The von-neumann-algebras tag has no usage guidance.

**12**

votes

**1**answer

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### Is Dependent Choice all we really need?

http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...

**9**

votes

**1**answer

341 views

### Do subgroups have “two sided bases”?

Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of
$$
E(g)=\begin{cases}
g &\text{if } g\in H\\\
0 ...

**8**

votes

**4**answers

476 views

### Does such a subgroup exist?

I am looking for a certain masa in a $II_1$ factor which is singular and has nontrivial Takesaki invariant.
For this I am looking for an example of an inclusion of groups $H\subset G$ such that:
$G$ ...

**5**

votes

**5**answers

1k views

### Measurable functions and unbounded operators in von Neumann algebras

How do you define unbounded measurable functions for a general von Neumann algebra?
For the commutative algebra $L^\infty(X,\mu)$, we can consider the space of all measurable functions that are ...

**7**

votes

**1**answer

273 views

### A coalgebraic description of the hyperfinite II_1 revisited

Back here I was asking for a coalgebraic characterisation of the hyperfinite $II_1$ factor. Recall the latter's construction by forming the inductive limit of a chain of matrix algebras $R \to M_2(R) ...

**9**

votes

**3**answers

2k views

### What is the relationship between algebraic geometry and quantum mechanics?

The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...

**4**

votes

**1**answer

270 views

### One-parameter semigroups of bimodules

Suppose M is a von Neumann algebra.
Consider a monoidal category of bimodules over M.
Here a bimodule is a Hilbert space with two normal representations of M.
The monoidal structure is given by ...

**7**

votes

**2**answers

578 views

### Do torsion-free groups give projectionless group ($C^\ast$) algebras?

One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...

**11**

votes

**5**answers

2k views

### Non-commutative geometry from von Neumann algebras?

The Gelfand transform gives an equivalence of categories from the category of unital, commutative $C^*$-algebras with unital $*$-homomorphisms to the category of compact Hausdorff spaces with ...

**8**

votes

**1**answer

284 views

### Is there a coalgebraic characterisation of the hyperfinite II_1 factor?

Peter Freyd showed that the real interval [0, 1] is a final coalgebra for a functor on sets equipped with two points, which sends such a set to the 'wedge' of two copies of itself, identifying the ...

**7**

votes

**1**answer

392 views

### Maximal localizations of von Neumann algebras

Suppose M is a von Neumann algebra.
Denote by L its maximal noncommutative localization,
i.e., the Ore localization with respect to the set of all left and right regular elements,
i.e., elements whose ...

**13**

votes

**3**answers

753 views

### Categorifying the Reals via von Neumann Algebras?

So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to ...

**4**

votes

**2**answers

646 views

### Operator Valued Weights

One of the basic tools in subfactors is the conditional expectation. If $N\subset M$ is a $II_1$-subfactor (or an inclusion of finite factors), then there is a unique trace-preserving conditional ...

**5**

votes

**7**answers

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### Hochschild/Cyclic Homology of von Neumann Algebras: Useless?

Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...

**10**

votes

**2**answers

476 views

### Ideals in Factors

One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I_n$, type $II_1$, and type ...

**8**

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**0**answers

545 views

### Pimsner-Popa Bases

Let $N\subset M$ be a finite index $II_1$-subfactor. Let $B=\{b_i\}$ be a finite orthonormal (Pimsner-Popa) basis for $M$ over $N$. Let $d=[M\colon N]^{1/2}$. It is well known that $B_1=\{d b_{i_1} ...

**7**

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**0**answers

292 views

### Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive?

Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.
There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < ...