Questions tagged [oa.operator-algebras]
Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry
1,884
questions
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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 $A$ ...
3
votes
0
answers
98
views
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 ...
0
votes
0
answers
60
views
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 $2$...
0
votes
1
answer
148
views
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 ...
0
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0
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28
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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 ...
0
votes
1
answer
29
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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 ...
4
votes
1
answer
209
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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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2
answers
144
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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 $H=H_s\oplus H_{ns}$ ...
2
votes
0
answers
52
views
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 ...
4
votes
2
answers
94
views
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
$$\...
3
votes
1
answer
96
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Impact of annihilators in C*-algebras
Let $A$ be a unital C*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$
Suppose that $A$ satisfies the following property:
For every $S\subseteq ...
2
votes
0
answers
131
views
Solvability and nilpotency for Banach algebras
Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...
5
votes
1
answer
176
views
An inequality in C*-algebras
Let $A$ be a non-unital C*-algebra, and let $\pi: A\to B(A)$ defined by $\pi(a)(x)=ax$ be the left representation of A. Is the following inequality correct?
$$\lVert I+ \pi(a) \rVert\ge 1$$
for all $a ...
3
votes
1
answer
316
views
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^*...
7
votes
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answers
87
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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, ...
1
vote
1
answer
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$ ...
1
vote
0
answers
178
views
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}$, ...
3
votes
0
answers
125
views
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$...
5
votes
1
answer
217
views
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 ...
9
votes
2
answers
172
views
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$ ...
2
votes
0
answers
141
views
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=...
4
votes
2
answers
239
views
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 $\...
1
vote
0
answers
32
views
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 ...
4
votes
0
answers
97
views
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$...
1
vote
0
answers
49
views
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 ...
2
votes
0
answers
146
views
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 ...
2
votes
0
answers
89
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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 ...
5
votes
0
answers
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
views
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\|}...
3
votes
2
answers
97
views
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 ...
5
votes
0
answers
106
views
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
vote
1
answer
120
views
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 \...
1
vote
1
answer
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 ...
1
vote
0
answers
108
views
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
votes
1
answer
347
views
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 ...
20
votes
8
answers
9k
views
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 ...
-1
votes
1
answer
206
views
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 ...
1
vote
0
answers
119
views
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 ...
5
votes
0
answers
97
views
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?
0
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0
answers
55
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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 $\...
-3
votes
1
answer
109
views
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
views
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
votes
3
answers
419
views
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 ...
0
votes
0
answers
83
views
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}...
0
votes
0
answers
65
views
$*$–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
views
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}_\...
1
vote
1
answer
78
views
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 ...
1
vote
0
answers
54
views
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
votes
0
answers
181
views
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
votes
0
answers
126
views
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 ...