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

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### The link between the subfactors and the motives as enriched Galois theories? [closed]

On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...

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**1**answer

39 views

### A Possible characterization of F.D or AF commutative $C^{*}$ algebras

By F.D or AF $C^{*}$ algebra,we mean finite dimensional or approximately finite dimensional $C^{*}$ algebra.
Let $A$ be a unital commutative $C^{*}$ algebra with the property that for every ...

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**1**answer

186 views

### Why does infinite tensor product associated with some vectors in the operator algebras?

I notice that in the definition of infinite tensor product in the operator algebras, such as C*-algebras and W*-algebras, every component in the product is associated with a vector(or s state) and the ...

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**1**answer

68 views

### A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra

Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued ...

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**1**answer

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### Eigenvalues and Compact Resolvent

For $A$ an unbounded (densely defined) operator on a separable Hilbert space, what conditions on its eigenvalues will show that, for $\lambda \notin $spec$(A)$, we have that $(A-\lambda)^{-1}$ is a ...

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**1**answer

113 views

### The coproduct on the 2-boxes space of the goup-subgroup subfactor planar algebras

Let $(H \subset G)$ be an inclusion of finite groups.
Let the subfactor $(\mathcal{R} \rtimes H \subset \mathcal{R} \rtimes G)$ with $\mathcal{R}$ the hyperfinite ${\rm II}_1$ factor, and its planar ...

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83 views

### Bounded operators with infinite matrix representations

I asked this question on StackExchange originally, but I'm giving it a go here as well.
Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and ...

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660 views

### Non-“weakly group theoretical” integral fusion categories?

Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices?
$\small{\begin{smallmatrix}
1 & ...

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298 views

### Projective modules over noncommutative tori?

It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...

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### The category of subfactors extending the category of groups?

This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...

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### Topology on the space of Schwartz Distributions

If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions ...

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### Morita Equivalence of Full Corners in $C^*$-algebras

Suppose $\mathcal{A}$ is a $C^*$-algebra with a unique normalized trace and $p \in \mathcal{A}$ is a projection so that $\mathcal{B} = p\mathcal{A}p$ is a full corner.
Does $\mathcal{B}$ have a ...

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**1**answer

62 views

### Definition of homotopy between Kasparov modules

I'm trying to understand the definition of homotopy between Kasparov modules as presented in Blackadar's book on K-theory for operator algebras. $A,B$ will be C*-algebras, while $E$ will denote a ...

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### Status of the analog of the Haar measure on quantum groups

In (Masuda, Nakagami, Woronowicz)'s paper, in the introduction, the authors mentioned the deficiency common to both their and (Kusterman, Vaes)'s approach regarding the Haar state (or Haar measure ...

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171 views

### A perturbation question for the intersection of C*-subalgebras

This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C* algebras".
Let M be a unital C*-algebra and let ...

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### One dimensional foliation of surfaces with prescribed graph of foliation

According to the definition of the graph of a foliation by Winkelnkemper we ask the following questions:
Let $G$ be one of the following non hausdorff 3 dim manifold
1) $G$ is a ...

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**1**answer

173 views

### not commutative symmetric vs strong symmetric spaces

Let M be a von Neumann algebra with semi-finite normal faithful trace $\tau$, $S(M)$ is space of all measurable operators introduced by I.Segal.
For the self-adjoint measurable operator $X\eta M$ ...

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**1**answer

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### Separability of the C*-algebra in the definition of K-homology

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book ...

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### Commuting nets for commuting projections

I think this should not be too difficult, but I am not an expert. I did not get an answer on stackexchange.
Let $A$ be a $C$*-algebra and let $p,q\in A^{**}$ be two commuting projections. Then there ...

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### The convolution on the finite dimensional weak Hopf $C^*$-algebras

Let $\mathbb{A}$ be a finite dimensional weak Hopf $C^*$-algebra, and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{A}}$ and the convolution $a * b ...

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184 views

### What's the relation between fusion and coproduct?

For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X_i)_{i \in I}$ of ...

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109 views

### An unconventional definition of the $ C^{*} $-algebraic reduced crossed product

Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system, i.e., $ A $ is a $ C^{*} $-algebra, $ G $ is a locally compact Hausdorff group and $ \alpha $ is a strongly continuous action of $ G $ on $ A $ by ...

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79 views

### Reference request on operator matrices [closed]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that
$$Tx = \begin{pmatrix}A & B \\
C & D
...

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374 views

### Continuity of the product map

Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...

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215 views

### Morphisms associated to measured spaces [duplicate]

In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...

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### Is a finite depth-index irreducible subfactor, intermediate of a depth ≤ 3 one?

Let $(N \subset M)$ be a finite depth-index irreducible subfactor.
Main question: Is $(N \subset M)$ the intermediate of a finite index depth $\le 3$ irreducible subfactor?
(In others words, is ...

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### Can we recover a von Neumann algebra from its predual?

