# Tagged Questions

**3**

votes

**1**answer

154 views

### Measure Preserving Transformation Induced by a $*$-automorphism on $L^\infty(X,\mu)$

The following excerpt is from Connes' Noncommutative Geometry
Let $(X, \mathcal{B}, \mu)$ be a standard Borel space equipped with a probability measure $\mu$, and let $\ T$ be a Borel ...

**12**

votes

**0**answers

249 views

### Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?

I recently gave a talk, where I talked about the tensor category
of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor.
Vaughan Jones, who was in the audience, later told me ...

**3**

votes

**1**answer

195 views

### References for von Neumann Algebras

I have some -possibly- simple but broad questions: Where to begin the study of von Neumann Algebras? Which are the important questions in the field that guide current research? I'm interested in ...

**2**

votes

**1**answer

105 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 ...

**8**

votes

**3**answers

287 views

### reference request : constructive measure theory

As the title said, I would like to know if constructive measure theory has been developed somewhere ?
I am more precisely interested in the (constructive) theory of completely continuous valuation on ...

**3**

votes

**1**answer

302 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?

**11**

votes

**1**answer

515 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 ...

**10**

votes

**1**answer

623 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 ...

**5**

votes

**1**answer

294 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 ...

**18**

votes

**0**answers

644 views

### local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group
$$
L_IG ...

**9**

votes

**4**answers

870 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 ...

**5**

votes

**1**answer

375 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 ...

**9**

votes

**1**answer

747 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**

votes

**0**answers

270 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 ...

**0**

votes

**1**answer

191 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) ...

**2**

votes

**0**answers

218 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.
...

**7**

votes

**1**answer

765 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**

votes

**0**answers

253 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},..., ...

**5**

votes

**1**answer

524 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 ...

**4**

votes

**2**answers

416 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; ...