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5
votes
1answer
281 views

representing tensor-C*-categories in BIM

Given a factor M (=von Neumann alg. with center ℂ), let us write BIM for the ⊗-C*-category of M-M-bimodules. Which ⊗-C*-categories can one faithfully embed into BIM? ⓵ Are ...
18
votes
0answers
636 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 ...
4
votes
1answer
455 views

topologies on U(H)

There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space: the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some ...
0
votes
0answers
137 views

For a separable v-N algebra M, how to see $M \rtimes \mathbb{R}$ as a subalgebra of $M \otimes B(L^2)$?

For a v-N algebra $M$ acting as bounded operators on a separable Hilbert space $H$, how to see $M \rtimes \mathbb{R}$ as a subalgebra of $M \otimes B(L^2(\mathbb{R})$? Why I am confused is because ...
2
votes
2answers
360 views

Spatial isomorphisms of tensor product of factors

Suppose $N \subset M$ are two factors, neither of them Type I, acting on a separable Hilbert space $H$. Let $\pi_1$ be a faithful normal representation of $N$ and $\pi_2$ a faithful normal ...
3
votes
3answers
730 views

What is the difference between a primary representation and a irreducible representation?

I am currently reading some of Mackey's work on unitary representation. Given a locally compact group $G$ and a unitary representation $\pi : G\rightarrow U(H)$. As far as I understood it, the ...
1
vote
1answer
198 views

when is an element of $M_n(M)$ $\ast$-free from $M_n(\mathbb{C})$ for a $\ast$-non-commutative probability space $M$.

From "Lectures on the combinatorics of free probability" by Nica and Speicher we have a necessary sufficient criteria for an element of $M_n(M)$ being free from $M_n(\mathbb{C})$ for a non-commutative ...
4
votes
2answers
278 views

v-Na generated by

Given two free semicirculars X_1 and X_2 and a projection h in the von-Neumann algebra generated by X_1, how does one show that the von-Neumann algebra generated by {X_1, hX_2(1-h)} is a factor? It is ...
5
votes
2answers
459 views

Presenting the Hyperfinite II_1 Factor

It's well known that all hyperfinite $\mathrm{II}_1$ factors are isomorphic. I risk the wrath of MathOverflow elders to ask if a particular isomorph is easier than others to handle. In particular, is ...
5
votes
1answer
694 views

Subspaces of a Subfactor

Is the following true? Let $\mathcal N \subset \mathcal M$ be a subfactor. There is a bijective correspondence between the ultraweakly closed subspaces of $\mathcal M$ that are bimodules over ...
9
votes
4answers
851 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
1answer
373 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
1answer
733 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 ...
2
votes
0answers
309 views

Connes fusion and the composition of completely positive maps

Let $N$ be a type $II_{1}$-factor with trace $\tau$. An $N-N$ correspondence is a Hilbert $N$-bimodule $H$ where the left and right actions are both ultraweakly continuous. Equivalently, a ...
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 ...
0
votes
1answer
185 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) ...
10
votes
4answers
652 views

What is the right definition of “real von Neumann algebra”?

Recall that a real C*-algebra is a Banach $\ast$-algebra $A$ over $\mathbb{R}$ which satisfies the standard C* identity and which also has the property that $1 + a^{\ast}a$ is invertible in the ...
2
votes
0answers
212 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
1answer
747 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 ...
5
votes
0answers
195 views

When can't spaces of correspondences distinguish type $II_{1}$ factors?

If $M$ is a type $II_{1}$ factor with trace $\tau$, let $Corr(M)$ denote the space of unitary equivalence classes of $M-M$ correspondences (binormal $M-M$ bimodules) equipped with Popa's analogue of ...
5
votes
2answers
328 views

Derivation of von Neumann algebra which is zero on MASA

Are there any example of $II_1$-factor $M$ with maximal abelian von Neumann subalgebra $A$ and non-zero derivation $\delta:M\rightarrow B(H)$ such that $\delta(a)=0$ for every $a\in A$?
7
votes
1answer
574 views

Can we characterize the spatial tensor product of von Neumann algebras categorically?

