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Questions tagged [oa.operator-algebras]

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

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Value of Plancherel weight on convolutions (Takesaki VII Section 3 Theorem 3.4)

In this post, I follow conventions and notations from Takesaki's second volume "Theory of operator algebras" (chapter VII sections 2 and 3). Let $G$ be a locally compact group with left Haar ...
Andromeda's user avatar
-2 votes
1 answer
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Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras

A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor. They form a category with usual structures. Question. Is this category equivalent to the category of $C^*$ algebras? ...
Ali Taghavi's user avatar
1 vote
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What is the status of The Halmos Similarity Problem?

What is the general status of "The Halmos Similarity Problem"(HSP) in Operator theory?For What conditions ,HSP has been solved?
P.Styles's user avatar
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2 answers
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Tensor product of operator values weights (in the theory of locally compact quantum groups)

Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object $$\iota \otimes \psi: (M\overline{\otimes} M)_+ \to \widehat{M_+}.$$ ...
Andromeda's user avatar
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Given $\sigma(AB-BA) = \{0\}$, what can be said about $\sigma(A)$ and $\sigma(B)$?

Let $\mathcal H$ be a separable Hilbert space, and $\mathfrak B(\mathcal H)$ denote the algebra of bounded linear operators on $\mathcal H$. Furthermore, let $A,B \in \mathfrak B(\mathcal H)$ be two ...
stoic-santiago's user avatar
4 votes
0 answers
94 views

Extreme points in the space of ucp maps

Suppose $M$ and $N$ are $\mathrm{II}_1$ factors. Let $\tau\mathrm{UCP}(M,N)$ be the convex space of trace-preserving UCP maps from $M$ to $N$, equipped with the topology of pointwise weak* convergence....
David Gao's user avatar
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1 answer
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A subalgebra of $B(H)$ which does not contain a commutator element

Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property: The algebra $A$ has trivial intersection with the set of commutator ...
Ali Taghavi's user avatar
1 vote
0 answers
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An example of a non rigid Banach algebra

A Banach algebra $A$ is called a rigid Banach algebra if for every injective Banach algebra morphism $J:A\to A$ we have either $\overline{J(A)}$ is ismorphic to $A$ or it does not contain ...
Ali Taghavi's user avatar
1 vote
1 answer
120 views

Intersection of von-Neumann algebra factors

Given two von-Neumann algebra factors $\mathcal M,\mathcal N$, is $\mathcal M\cap\mathcal N$ a factor? And how about the intersection of infinitely many factors? Notes: I know that the intersection ...
Dominique Unruh's user avatar
3 votes
1 answer
169 views

Is the weighted shift strong frequently hypercyclic?

One sided Shift Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum_{i=1}^\infty |x_i-y_i|/2^i$ . Define the shift map $\...
Eduardo's user avatar
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Property A, Higson-Roe condition and its applications

Recently I have been studying amenability of groups and property A, and I came across the Higson-Roe condition: Let $X$ be a uniformly discrete metric space with bounded geometry. $X$ has property $A$ ...
Ken's user avatar
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3 votes
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The centralizer of a normal state on a type III$_1$ factor

Let $M$ be any type III$_1$ factor. Does there must exist a normal state $\rho$ on $M$ such that the centralizer $M_{\rho}$ of $\rho$ is a factor?
mathbeginner's user avatar
0 votes
1 answer
56 views

Cyclic vectors and subfactor inlcusion

Let $N\subset M$ be a be factors acting on a Hilbert space $H$. Denote by $\mathrm{Cyc}(A)\subset H$ the set of cyclic vectors for $A = M,N$. I am interested in the equality case of the inclusion $\...
Lauritz's user avatar
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What's the purpose of the operator $(\Delta^{-1}+\lambda)^{-1}$ in Tomita-Takesaki modular theory?

I was reading Tomita-Takesaki modular theory (from all the books, and articles), the goal is to relate a von Neumann algebra $\mathcal{A}$ with its commutant $\mathcal{A}'$ on a Hilbert space $\...
MrPajeet's user avatar
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1 vote
1 answer
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Convergence of the partial sum of a sequence strictly converging to zero

The following question comes from a statement in Lemma 16.4 in K-theory and $C^{\ast}$-Algebras written by N.E. Wegge-Olsen. Let $A$ be a non-unital $C^*$-algebra, $\{p_n\}_{n\in\mathbb{N}}$ be a ...
Sanae Kochiya's user avatar
0 votes
1 answer
179 views

Compactly supported continuous functions as a Tomita algebra

Let $G$ be a locally compact group with modular function $\delta_G$ and consider $\mathcal{K}(G)$, the set of compactly supported continuous functions $G\to \mathbb{C}$, with the $*$-algebra structure ...
Andromeda's user avatar
0 votes
1 answer
87 views

Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$?

Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that $${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$ $${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...
user92646's user avatar
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4 votes
0 answers
120 views

Questions about the $K$-theory of the algebraic standard Podleś sphere

Given $\theta \in \mathbb{R}$ irrational, the $K$-theory of the smooth noncommutative $2$-torus $C^\infty_\theta(\mathbb{T}^2)$ is well understood in relation to that of the corresponding $\mathrm{C}^\...
Branimir Ćaćić's user avatar
4 votes
0 answers
168 views

Does SO(n) have Lafforgue's Strong Property (T)?

On page 13 of the monograph of Bekka, de la Harpe, and Valette on Kazhdan's property (T), it is written "for $n \geq 3$, the compact group $\mathrm{SO}(n)$ has the strong property (T)," ...
Aleksander Skenderi's user avatar
0 votes
0 answers
43 views

existence of a partial isometrie satisfying special property in a von Neumann algebra factor

Let $M$ be any factor and $\varphi$ be any faithful normal state on $M$.Suppose that $M_{\varphi}$ is a type II$_1$ factor and there exists $0<\lambda<1$ such that $S:=\{x\in M:\sigma_t^{\varphi}(x)=\...
mathbeginner's user avatar
3 votes
1 answer
83 views

Least upper bound of type I factors

Given two type I (von-Neumann algebra) factors $\mathcal M,\mathcal N$, is there a smallest type I factor containing both $\mathcal M,\mathcal N$? Notes: $\mathcal M,\mathcal N$ are over the same ...
Dominique Unruh's user avatar
3 votes
1 answer
224 views

Takesaki: question about lemma in section "Left Hilbert algebras and weights"

To make this question relatively self-contained, this post is quite long, but the question itself is rather short. Consider the following fragments in Takesaki's second volume "Theory of operator ...
Andromeda's user avatar
1 vote
1 answer
167 views

Adjunction via Gelfand duality

$\DeclareMathOperator\Hom{Hom}$For which unital $C^{\ast}$-algebras $A$ does it hold that for all compact Hausdorff $S$ we have the bijection: \begin{align*} \Hom(A, C(S)) \cong \Hom(S, \Hom (A, \...
Luiz Felipe Garcia's user avatar
4 votes
1 answer
253 views

Is the sigma-strong topology generated by bounded sets?

Let $V$ be a Banach space and $B(V)$ be the bounded operators on $V$. Now, the space $B(V)$ has a norm, the strong topology (which is the topology of pointwise convergence) and it has a stronger ...
Sebastian Meyer's user avatar
5 votes
1 answer
200 views

Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity

Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...
Andromeda's user avatar
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37 views

In the proof of Neural Tangent Kernel stays constant in infinite width limit, why the norm of the dual mapping operator equals operator norm of kernel

For a fixed distribution $p^{in}$ on the input space $ \mathbb{R}^{n_0}$, consider a function space $\mathcal{F}$ defined as $\{{f: \mathbb{R}^{n_0} \rightarrow \mathbb{R}^{n_L}}\}$. On this space, ...
Shuofeng Zhang's user avatar
2 votes
1 answer
402 views

Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?

Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123. Why is it possible to choose an ...
Andromeda's user avatar
2 votes
1 answer
137 views

inclusion of von Neumann algebras implies reversing inequality of its modular operators

I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999) Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...
Gabriel Palau's user avatar
2 votes
2 answers
350 views

Takesaki II "Connes cocycle derivative"

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108: Why are the second and third ...
Andromeda's user avatar
1 vote
0 answers
38 views

A question on the domain of a non-commutative Radon-Nikodym derivative

Let $\omega$ be a state on a $C^*$-algebra $\mathfrak{A}$ and $\pi_\omega:\mathfrak{A}\rightarrow B(\mathcal{H}_\omega)$ its GNS representation. Suppose in addition that $\omega$ has central support, ...
Stefano Rossi's user avatar
1 vote
1 answer
286 views

Takesaki lemma: existence Gelfand-Pettis integral

Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras"). In another post, it was explained ...
Andromeda's user avatar
1 vote
0 answers
138 views

Literature on Lyndon words and the Lie commutator

Since I lost my paper notes in a domestic conflagration in Japan some ten years ago, I've occasionally tried to recall one particular author who wrote in the 1900s about Lyndon words / strings, or ...
Tom Copeland's user avatar
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1 vote
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92 views

