All Questions
29 questions
14
votes
1
answer
1k
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Connes's unpublished manuscript on correspondences, anyone?
There exist unpublished notes on correspondences of von Neumann algebras due to Connes.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 ...
- 7,057
13
votes
1
answer
1k
views
Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?
Regarding the hyperfinite $II_{1}$ factor $R$ as $C^{*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of ...
- 7,057
13
votes
1
answer
1k
views
Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?
A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
- 7,057
11
votes
4
answers
2k
views
What kind of completion is this?
Let $X$ be a compact Hausdorff space, and $C(X)$ the unital commutative C*-algebra of continuous functions on it. The double Banach dual $C(X)^{**}$ is a commutative von Neumann algebra and hence has ...
- 3,937
9
votes
1
answer
1k
views
Ping Pong and Free Group Factors
This question concerns alternative characterizations of free group factors. The ping pong lemma is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like ...
- 7,057
8
votes
1
answer
426
views
Is $L(\mathbb{Z}*\mathbb{Z}_{2})$ a free group factor?
This is a reference request for something that is likely to be well-known to operator algebraists. I will not, therefore, include the technical definition of free product of finite von Neumann ...
- 7,057
8
votes
0
answers
339
views
Canonical Time Evolution for Type $II_{1}$-Factors?
This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor (...
- 7,057
7
votes
1
answer
990
views
Subfactor theory and Hilbert von Neumann Algebras
There seem to be intimate connections between the different definitions of von Neumann module. The two that I'm aware of are Hilbert von Neumann modules and correspondences (in the sense of Connes). I ...
- 1,411
7
votes
0
answers
127
views
Is there an adaptation of the theory of standard forms and Tomita-Takesaki theory to the $\mathbb{Z}_{2}$-graded case?
Let $A$ be a von Neumann algebra acting on a Hilbert space $H$, and suppose that $\Omega \in H$ is a cyclic and separating vector for $A$. Then in Tomita-Takesaki theory one defines an unbounded ...
- 556
7
votes
0
answers
322
views
Is it known that "hyperfinite length" cannot distinguish free group factors?
Given a type $II_{1}$ factor $M$, Popa and Ge defined the hyperfinite length $l_{h}(M)$ of $M$ to be the minimum natural number $n$ such that there are hyperfinite subalgebras $R_{1}, R_{2},..., R_{n}$...
- 7,057
6
votes
1
answer
486
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 ...
- 388
6
votes
0
answers
122
views
Premeasurability of affiliated operators for type $\textrm{III}$ von Neumann algebras
$\DeclareMathOperator\dom{dom}$If $M\subset B(H)$ is a semifinite von Neumann algebra with faithful, normal, semifinite trace $\tau$, then a closed operator $T:H\rightarrow H$ intertwining the action ...
- 7,057
5
votes
1
answer
510
views
Reference for embedding an infinite direct product of matrix algebras into the hyperfinite $II_1$ factor
In some calculations I am writing up,
$\newcommand{\cR}{{\mathcal R}}$
I want to add - as a fairly throwaway remark - that any countable product (= $\ell^\infty$-direct sum) of matrix algebras can be ...
- 25.8k
5
votes
2
answers
545
views
range projection of an unbounded idempotent affiliated to a finite von Neumann algebra
To be slightly more precise: let $M\subset B(H)$ be a finite von Neumann algebra equipped with a faithful normal trace $\tau$, and let $L^0(M,\tau)$ be the completion of $M$ in the measure topology; ...
- 25.8k
5
votes
1
answer
577
views
Is there a trivial construction of the trace on the Jones basic construction?
Let $N$ be a type $II_{1}$-factor with trace $\tau$, and $B$ a von Neumann subalgebra. The existence of the semifinite trace on the Jones basic construction $\langle N, e_{B} \rangle$ is reasonably ...
- 7,057
5
votes
0
answers
254
views
Examples of non-proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors
Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\...
- 7,057
4
votes
0
answers
263
views
Approximately inner conditional expectations of $II_{1}$ factors
In many contexts it is helpful to think of conditional expectations as averages of unitary conjugates, a standpoint vindicated by many standard techniques in the theory of finite von Neumann algebras. ...
- 7,057
4
votes
0
answers
282
views
If the flip automorphism of a finite factor can be connected to the identity is it approximately inner?
This intriguing question is due to Sorin Popa, so I'm marking it community wiki. I'd like to know the status of the question, and if it is still open, perhaps gather some new ideas for attacking it.
...
3
votes
1
answer
621
views
Injective von Neumann algebra
Let $G$ be a non-amenable countable discrete group. How can I show that the group von Neumann algebra $L(G)$ has no injective direct summand?
- 1,702
3
votes
1
answer
254
views
Reference request for c*-algebra embedding into the atomic part of its double dual
Let $A$ be a C*-algebra. According to operator algebraists, it is well known that $A$ embeds into the atomic part of its double dual in the following sense: if $z$ is the central projection in $A^{**}$...
- 388
3
votes
0
answers
274
views
Decomposition of a von Neumann Algebra into Factors
I know that a von Neumann algebra on a separable Hilbert space can be (uniquely) decomposed into factors. But is there any non-trivial example showing how we can explicitly compute it?
Non-trivial: ...
- 241
2
votes
1
answer
101
views
Hyperexpectations from injective subfactors of a type $II_1$ factor
Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $...
- 7,057
2
votes
1
answer
158
views
Reference request for a type III action of a group on a manifold
Let an action of a group $\Gamma$ on a manifold $M$ such that $L^{∞}(M)⋊Γ$ is a type $III$ factor.
André Henriques posted here the following comment :
I don't know the literature, so I can't point to ...
2
votes
0
answers
201
views
An example of non trivial projections in a group von Neumann algebra
Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
- 2,118
2
votes
0
answers
412
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 ...
- 75
1
vote
1
answer
287
views
How coarse is the coarse correspondence?
Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) \overline{\...
- 7,057
1
vote
0
answers
86
views
Inner product on Standard form of von Neumann algebra
Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that
$$\langle x,yz\rangle=\langle zx,y\rangle$...
- 131
1
vote
0
answers
83
views
Are these kinds of "crossed product" studied?
Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...
- 2,118
0
votes
1
answer
119
views
Reference request: hyperfinite cross product
Given a countable essentially free ergodic non-singular group action $G \curvearrowright (X, \mu)$ on a standard measure space, suppose $\mu$ is a non-atomic probability measure and $\alpha: G \...
