# Questions tagged [oa.operator-algebras]

Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

1,884
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### A locally convex $C^*$ algebra without zero divisor

Let we have a locally convex $C^*$ algebra $A$. That is $A$ is a TVS equipped with an algebra and an involution structure such that all operations are continuous. moreover the topology of $A$ ...

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### Can any POVM be induced by a quantum instrument?

I suspect this is the obvious result of something in operator algebras, but that's far outside my field.
Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...

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### 2-positivity to 3-positivity

Let $B\in M_3(\mathbb{C})$ and $S_3=
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{pmatrix}
$. Suppose $I_3\otimes I_2+B\otimes X+B^*\otimes X^*\geq 0$ for all $2$...

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### Semi-commutative von Neumann algebras

Suppose $\Omega$ is a $\sigma$-finite measure space with measure $\mu.$ Let $\mathcal M\subseteq B(H)$ be a von Neumann algebra.
Can an element of $L_\infty(\Omega)\overline{\otimes}\mathcal M$ be ...

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### essential numerical range of an idempotent

Notation: $W_e()$ denotes the essential numerical range of an operator in $L(H)$ and $\Bbb D$ is the unit disk of $\Bbb C$.
I tried to prove the conclusion : $\Bbb D\subset W_e(Q)$ iff $\Bbb D\subset ...

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### Closure of the point spectrum of an unbounded diagonalizable operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...

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### Takesaki lemma 1.16 (volume II, chapter VII)

I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following:
The relevant setting ...

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### Decompose a Hilbert space into two invariant subspaces

The following conclusion is from Bourin, Lee, Pinchings and positive linear maps arXiv:1505.02341 [math.FA] zbMath
Let $Q$ be an idempotent in $L(H)$.Then we have a decomposition $H=H_s\oplus H_{ns}$ ...

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### What about the structure theory in Baer *-rings?

In the literature, Baer *-rings are called as the algebraic analogue of von Neumann alegars.
It is well-known that
Theorem. Every von-Neumann algebra is decomposed into a direct sum of the algebras of ...

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### Is a unital $*$-morphism from a unital $C^*$-algebra $A$ to $\operatorname{End}_{\mathbb{C}}(K)$ automatically contractive?

Let $A$ be a unital $C^*$-algebra and let $K$ be an inner product space (not necessarily complete!). Let $\pi: A \to \operatorname{End}_{\mathbb{C}}(K)$ be a unital algebra homomorphism such that
$$\...

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### Impact of annihilators in C*-algebras

Let $A$ be a unital C*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$
Suppose that $A$ satisfies the following property:
For every $S\subseteq ...

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### Solvability and nilpotency for Banach algebras

Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...

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176
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### An inequality in C*-algebras

Let $A$ be a non-unital C*-algebra, and let $\pi: A\to B(A)$ defined by $\pi(a)(x)=ax$ be the left representation of A. Is the following inequality correct?
$$\lVert I+ \pi(a) \rVert\ge 1$$
for all $a ...

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answer

316
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### Completions of $C(X)$ with respect to the topologies generated by states

I have no intuition in this field so excuse me if this is trivial.
Let $X$ be a compact Hausdorff space, and $C(X)$ the algebra of continuous functions on $X$ with the usual $\sup$-norm. This is a $C^*...

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### Is every quasi-nilpotent element in a C$^*$-algebra a norm-limit of nilpotent elements?

Let $A$ be a C$^*$-algebra. I have seen theorems either stating or implying that if $A$ is the algebra of bounded linear operators on a separable Hilbert space (Herrero et al.), or the Calkin algebra, ...

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264
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### A particular commutator of the discrete Fourier matrix

For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...

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### Contractive projections on operator algebras

This is a follow up on an earlier question.
In [Lau&Loy, 2008] a Banach algebra $\mathcal{U}$ was called to have the Tomiyama property if any contractive projection $P:\mathcal{U}\to \mathcal{U}$, ...