Intersection of finitely many type-I von-Neumann algebra factors

If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M\cap\mathcal N$ a type-I von-Neumann algebra factor? Notes: An elementary ...
Dominique Unruh's user avatar
0 votes
0 answers
70 views

Intersection of type-I von-Neumann algebra factors

Is the intersection of a (possibly infinite) family $\{\mathcal M_i\}$ of type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$) again a type-I von-Neumann algebra factor?
Dominique Unruh's user avatar
1 vote
0 answers
71 views

Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$

Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
DeltaEpsilon's user avatar
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0 answers
68 views

Extension of $\{f\in C([0, 1], B)\,\vert\, f(0)=f(1)=0\}$ by $A$ with $\ast$-homomorphism $\phi:A\rightarrow B$

The following question is from An Introduction to $K$-theory for $C^{\ast}$-Algebra and an e-copy can be found here. Below is the question (since I do not know how to create a diagram in MS ...) By ...
Sanae Kochiya's user avatar
12 votes
0 answers
104 views

Existence of more than two C*-norms on algebraic tensor product of C*-algebras

Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$. If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
Alcides Buss's user avatar
4 votes
0 answers
97 views

Maximally fine topologies on $B(H)$ making the unit ball compact

Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
Aareyan Manzoor's user avatar
0 votes
1 answer
101 views

Differential form of the multidimensional "orthogonal dilation" operator

For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion. ...
Kanghun Kim's user avatar
2 votes
1 answer
115 views

Question on tensor product of von Neumann algebras and subfactors

Let $M_1$ and $M_2$ be von Neumann algebras acting on Hilbert spaces $H_1,H_2$ and consider $M=M_1\overline\otimes M_2$ acting on $H_1\otimes H_2$. Let $K$ be an $M$-invariant subspace (so that $P_K\...
Lauritz's user avatar
  • 379
9 votes
0 answers
211 views

Why are projectionless $C^*$-algebras important (Kadison's conjecture)

It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
Alexandar Ruño's user avatar
6 votes
1 answer
507 views

Integration in Banach algebra

Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a ...
user72829's user avatar
  • 508
3 votes
1 answer
309 views

Takesaki II Lemma 1.13: stuck in proof

Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"): Here, we associate with an ...
Andromeda's user avatar
7 votes
1 answer
227 views

When is the multiplication map of the algebraic tensor product of C*-algebras injective?

A classic result, of Murray and Von Neumann I believe, is that if $\mathcal M\subseteq B(H)$ is a factor then the $*$-homomorphism $\pi : \mathcal M \odot \mathcal M' \rightarrow B(H)$ given by $\pi(...
Chris Ramsey's user avatar
  • 3,894
3 votes
0 answers
92 views

subfactors and trace

For every factor $M$ there is an (essentially unique) dimension function (or trace) $\tau_M : M_p \to [0,\infty]$ (where $M_p$ is the lattice of orthogonal projections in $M$). My question is: Under ...
Lauritz's user avatar
  • 379
1 vote
2 answers
152 views

Limit of a countable separable is countable separable?

Let $\rho$ be a positive trace class operator on $H\otimes H$, where $H$ is a separable Hilbert space (not necessarily finite dimensional). We say that $\rho$ is countable separable if $\rho=\sum_{i=...
Deva's user avatar
  • 11
5 votes
0 answers
427 views

Watatani's theorem for tensor categories

We refer to [JS97] for the notion of subfactor. Yasuo Watatani proved the following result [W96, Theorem 2.2]: Theorem: An irreducible finite index subfactor of a type $\mathrm{II}_1$ factor has ...
Sebastien Palcoux's user avatar
5 votes
1 answer
164 views

Continuity of the extension of a tracial state with respect to the strong operator topology

Problem: Let $M\subseteq B(H)$ be a finite von Neumann algebra with a faithful tracial state $\tau$. Let $\widetilde{M}$ be the $\tau$-measurable operators on $M$ (recalled below). Extend the trace $\...
John's user avatar
  • 45
3 votes
1 answer
120 views

$\tau$-measurable operator

Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...
John's user avatar
  • 45
0 votes
0 answers
63 views

Necessary conditions for $K_0(I_x\bigotimes A)$ to be the trivial group

Let $A$ be a unital $C^*$-Algebra with non-trivial $K_0$ group. Define $CA = \{f\in C\big([0, 1], A\big)\,\vert\,f(0) = 0\}$. It can be shown that $CA$ is homotopic equivalent to the set $\{0\}$ and ...
Sanae Kochiya's user avatar

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