**3**

votes

**1**answer

113 views

### Elements in the group von Neumann algebra which are not summable

Let $G$ be a discrete group. I am wondering if there is a recipe which can be applied to find elements in the group von Neumann algebra which are not absolutely summable, i.e. $T \in VN(G)$ while $T \...

**11**

votes

**2**answers

286 views

### Is a C*-algebra with an isomorphic predual a von Neumann algebra?

It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...

**4**

votes

**1**answer

127 views

### Dye's Theorem for real von Neumann algebras

Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically ...

**11**

votes

**0**answers

193 views

### Countably decomposable von Neumann algebras

A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...

**11**

votes

**0**answers

118 views

### On spatial tensor products of von Neumann algebras

Let $H$ be a Hilbert space, and let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras.
We write $\odot$ for the algebraic tensor product,
and $\bar\otimes$ for the spatial tensor ...

**4**

votes

**1**answer

266 views

### Do irreducible characters form a closed set?

A character on a discrete group $\Gamma$ is a conjugation-invariant function $\tau$ which is of positive
type, and is normalized so that $\tau(e) = 1$, where $e$ is the identity element of $\Gamma$. A ...

**8**

votes

**3**answers

330 views

### What are the intermediate subfactors of the tensor product of two maximal subfactors?

Let $(N_1 \subset M_1)$ and $(N_2 \subset M_2)$ be two maximal subfactors.
Their tensor product, the subfactor $(N_1 \otimes N_2 \subset M_1 \otimes M_2)$, admits four obvious intermediate ...

**10**

votes

**5**answers

764 views

### If two projections are close, then they are unitarily equivalent

Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$.
The proof that immediately occurs to me uses comparison of ...

**4**

votes

**1**answer

182 views

### Can we solve the FGF problem by finding an appropriate action?

If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...

**3**

votes

**3**answers

471 views

### Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then :
$$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$
Question : Is it singly generated (as von Neumann algebra)? how ?
...

**1**

vote

**0**answers

61 views

### weak convergence in operator space structure

Let $M$ be von Neumann algebra and $B(H)$ be it's universal representation. Let $(e_i)$ be a Hilbert basis of $H$ and $\zeta_n\xrightarrow{w}\zeta $ in $H$. I know that $[w_{\zeta_n ,e_i}]_{1\times I}\...

**4**

votes

**0**answers

107 views

### Good reference for noncommutative $L^p$ spaces

I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...

**9**

votes

**0**answers

57 views

### Is the domain of an operator valued weight closed under Hahn-Jordan decomposition?

Let $N\subseteq M$ be an inclusion of semi-finite factors with normal faithful semi-finite traces $\operatorname{Tr}_N$ and $\operatorname{Tr}_M$ respectively. Let $T: M^+\to \widehat{N^+}$ be the ...

**13**

votes

**1**answer

544 views

### A left inverse for the comultiplication on a Hopf von Neumann algebra

Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments.
$\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following ...

**6**

votes

**1**answer

128 views

### The quantum group SUq(n) as von Neumann algebra

i have a question about a "presentation" of the quantum $SU(n)$. Here presentation means the following. Let $(M,\Delta)$ be a quantum group in the sense of Kustermanns and Vaes. One can show that the ...

**2**

votes

**0**answers

84 views

### Quantum Groups and quantum spaces - From algebra to Analysis

My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic ...

**9**

votes

**3**answers

480 views

### Separable von Neumann algebra

What is the simplest argument which shows that each infinite dimensional von Neumann algebra is not separable (in the norm topology)? It seems that this is a kind of folklore: at least I never saw the ...

**7**

votes

**2**answers

655 views

### subfactor of finite rank but infinite index: is this possible?

A subfactor $N\subset M$ is essentially the same thing as an $N$-$M$-bimodule.
I'll recall the basic definitions in the language of bimodules, and I hope that subfactor people will excuse me.
...

**2**

votes

**1**answer

388 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

**0**answers

118 views

### The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment.
Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...

**20**

votes

**3**answers

2k views

### What's a noncommutative set?

This issue is for logicians and operator algebraists (but also for anyone who is interested).
Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]):
Let $H$ ...

**7**

votes

**1**answer

358 views

### Is the fundamental group of $II_{1}$ factors invariant under a relation?

In order to define the equivalence relation, let's first recall the Tomita-Takesaki modular theory and conditional expectation for von Neumann algebras.
Let $H$ be a separable Hilbert space and $B(H)$...

**3**

votes

**1**answer

104 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**

vote

**1**answer

68 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 = 1-...

**1**

vote

**0**answers

58 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**

votes

**2**answers

132 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) ...

**1**

vote

**1**answer

86 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**

votes

**1**answer

102 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 $q=u^*u$....

**7**

votes

**1**answer

149 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**

votes

**0**answers

131 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**

votes

**2**answers

90 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 ...

**10**

votes

**1**answer

923 views

### Connes's unpublished manuscript on correspondences, anyone?

There exist unpublished notes on correspondences of von Neumann algebras due to Connes. This is often cited, but I've never seen a copy. It would be nice to have this, say, to maybe look further into ...

**0**

votes

**0**answers

84 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 ...

**3**

votes

**1**answer

144 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**

votes

**1**answer

90 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**

votes

**1**answer

159 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 $II_1$ ...

**4**

votes

**1**answer

89 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 $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?

**5**

votes

**0**answers

117 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 (...

**3**

votes

**0**answers

193 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 $y\in\...

**7**

votes

**1**answer

298 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 ...

**2**

votes

**0**answers

126 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**

votes

**1**answer

221 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

**1**answer

166 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 $\mathcal U$?

**0**

votes

**0**answers

67 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**

votes

**2**answers

291 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 ...

**4**

votes

**1**answer

136 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**

votes

**1**answer

131 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 $l^2(\...

**1**

vote

**0**answers

73 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 $...

**0**

votes

**0**answers

63 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 ...

**5**

votes

**0**answers

284 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. ...