13
votes
2answers
850 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
0answers
267 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 ...
5
votes
2answers
691 views

Universal characterization and explicit description of elements of the group von Neumann algebra and the crossed product

Group von Neumann algebras and crossed products for a locally compact group G can be constructed in many different ways. For example, one can take the von Neumann algebra generated by certain ...
13
votes
3answers
1k views

Is the group von Neumann algebra construction functorial?

Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...
7
votes
3answers
1k views

Definition of a von Neumann algebra

Is there a way to equip every C*-algebra A with a functorial topology such that the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra? Here A** denotes the dual of A* in ...
4
votes
1answer
252 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 ...
11
votes
5answers
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
1answer
366 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 ...