**12**

votes

**1**answer

784 views

### W*-completion of a C*-algebra?

tl;dr: Is there such a thing as a W*-completion of a C*-algebra, and if so, where can I read about it?
I'm wondering about the relationship between (abstract) C*-algebras and W*-algebras. On the one ...

**17**

votes

**0**answers

538 views

### The Mackey Topology on a Von Neumann Algebra

Every von Neumann algebra $\mathcal M$ is the dual of a unique Banach space $\mathcal M_* $. The Mackey topology on $\mathcal M$ is the topology of uniform convergence on weakly compact subsets of ...

**16**

votes

**0**answers

655 views

### Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?

Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...

**7**

votes

**1**answer

464 views

### How does the right regular of GL(n, R) and GL(n,Qp) decompose?

The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right ...

**8**

votes

**1**answer

457 views

### Lifting of a ucp map with values in a von Neumann algebra ultraproduct of matrix algebras

Let $u:A \to \prod_{\mathcal U} M_n$ be a unital completely positive map (ucp) from a unital separable $C^*$algebra into the von Neumann algebra ultraprodut $\prod_{\mathcal U} M_n$.
Here $\mathcal ...

**15**

votes

**4**answers

998 views

### Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?

I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{Tr(A^* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost commuting (with ...

**24**

votes

**1**answer

1k views

### Tomita-Takesaki versus Frobenius: where is the similarity?

I've often heard Alain Connes say that the modular flow of Tomita-Takesaki theory should be thought of as a characteristic zero analog of the Frobenius endomorphism. ... can anyone justify this ...

**5**

votes

**1**answer

296 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

**0**answers

739 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

**1**answer

468 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

**0**answers

140 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

**2**answers

418 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 ...

**5**

votes

**3**answers

1k 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

**1**answer

210 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

**2**answers

301 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

**2**answers

548 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

**1**answer

701 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

**4**answers

977 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

391 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 ...

**10**

votes

**1**answer

909 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 ...

**4**

votes

**0**answers

351 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

**0**answers

288 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

205 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) ...

**12**

votes

**4**answers

772 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

**0**answers

229 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.
...

**8**

votes

**1**answer

855 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

**0**answers

203 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

**2**answers

343 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

**1**answer

733 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 ...

**22**

votes

**3**answers

2k 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

**2**answers

580 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

**1**answer

308 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

**1**answer

417 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

**1**answer

984 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 ...

**7**

votes

**0**answers

268 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},..., ...

**6**

votes

**1**answer

582 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

**2**answers

412 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 ...

**7**

votes

**3**answers

505 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

**2**answers

584 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

**1**answer

447 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

**1**answer

625 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.
...

**6**

votes

**1**answer

643 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

**1**answer

409 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

**1**answer

452 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

**0**answers

238 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 ...

**11**

votes

**2**answers

680 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

**3**answers

396 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

**2**answers

439 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; ...

**7**

votes

**1**answer

403 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

**3**answers

419 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 ...