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125
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### Projections in von Neumann algebra tensor product

Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra with normal faithful semifinite trace $\tau.$ Consider the von Neumann algebra $\mathcal N:=L_\infty([0,1])\overline{\otimes}\mathcal M$...

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217
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### A trace inequality between self-adjoint operators

Let $A$ and $B$ be self-adjoint operators on some Hilbert space and $B$ is postive. Suppose we have $-B\leq A\leq B$.Is it true then that $\|A\|_p\leq\|B\|_p$ where $\|.\|_p$ is the Schatten-$p$ norm ...

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### Lifting quasi-nilpotent elements in C$^*$-algebras

Let $A$ be a C$^*$-algebra with closed two-sided ideal $I$. Set $B=A/I$ and let $\pi:A\to B$ be the quotient map. Suppose that $b\in B$ is quasi-nilpotent. Does there exist quasi-nilpotent $a\in A$ ...

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141
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### A closed ideal in $L^1(T)$

Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$.
Let $I=...

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### Takesaki volume II chapter VII lemma 1.15

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...

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### Examples of TRO $V $ and $C^{\ast} $-algebra $B $ for which $V\otimes^hB $ is a TRO

Let $V $ be a ternary ring of operator(TRO) and $B $ be a $\mathbb {C}^{\ast} $-algebra. Let $V \otimes^hB $ denotes the Haagerup tensor product of $V $ and $B $. Obviously if $V $ or $B $ is $\mathbb ...

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97
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### Nuclear approximations preserving the Cartan sub-algebra structure

Let $D \subset A$ be a unital Cartan inclusion, that is, $A$ is a separable $C^*$-algebra, and $D$ is a maximally abelian sub-$C^*$-algebra that contains the identity of $A$, with the property that $A$...

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49
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### Types of normal states in the injective III$_1$ factor

The injective type III$_1$ factor is isomorphic to the Araki-Woods factor $R_{\infty}$.
I wonder how many types of normal states on $R_{\infty}$.
Haagerup mentions that there exist two different ...

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146
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### Irreducible group representation(algebraic and topological irreducibility)

In page 280 of "C^* algebra" by Dixmier, in the context of group representation, it is written 'We never encounter the concept of algebraic irreducibility except in finite dimensional ...

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### Does the convolution $C^*$-algebra of locally compact Hausdorff groupoids recover back the respective groupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.
Let $X$ be a locally compact Hausdorff groupoid (or Lie ...

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111
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### Trying to prove a seemingly easy fact on ideals of ternary C*-algebras

Currently I'm reading the paper by Abadie and Ferraro titled Applications of ternary rings to $C^*$-algebras.
Recall that a $C^{\ast}$-ternary ring is a complex Banach space $M$, equipped with a ...

5
votes

1
answer

297
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### Permanent invertible elements

Let $A$ be a unital complex algebra with the unit $\bf1$. Let $\mathcal{N}$ be the family of all norms on $A$ making it a unital normed algebra with the same unit $\bf1$. Let us put
$B_{\|\cdot\|}...

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97
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### Unicellular compact operators

An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...

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106
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### Generalized commutator

A well-known generalization of the commutator for operators is the so-called q-commutator defined as
$$[A,B]_q=AB-qBA.$$
I was wondering if the case where $q$ is not a number but other operator has ...

1
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1
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120
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### What is a C*-algebra generated by a subset of a direct sum of C*-algebras equal to?

I'm studying C-algebras and I don't know how to address the following question: let $(A_k)_{k\in \mathbb{N}}$ a family of C-algebras and let $\mathcal{G}$ a subset of $\displaystyle\bigoplus_{k\in \...

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166
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### Takesaki proposition 7.4 chapter 4 volume I

I initially asked on MSE, but did not get an answer there.
Consider the following proposition from chapter IV of Takesaki's "Theory of operator algebras I" (more context/definitions in the ...

