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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Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
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Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A $ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...
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How to prove the following map is a c.c.p map

Here is a quotation in a book "C*-algebrass and Finite-Dimensional Approximations" by Nate and Taka (P122). Let $A$ be a C*-algebra, $\Gamma$ be a discrete group and the $\alpha$ is an action of ...
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Is $vN(M_1,M_2) \cap M_3= vN(M_1,M3) \cap vN(M_2,M3)$?

Let $M_1,M_2,M_3$ be von Neumann algebras (i.e. weakly closed subalgebras of $B(H)$ where $H$ is a Hilbert space). Let $vN(M_1,M_2)$ denote the von Neumann algebra generated by $M_1$ and $M_2$ inside ...
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37 views

An exercise about reduced crossed product of a C*-dynamical system

Here is an Exercise in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Exercise 4.1.3. Let $A$ and $B$ be two C*-algebras and $\Gamma$ be a discrete group. If ...
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A question about cross product of C*-dynamical system

Here is a Theorem (P123) in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Theorem 4.2.4. Let $A$ be a C*-algebra and $\mathbb{Z}$ be the set of integers. ...
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Proof of operator norm with the equality of involution

Given the equality $$\|A^*A\| = \sup_{\| x \| = \| y \| = 1} | (Ax, (A^*)^*y) | $$ How do show that it is equal to $\|A\|^2$ Is it by using Cauchy Schwartz inequality such that $(x, y) \leq ...
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Why is the Gelfand transform injective?

There is a theorem that proves that if $A$ is a commutative C*-Algebra, the Gelfand map is an isometric *-isomorphism of $A$ onto $\hat{A}$ i.e. the spectrum of $A$. (Theorem 1.1 in Averson's "An ...
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Abelian projection in the von-Neumann algebras

As we know, a minimal projection must be Abelian, An Abelian projection must be finite. A minimal projection correspond to a rank one operator, a finite projection correspond to a finite rank ...
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Show the algebra of all bouned operators is a $W^*$-algebra

Given that the algebra of all bounded operators $B(H)$ on Hilbert space is a $C^*$-algebra such that $\| A^*A \| = \|A\|^2$. Then how can we show that it is also a $W^*$-algebra, where a $W^*$-algebra ...
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$C^{*}$ algebras positively dominated by finite dimensional algebras

Assume that $A$ is a $C^{*}$ algebra and $B$ and $C$ are two sub $C^{*}$ algebras of $A$ such that: $B$ is finite dimensional algebra. For all positive $c\in C$, there exist a positive $b\in ...
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$R\mbox{ is a right multiplier and }R(a)b=a\overset{?}{\implies} A\mbox{ is unital }$

Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that $$ \exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad $$ implies $A$ is unital. I know this is true if A is a ...
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Spectrum of projection

I have a question concerning the spectrum of a projection in an abstract $C^*-$algebra. Let $A$ be $C^*-$algebra, $p\in A$ a projection (i.e. a selfadjoint idempotent). Can we say what is the spectrum ...
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Topologizing $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$

Given a separable Hilbert space $\mathcal{H}$ I would like to know how one could topologize the quotient algebra $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$? Here ...
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Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.

If any of the terminology is unclear then please don't hesitate to point it out. My question is: is it true that when $G$ is a locally compact second countable group then: \begin{equation*} C_0(G) ...
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II$_1$-factors with finite commutant and trivial intersection generate $B(H)$? [migrated]

Let $H$ be an $\infty$-dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Let $\mathcal{A}$, $\mathcal{B} \subset B(H)$ be II$_1$-factors such that $\mathcal{A}'$, ...
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Do two II$_1$-factors with trivial intersection generate $B(H)$?

Let $H$ be an infinite dim. separable Hilbert space and $B(H)$ the algebra of bounded operators. Let $A$, $B \subset B(H)$ be II$_1$-factors such that $A \cap B = \mathbb{C}I$. Examples: (1) Take ...
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Question on amenable direct summand

Given a finite Von Neumann algebra $(N,\tau)$, and Von Neumann subalgebras $A\subset B$ with the same identity, I came across the fact saying that $B$ has an amenable direct summand implies $A$ has an ...
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Where is the most clear and concise exposition of the spectral theorem for self-adjoint operators on Hilbert space?

This question is certainly subjective, which may warrant votes to close. I'm simply looking to find the "best" written exposition of the spectral theorem for possibly unbounded self-adjoint operators ...
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“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
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find element in relative commutant of a matrix subalgebra

Let $M=A*P$ to be a free product von Neumann algebra, and $A$ is a finite dimensional subalgebra, for simplicity, we may assume $A=M_2(\mathbb{C})$, and $P\neq \mathbb{C}$. A standard fact is that ...
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Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
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Understanding minimal projections

This might be very easy but it is not quite clear for me. Detailed explanation appreciated! I went through the commutative case but beyond that I lack intuition. Let $A$ be a C*-algebra and let ...
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1answer
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If $A$ is a primitive / irreducible C*-algebra, then $M(A)$ has trivial center.

