von Neumann Algebras and Masas the authors write "Throughout this section, let $M$ denote a finite von Neumann algebra with centre $Z$ and centre-valued trace $\mathbb{T}$...
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What does it mean for an algebra to be block diagonal?

Trying to understand the proof of the following: Let $\Phi: B(H) \to B(H)$ be a unital quantum channel with Kraus operators $A_i$ for $i = 1, \dotsc, n$. Then the algebra generated by $A_1, \dotsc A_n$...
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Higson's homotopy invariance result

I am learning about operator algebras and $KK$-theory, a result I find very striking is the following : Any split-exact $K$-stable functor $F : C^*\text{-alg} \to \text{Ab}$ is necessarily homotopy ...
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Show that every finite-dimensional $C^*$-algebra can be faithfully represented on a finite-dimensional Hilbert space

Show that every finite-dimensional $C^*$-algebra can be faithfully represented on a finite-dimensional Hilbert space. [Hint: Show that finitely many states suffice for the Gelfand-Naimark theorem.] ...
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Question about continuous functional calculus and its application

I recently started learning about the topic functional calculus. My problem is that I have no idea on how to use it for, say, solving problems, exercises etc. Here is a short review of what I learned ...
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A circular argument in the proof of Proposition 2.3.1 of Bratteli-Robinson

I think the proof of Proposition 2.3.1 of Operator Algebras and Quantum Statistical Mechanics 1 by Bratteli and Robinson, contains a circular argument. The book uses the following inequality \begin{...
1 vote
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Multiplicative linear functional on noncommutative unital Banach algebra is bounded and norm-decreasing

It is a standard theorem that, for an abelian unital Banach algebra, every nonzero multiplicative linear functional is bounded and has norm at most $1$. I don't see why the proof can't work word-by-...
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For $C^*$-algebras A and B is the map $A\times B \to A\otimes_{\min} B$ continuous?

For $C^*$-algebras A and B is the map $A\times B \to A\otimes_{\min} B$ continuous? Where $A\times B$ has product topology and $A\otimes_{\min} B$ the norm topology. If yes, a proof (or reference) ...
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$C^*$ algebra and deformation quantization

I heard that (from ref ) Classical observables : The set of observables $\mathcal{O}$ of a classical systems are exactly the self-adjoint elements of a separable commutative unital $C^*$-algebra. ...
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If $A^*A=A^*B=B^*A=B^*B$, prove that $A=B$.

Suppose that $H$ is a Hilbert space and $A,B\in B(H)$ are such that $A^*A=A^*B=B^*A=B^*B$,then $A=B$.I'm having difficulty trying to read a proof that solves this problem. First, the author decomposes ...
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Is the cut-down of a positive invertible element postive and invertible?

Let $\mathfrak A$ be a unital C*-algebra; let $A \in \mathfrak A$ be such that $\sigma_{\mathfrak A}(A) \subset [a,\infty)$ for some $a > 0$; let $p \in \mathfrak A$ be a (self-adjoint) projection. ...
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Does $f(xy)=f(x)f(y)$ for a continuous function $f$ and normal elements $x,y$ in a C*-algebra?

In words, does a function in the continuous functional calculus behave like a homomorphism? I know that $f^n(x)=f(x)^n$ but that does not help here.
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Question about the support of a normal weight on a von Neumann algebra

Consider the following fragment from Stratila's book "Modular theory in operator algebras": I understand everything in this fragment, except that equation $(3)$ holds. How can one show this ...
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$s(e+f) = e\lor f$ for projections $p,q\in B(H)$

Let $H$ be a Hilbert space and $p,q\in B(H)$ orthogonal projections, i.e. $$p=p^*= p^2, \quad q=q^* = q^2.$$ I want to show that $s(p+q) = p\lor q$ where $s(p+q)$ is the support projection of $p+q$ ...
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states and density matrices, compactness

I just started reading about the $C^*$-algebras so I'm hoping that what I'm writing won't be awfully wrong. I had a doubt regarding the connection between states in an algebra and density operators in ...
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Existence of uniformly convergent sequence of polynomials converging to an analytic function

Let $\Delta_1,\Delta_2,\dotsc,\Delta_n$ be mutually disjoint open discs in the complex plane. Question part 1: Does there exists a sequence of polynomials $(p_n)_{n\in \mathbb N}$ converging uniformly ...
1 vote
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Doubt on exercise on Von Neumann factor

I am trying to solve the last point of the following exercise, but I don't know how to approach it.The exercise says: Let $M \subseteq B(\mathbb{H})$ be a type $\mathbb{II}_1$ factor Von Neumann ...
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Nonunital weak operator closed self adjoit subalgebra has a projection that acts like an identity operator?

Let $A$ be weak operator closed self adjoint subalgebra of the bounded operators for a Hilbert space $H$(may not have identity so cannot use usual von neumann alegbra properties). I have shown that if ...
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1 vote
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double commutant of unitary operators of a Von Neumann algebra.

Let $M$ be a Von Neumann algebra. I am having trouble showing that the double commutant of unitary operators for $M$ is equal to $M$. I have shown it with projections using borel functioncal calculus. ...
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