Subtag of [tag:oa.operator-algebras] for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.

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88 views

Is the module action $M\times M^*\to M^*$ jointly continuous?

Let $M$ be a W*-algebra and consider the following map: $$\gamma: M\times M^*\to M^*: (a,f)\to af$$ where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
1
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1answer
58 views

A relation among projections of a von Neumann algebra

This is a follow-up question on this. Let $A$ be a von Neumann algebra and $P$ be its projection lattice. For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = ...
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0answers
41 views

Support vectors and relative modular operator

I'm studying the relative modular operator and I'm looking for o good text to do it. Until now I'm using Araki's papers but I don't know how to deal with the support of a vector, $s^M(\xi)$, which is ...
4
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2answers
126 views

Collection of projection operators in finite dimension and algebraic techinques

Consider a set of linearly independent vectors $\{x_1,\dots,x_n\}$ in some finite-dimensional Hilbert space $H$. For any subset $S \subset [n]$, let $P_S$ be the (orthogonal) projection (operator) ...
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1answer
65 views

Minimal central projection in W*-algebras [closed]

Let $M$ be a W*-algebra. I am looking for the proof of the following fact: Let $z$ be the supremum of minimal projections in $M$. Then $z$ is central.
0
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1answer
78 views

Equivalent projections in von Neumann algebras

Let $M$ be a von Neumann algebra in $B(H)$. Let $p$ and $q$ be projections in $M$. Assume that they are equivalent in $B(H)$, i.e there is a partial isometry $u$ in $B(H)$ with $p=uu^*$ and ...
6
votes
1answer
126 views

The positive cone of the standard representation of a Von Neumann algebra

Let $A$ be a von Neumann algebra, let $L^2(A)$ be the underlying Hilbert space of the standard form of $A$, and $P \subset L^2(A)$ the canonical positive cone (see for example this paper by Haagerup). ...
4
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0answers
123 views

Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra

While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove: Let $H$ be a Hilbert space, $A$ and $B$ trace von Neumann subalgebras ...
0
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2answers
80 views

rank 1 projections of finite dimensional von Neumann algebra have the same traces?

Let M be a finite dimensional von Neumann algebras with a normal faithful trace. Let e and f be two projections with rank 1. I want to know if e and f have identical traces. (This is obviously true if ...
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79 views

semifinite projection

Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$. ( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
2
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1answer
82 views

Approximation of the central support

Let $(M,\tau)$ be a tracial von Neumann algebra, i.e. a unital subalgebra $M=M''\subset \mathbb{B}(H)$; a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ ...
0
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1answer
132 views

types of crossed product von Neumann algebras

Let $M$ be a type $II_1$ factor von Neumann algebra, and let $G$ be a discrete group acting on $M$ which is free and ergodic. Is the crossed product von Neumann algebra $M \rtimes G$ type $II_1$ ...
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0answers
57 views

On the tensor product of irreducible finite index subfactors

This post extends this question on maximal subfactors, admitting this answer of Feng Xu. Statement: A tensor product of irreducible finite index subfactors has a non-obvious intermediate if and only ...
4
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1answer
76 views

second dual of minimal tensor products of $C^*$-algebras

Let $A$ be a unital $C^*$-algebras and $K(H)$ is $C^*$-algebras of compact operators on separable Hilbert space $H$. Is it true that $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?
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174 views

Infinite number of non-isomorphic von Neumann algebras with property Gamma?

A II$_1$ factor $\mathcal M$ with trace $\tau$ has property Gamma if for every $\epsilon > 0$ and finite set $\{x_1,\cdots, x_n\} \subset \mathcal M$ there exists a trace 0 unitary element ...
3
votes
1answer
136 views

What are the applications of the depth 2 reduction to the subfactors theory?

Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth $2$ ...
2
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0answers
120 views

Rank–nullity theorem for finite von Neumann algebras

The rank-nullity theorem states that for $U, V$ finite dimensional vector spaces and $T:U \to V$ a linear map $$\dim(U) = \dim(im(T)) + \dim(ker(T)) $$ Let $M \subset B(H) $ be a finite von Neumann ...
6
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1answer
200 views

Connes Embedding Conjecture and Fusion Categories

I was recently introduced to Connes' Embedding Conjecture (CEC) which states: Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on ...
3
votes
1answer
158 views

A relative property gamma and $L(\mathbb F_2)$

Given any unital non-commutative subalgebra $\mathcal M$ of $L(\mathbb F_2)$ is it true that $\mathcal M' \bigcap L(\mathbb F_2)^\mathcal U = \mathbb C I$ for any free ultrafilter $\mathcal U$?
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65 views

An equality for the trace of the join of a non-degenerate indecomposable system of projections in a finite factor