The tensor product of commutative algebras is exactly their coproduct in the category of commutative algebras. In other words, if A and B are two commutative algebras, then the covariant functor that ...
15
votes
3answers
1k views

Can we recover a von Neumann algebra from its predual?

By definition, a von Neumann algebra is a C*‑algebra A that admits a predual, i.e., a Banach space Z such that Z* is isomorphic to the underlying Banach space of A. (We require that isomorphisms in ...
1
vote
2answers
505 views

Decomposition of an abelian von Neumann algebra

Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance. "Let $\pi$ be a non-degenerate ...
4
votes
1answer
290 views

When can the group von Neumann algebra of a one-relator group be isomorphic to a free group factor?

Let $G=\langle a,b | R \rangle$ be a one-relator group. When can the left group von Neumann algebra $LG$ be isomorphic to a free group factor? Jesse and Andreas have "trapped the lion" pretty well ...
3
votes
1answer
382 views

Weakly solid factors?

A type $II_{1}$ factor $\mathcal{M}$ is solid if for every diffuse von Neumann subalgebra $\mathcal{A}$ of $\mathcal{M}$, the relative commutant $\mathcal{A}'\cap \mathcal{M}$ is injective. (See ...
13
votes
1answer
968 views

Endomorphism of type III factor: can it satisfy $\phi\circ\phi = \phi\oplus\phi$?

I'm still trying to get some feeling about this question... Given Jesse Peterson's answer to this question (he showed that $\phi\circ\phi\sim\phi$ is impossible), I suspect that the following is also ...
6
votes
0answers
249 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
1answer
509 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 ...
7
votes
2answers
406 views

endomorphism of factor: can it be idempotent up to congugacy?

Let $M$ be a factor, and let $\phi:M\to M$ be an irreducible endomorphism ("irreducible" means that the relative commutant of $\phi(M)$ in $M$ is trivial). Let's also assume that $\phi$ is not ...
5
votes
3answers
427 views

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 ...
6
votes
2answers
547 views

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, ...
9
votes
1answer
431 views

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 ...
6
votes
1answer
604 views

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. ...
5
votes
1answer
523 views

“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. ...
12
votes
1answer
397 views

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 $Ind(E)$ ...
7
votes
1answer
415 views

Are the compact and Haagerup approximation properties equivalent?

The following essentially implies the equivalence of Anantharaman-Delaroche's compact approximation property (page 337 of ...
7
votes
0answers
207 views

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 ...
9
votes
1answer
516 views

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 ...
5
votes
3answers
360 views

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 ...
4
votes
2answers
408 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; ...
6
votes
1answer
384 views

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 ...
4
votes
3answers
372 views

Ideal of “Compact Operators” in a W*-algebra which gives the sigma-strong-* topology.

In the case of bounded operators on a Hilbert space $\mathcal{H}$, $L(\mathcal{H})$, there are multiple descriptions of the $\sigma$-strong-* topology, namely: 1) As given by seminorms ...
20
votes
0answers
1k views

Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...
6
votes
1answer
575 views

Proof of PCT theorem for Haag-Kastler nets in QFT

This question is about a theorem in the Haag-Kastler axiomatic approach to quantum field theory (QFT), also known as axiomatic or algebraic or local QFT. PCT stands for parity, charge and time, a ...
12
votes
0answers
658 views

Must we close weakly to apply the spectral theorem?

Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate. The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...
3
votes
2answers
405 views

Reference for von Neumann algebras coming from a group algebra twisted by a 2-cocycle?

I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce. Does someone know some good references (article, book)? It would be very helpful for me. To be more precise, ...
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 ...
4
votes
2answers
265 views

Lifting surjective von Neumann algebra homomorphisms

Is the following true? What's a nice proof? Let $M$ and $N$ be von Neumann algebras, and let $\phi:M\rightarrow N$ be a normal, surjective, *-homomorphism. Is there a normal *-homomorphism ...
12
votes
4answers
969 views

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 ...