By definition, a von Neumann algebra is a C*‑algebra A
that admits a predual, i.e., a Banach space Z such that
Z* is isomorphic to the underlying Banach space of A.
(We require that isomorphisms in ...

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**1**answer

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### What is about nonassociative geometry?

At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below:
Audience: You showed ...

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**1**answer

263 views

### Are all the $R$-$R$ bimodules completely reducible?

Let $R$ be the hyperfinite $II_1$ factor and let $X$ be any $R-R$ bimodule.
Question : Is $X$ completely reducible (i.e. a direct integral of irreducible $R-R$ bimodules) ?
Example : Let $N ...

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135 views

### Is this left ideal of C*-algebra principal?

This is a follow up of this question. Let $I$ be closed left ideal of $C^*$-algebra $A$.
Assume we are given a sequence of left $A$-module morphisms $R_n:I\to A$ with $\sum_n \Vert ...

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421 views

### Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant.
Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...

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### What are the first non-maximal non-group-subgroup simple irreducible subfactors?

Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$
is normal if the biprojections ...

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### Are the integer index finite depth irreducible subfactors Kac-coideal?

Is it known whether or not the integer index finite depth irreducible subfactors (planar algebra) are Kac-coideal subfactors: $(R^{\mathbb{A}} \subset R^{\mathbb{I}})$, with $\mathbb{A}$ a finite ...

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### Matrices whose entries are essentially traces of products of unitary matrices.

A $n\times n$ matrix $A=[a_{ij}]$ is called "good", if there exists some $k$ and a set of $k\times k$ complex unitaries $U_i$, $1\leq i\leq n$ , such that $tr(U_i^{+}U_j)=ka_{ij}$, where $U^{+}$ ...

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### Uniqueness of the tensor product decomposition of subfactors

A subfactor $(N \subset M)$ is indecomposable if (for $N_i \subset M_i)$: $$(N \subset M) = (N_1 \otimes N_2 \subset M_1 \otimes M_2) \Rightarrow \exists i \ N_i = M_i$$
Then, a subfactor $(N ...

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### Completely positive maps-equivalent definition

The most usual definition of the completely positive map (c.p.) between two C*-algebras (say, unital) is the following: $\sigma: A \to B$ should satisfy $\sigma(1)=1$ and for each $n \in \mathbb{N}$ ...

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### $S^{3}$-valued harmonic analysis

Edit:
Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider
$$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid ...

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### Kaplansky Idempotent conjecture and Extension theory

We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...

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563 views

### Question about hereditary $C^*$-algebra.

Can anyone give me a relatively simple proof or Some reference for the following fact.(I know that there is a proof of this theorem in Gerard J. Murphy'book: "$C^*$-Algebras and Operator Theory", but ...

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### Invertible operator with countable spectrum

Let $H$ be a separable Hilbert space and $A$ is an invertible bounded operator on $H$. Can we approximate $A$ with an invertible operator $B$ such that $sp(B)$ is a countable set?
Motivation:
...

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### A special Lie subalgebra

Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra ...

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### $R$ is a right multiplier and $R(a)b=a\overset{?}{\implies} A$ 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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### A question on unbounded operators

Assume that $H$ is a separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)>1$ with the following property?:
Every densely defined operator $A:D(A)\to ...

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### Hyperfinite type II_1 factor as the Clifford algebra

In Connes' book Noncommutative geometry, there is a presentation of all hyperfinite factors. He reffers to type $II_1$ as the Clifford algebra of infinite dimensional Euclidean space.
This factor can ...

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**1**answer

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### Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...

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### Two Definitions of Non-commutative $L^p$ space

Throughout, let $(\mathcal{M},\tau)$ be a von Neumann algebra $\mathcal{M}$, acting on a Hilbert space $H$, with normal semifinite faithful trace $\tau$.
In the survey article by Pisier and Xu, the ...

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154 views

### Are two $W^*$-algebras $A$ and $B$ that are Morita equivalent (in the sense of Rieffel) also Morita equivalent in the algebraic sense (as rings)?

Let $A$ and $B$ be $C^*$-algebras, I had a question about Morita equivalence for $C^*$-algebras and $W^*$-algebras, I've come across 3 concepts:
Morita equivalence for $C^*$-algebras: Equivalence ...

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### Positive operators - norm equality

I hope that somebody can help me with the following problem:
Let $A$ be a positive operator on $\mathbf{B}(\mathcal{H})$, ( $\mathcal{H}$ is a Hilbert space) with its spectral measure $E$. Show that ...

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### Do Burnside Group Factors have Gamma?

The Free Burnside group $G=B(2,665)=\langle a,b|g^{665} \rangle$ is infinite, by the work of Adyan and Novikov. Furthermore, the centralizer of any nonidentity element in $G$ is finite cyclic, and so ...

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### Infinite amenable group subfactors

Let amenable groups $\Gamma$ and $\Gamma'$. They act outerly of only one manner on the hyperfinite ${\rm II}_1$-factor $\mathcal{R}$.
Question: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) ...