Recall some definitions: a sub-C*-algebra $A$ of $B(H)$, the algebra of bounded operators on a Hilbert space $H$, is called irreducible if the only closed $A$-invariant subspaces of $H$ are $0$ and ...
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Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
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von Neumann Algebras and measures

I read that any abelian von Neumann algebra is isomorphic to $L^\infty(X,\mu)$ for some $X$ and $\mu$. This seems to be reasons, to consider any von Neumann Algebra as non-commutative measurable ...
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A question on nuclearity

Definition 2.1.1. If $A$, $B$ are C*-algebra, a map $\theta: A\rightarrow B$ is called nuclear if there exist contractive completely positive maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ and ...
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55 views

A simple lemma in tensor product

Here is a quotation of a book: ($\otimes$ denotes the minimal tensor product) Lemma 3.9.2. Let $A$ be a C*-algebra. If $E\subset A$ is an operator system and $J\triangleleft B$ is an ideal, then ...
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Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
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Faithful Representations of C*-algebras

Can anyone give me an example of a represetation of the algebra $M_n(\mathbb{C})$ that is not faithul? If it's not possible, could you explain me why it is not?
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No trace on $B(H)$ if $H$ is infinite dimensional

Let $H$ be an infinite dimensional Hilbert space and $B(H)$ the bounded linear operators on $H$. Then thre is no ultra weakly continous non-zero positve trace $tr:B(H)\rightarrow \mathbb{C}$. I ...
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Ultra weakly continuous trace on a von Neumann Algebra

Let $M$ be a infinite dimensional von Neumann Algebra with a positive, faithful, ultra weakly continuous trace $tr:M\rightarrow \mathbb{C}$. Is it possible to show that $tr$ is strongly continuous?
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Maximal ideals and maximal subspaces of normed algebras

This is a kind of "prove or give a counter-example" question, and I'm having some difficults with it: By a maximal ideal $I$ of an algebra $A$, we mean an ideal $I\neq A$ which is not properly ...
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1answer
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A question on a lemma about the product map

Here is a Lemma in the book “C*-algebras and Finite-Dimensional Approximations”: Lemma 3.8.4. Let $A$ be a C*-algebra, $M\subset B(H)$ be a con Neumann algebra and $\phi: A\rightarrow M$ be a ...
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1answer
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Why is $K_{0}(C(\mathbb{D}))\rightarrow K_{0}(C(\mathbb{T}))$ injective?

There is the restriction map $\pi:C(\mathbb{D})\rightarrow C(\mathbb{T})$ where $\mathbb{D}$ is the closed unit disk and $\mathbb{T}$ is the unit circle. Why is $\pi_*:K_0(C(\mathbb{D}))\rightarrow ...
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1answer
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Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continious normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
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1answer
16 views

Lifting a unitary to a partial isometry

What is an example of a unital $C^*$-algebra $A$ and an ideal $I$ such that some unitary element in $A/I$ cannot be lifted to a partial isometry in $A$? Or can it be shown using general properties of ...
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Examples of hyperstonean space

From the abelian von Neumann algebra, I see the hyperstonean space as its spectrum(analogy with the C*-algebra). Now I want to see some examples of hyperstonean space. 1) Can you give me a ...
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75 views

Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 ...
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1answer
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Minimal projections and Type II von Neumann Algebras.

Let $M \subseteq B(H)$ be a type $II_1$ factor. Can it contain a minimal projection? If it can't, what would go wrong? I assume something about the trace being faithful?
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Weyl Operators: Discontinuity

Let $\mathcal{A}_{CCR}(\mathcal{H})$ be a CCR algebra over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary and therefore $\sigma(W(f))\subseteq \mathbb{S}$ so by the spectral ...
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Bogoliubov Transformation

Let $\mathcal{A}_{CAR}(\mathcal{H})$ be a CAR algebra over a Hilbert space $\mathcal{H}$. Consider a linear $S$ and an antilinear $T$ both bounded operators acting on $\mathcal{H}$ satisfying: ...
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Ultra weakly closed *-subalgebra of B(H)

I'm currently working on a text about von Neumann algebras and the author used without further clarifying that any ultra weakly closed *-subalgebra of $B(H)$ contains a largest projection. Could ...
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Short exact sequence involving mapping cone, cone, suspension of $C^*$-algebras

This is part of exercise 6.N in Wegge-Olsen's book '$K$-theory and $C^*$-algebras'. In the following, $A$ and $B$ are $C^*$-algebras, $\alpha:A\rightarrow B$ is a surjective $C^*$ morphism with kernel ...
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Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
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Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentialky selfadjoint if it contains a dense subset of analytic vectors?
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A question on tensor product of $C^{*}$ algebras

Let $A$ and $B$ be two $C^{*}$ algebras. Assume that every element of the minimal tensor product $A\otimes_{min} B$ is a finite linear combination of simple tensors $a\otimes b$. Can we say that ...
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Idempotent operators.

Apologies first. I am a physicist and my notations and rigour is probably lousy. If $P$ is an idempotent operator, $P^2 = P$, $P\neq \mathbb1$ and we have $\forall |\psi\rangle$ the relation, $P.L ...
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61 views

C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
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1answer
40 views

Kadison's Inequality

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$\overline{\omega(A)}\omega(A)\leq\omega(A^*A)$$