# 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 ...

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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?
...

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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?

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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
$$\iota \otimes \psi: (M\overline{\otimes} M)_+ \to \widehat{M_+}.$$
...

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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 ...

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### 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....

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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 ...

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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 ...

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### 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 ...

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### 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 $\...

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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 $A$ ...

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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?

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### 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 $\...

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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 $\...

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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 ...

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### 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 ...

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### 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 +\...

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### 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}^\...

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### 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)," ...

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### 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)=\...

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### 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 ...

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### 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 ...

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### 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, \...

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### 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 ...

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### 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 $...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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):=\...

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### 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 ...

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### 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 $(A,B)$...

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### 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 ...

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### 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.
...

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### 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\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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(...

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### 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 ...

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### 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=...

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### 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 ...

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### 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 $\...

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### $\tau$-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 $m$ ...

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### 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 ...