Let $M \subset B(H)$ be a finite factor (see for example here p2, or there) with a trace $tr$. The subset of projections of $M$ is naturally a lattice, noted $(\mathcal{P}(M), \wedge, \vee)$. A ...
3
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2answers
253 views

Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages. I read some pages such as Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
3
votes
1answer
123 views

Tomita Takesaki theory and boundeness of $S$

Let $M$ be a von Neumann algebra, $\xi$-separating and cyclic vector for $M$. Let $S$ be antilinear operator acting as $x \xi \mapsto x^* \xi$ where $x \in M$. Then one can show that $S$ is closable ...
5
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1answer
123 views

The (Hecke) double coset von Neumann algebra

It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on ...
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65 views

Examples of non-extremal subfactors

Every subfactor $(N \subset M)$ in this post are supposed to be finite index inclusion of ${\rm II}_1$ factors. Definition (here p64): Such a subfactor is called extremal if $tr_{N'} = tr_M$ on ...
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59 views

The completely reducible bimodules coming from subfactors

This post is a sequel of: Are all the R-R-bimodules completely reducible? Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...
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273 views

Koopman representation, weakly compact action, Ozawa Popa

Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
2
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0answers
107 views

Arveson spectrum for a unitary representation of a group on a Hilbert space

Although this is not research, I think the question is a little bit too specific for math.stackexchange Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a ...
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0answers
65 views

Why does one only consider one-parameter groups in Borchers-Arveson theorem?

(question from math.stackexchange) The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter ...
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0answers
63 views

Are there infinitely many amenable Hadamard-Petrescu subfactors?

The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by ...
7
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1answer
289 views

Can a finite von Neumann algebra be strongly morita equivalent to a properly infinite von neumann algebra?

Can a finite (by finite I mean when the projection $1$ is finite) von Neumann algebra be strongly morita equivalent to a properly infinite von Neumann algebra? (Strong morita equivalence is the same ...
4
votes
2answers
372 views

Question about projections in von Neumann algebras

Let $M$ be a von Neumann algebra, and let $\mathcal{P}$ be the set of nontrivial (not equal to $0$ or $e$) projections of $M$. Define $p,q \in \mathcal{P}$ to be equivalent if there exist projections ...
5
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0answers
129 views

Approximation in the tensor square of a weakly exact von Neumann algebra

Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...
5
votes
1answer
181 views

Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
0
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1answer
94 views

Is there an irreducible subfactor with an infinite homogeneous single chain lattice?

We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here). Now ...
8
votes
1answer
282 views

Embedding the group von Neumann algebra into an injective von Neumann algebra on the same Hilbert space

Let $\Gamma$ be a discrete group, $\newcommand{\VN}{\rm VN}$ and let $\VN(\Gamma)$ denote its von Neumann algebra, regarded as a subalgebra of ${\sf B}(\ell^2(\Gamma))$. It is well known that ...
6
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1answer
198 views

Connes' correspondences of two $L^\infty$-algebras

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu_X)$ and ...
8
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1answer
164 views

Is the center of the automorphism group of a von Neumann algebra M trivial whenever M is a factor?

Question: Is the center of the automorphism group of a von Neumann algebra $\mathscr{M}$ trivial (=$\{\mathrm{id}\}$) whenever $\mathscr{M}$ is a factor (=$\mathscr{M}$ has center $\{\lambda I; ...
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0answers
83 views

How the modular theory of von Neumann algebras, deal with generating C*-algebras?

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Suppose the existence of a bicyclic ...
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1answer
296 views

Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Let $p,q \in M_{\infty}(A)$ be ...
3
votes
1answer
136 views

Example of joint cyclic and separating vector

Let $\mathcal{H}$ be a separate Hilbert space and $\mathcal{B}(\mathcal{H}) \subset \mathcal{B}(\mathcal{H}) \otimes M_2(C)$ be a W$^*$-inclusion pairs. It is known that this pair share a joint cyclic ...
4
votes
1answer
170 views

Noncommutative version of Littlewood's First Principle

There are definitely noncommutative analogues for Lusin's theorem and Egoroff's theorem (found in Blackadar for example). I'm curious if there is a version of the first principle: Every Lebesgue ...
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2answers
2k views

Mysterious quotes (at least for me)

I heard two quotes, one from Alain Connes and an other one from Orlov. Alain Connes was talking about noncommutative geometry and he said the following: " a noncommutative algebra creates its own ...
3
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2answers
93 views

Characterization of hyperfinitness by the Effros-Marechal topology

Let $A$ be a von Neumann algebra and $A_*$ its predual. One can define a topology on the set $vN(A)$of all von Neumann subalgebras of $A$ called the Effros-Marechal topology. It is characterized as ...
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219 views

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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111 views

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

Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra? In other words, of the following form ...
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171 views

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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105 views

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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107 views

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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189 views

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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1answer
166 views

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) ...