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0
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108
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### Is it true that $\omega = \sum_{(k,l)\in I^2}\omega(p_k - p_l)$

Let $\{p_i\}_{i \in I}$ be a collection of projections in a $C^*$-algebra $A$ such that $\sum_{i \in I} p_i = 1$ in the strict topology (note here that $1$ is the unit of the multiplier algebra $M(A)$)...

9
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### Is a $*$-automorphism $M(A) \to M(A)$ automatically strictly continuous?

Let $A$ be a (non-unital) $C^*$-algebra with multiplier $C^*$-algebra $M(A)$. Let $\phi: M(A) \to M(A)$ be a $*$-automorphism. Is it true that $\phi$ is automatically strictly continuous (on bounded ...

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### Is it possible to start a PhD in mathematics at the age of 29? [duplicate]

I graduated with a bachelor’s degree in mathematics. I was initially focused on branches in analysis like operator algebra. At the third year of my undergraduate study, I experienced a financial ...

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votes

1
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206
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### Determine whether the center of a $C^*$-algebra is 0

Let $G$ be a locally compact group and $A$ be a non-unital $C^*$-algebra. )$(A,G,\alpha)$ is a $C^*$-dynamical sysytem. The space of all continous functions from $G$ to $A$ with compact support is ...

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0
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### Socle of an operator algebra

Let $H, K$ be Hilbert spaces.
Let $A\subseteq B(H)$ be a nonselfadjoint closed subalgebra such that the identity map is in $A$.
Let $C_A$ denote the $C^*$-algebra generated by $A$.
Q1: (this question ...

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### Finitely presentable group with purely infinite full group $C^*$-algebra?

Does there exist an example of a finitely presentable group whose full group $C^*$-algebra is purely infinite,
resp. is it known to be impossible?

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### Why is this norm called the maximal norm on the tensor product of ternary rings of operators?

Let $V$ and $W$ be ternary rings of operators (TROs). In section 5 of Kaur and Ruan - Local Properties of Ternary Rings of Operators and Their Linking $C^*$-Algebras, the maximal tensor product $\...

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### SU(2) and entangled particles [closed]

We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle_A\otimes \left| ...

2
votes

1
answer

261
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### A C*-algebra enjoying some different C*-norms

Does there exist any C*-algebra $(A,\|\cdot\|)$ enjoying the following property?
$\bullet$ There exists a norm $|\cdot|$ on $A$ with $\|\cdot\|\leq|\cdot|$ such that $(A,|\cdot|)$ is a pre C*-...

9
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3
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419
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### Defining the abstract tensor product of W*-algebras via a universal property

I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt:
It is easy to show that such an object ...

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83
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### Unitarily equivalent matrices that are also unitarily equivalent on orthogonal subspaces

Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$.
Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...

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65
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### $*$–homomorphisms of the center of $C^*$-algebras

Let $A$ and $B$ be $C^*$-algebras with centers $Z_A$ and $Z_B$. Suppose $\rho：A\rightarrow B$ is a surjective $*$- homomorphism. It is easy to check $\rho(Z_A)\subset Z(B)$.
I wonder how to assure ...

2
votes

2
answers

106
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### Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)

I originally asked this on MSE, but did not get an answer there.
Let $M$ be a von Neumann algebra. Let $\varphi: M_+ \to [0, \infty]$ be a weight on $M$. Consider
\begin{align*}&\mathfrak{p}_\...

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1
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78
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### The closure of selfadjoint elements of an algebra whose spectrum consist of rational numbers

Let $H$ be a seperable complex Hilbert space. What is the closure of the set of all self adjoint operators in $B(H)$ whose spectrum is a subset of the rational number $\mathbb{Q}$.
Apart from finite ...

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0
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54
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### A problem arising from Wiener-Levy theorem on the real line

Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...

2
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0
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181
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### Almost periodicity and approximation in tracial von Neumann algebra

Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. For a countable group $G$, let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action on $N$ which preserves the tracial ...

3
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0
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126
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### A question about Voiculescu's theorem

I saw many reference books, when the authors state the theorem, they assume that the representation spaces $H$ and $K$ are separable.
My question : If the representation spaces are non-separable, does ...