# Tagged Questions

Subtag of [tag:oa.operator-algebras] for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.

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### Why is every factor a tensor product of a $\sigma$-finite factor and a factor of type I?

Here I am not assuming the factor is represented on a separable Hilbert space. This is quoted on page 370 of Takesaki II, then a bit later on page 381, and I haven't been able to find a proof prior to ...
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### Would a supersymmetric theory of von Neumann algebras be useful?

While looking over the first chapter of 1) Quantum Fields and Strings: A Course For Mathematicians (P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, ...
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### Transfinite induction, a theorem of Pedersen, and chains of subalgebras of $B(H)$

This post is closely related to this one. (In fact I copied some of its content.) Let $H$ be an infinite dimensional separable complex Hilbert space. All $C^{\star}$-subalgebras of $B(H)$ are assumed ...
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### 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. ...
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### “Averaging” in von Neumann algebras

I've been reading recently about various "averaging" tricks in von Neumann algebras. For example, Christensen and Sinclair, in "On von Neumann algebras which are complemented subspaces of $B(H)$." J. ...
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### Can the minimal index of a subfactor take all values in {4cos^2(pi/n);n=3,4,5,…} u [4,infinity]?

Given a subfactor $N\to M$ and a conditional expectation $E:M\to N$, there is a numerical invariant $Ind(E)$ associated to to this situation, called the index of $E$. The possible values of $Ind(E)$ ...
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### Are the compact and Haagerup approximation properties equivalent?

The following essentially implies the equivalence of Anantharaman-Delaroche's compact approximation property (page 337 of http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=...
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### Is the “Laplacian” a MASA in a Burnside Factor?

It is a basic fact that every type $II_{1}$ factor posesses a maximal abelian $*$-subalgebra (MASA). My question concerns the concrete realization of such subalgebras. For example, in the free group ...
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### Do Burnside Group Factors have Gamma?

The Free Burnside group $G=B(2,665)=\langle a,b|g^{665} \rangle$ is infinite, by the work of Adyan and Novikov. Furthermore, the centralizer of any nonidentity element in $G$ is finite cyclic, and so ...
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### Relative Bicommutant

If $A \subseteq \mathcal B(\mathcal H)$ is an algebra of operators that is closed under adjoint, then its bicommutant $A''$ is a von Neumann algebra, and is the ultraweak closure of $A$; this is one ...
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### 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; ...
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### Finite dimensionality of certain $C^{\star}$-algebras

In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up: Let $H$ be a Hilbert space and $a_1, \dots, a_n \in B(H)$ be self-...
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### Monoidal structures on von Neumann algebras

My question is based on the following vague belief, shared by many people: It should be possible to use von Neumann algebras in order to define the cohomology theory TMF (topological modular forms) in ...
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### Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras

The Gelfand-Neumark theorem for commutative von Neumann algebras states that the following three categories are equivalent: (1) The opposite category of the category of commutative von Neumann ...
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### 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 ...
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### 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 Connes'...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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. ...
### 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 < ...