# Questions tagged [von-neumann-algebras]

Subtag of the [oa.operator-algebras] tag 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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### Tomita algebra associated to nsf weight on von Neumann algebra

Let $M$ be a von Neumann algebra, $\varphi$ an nsf weight on $M$ and $\{\sigma_t\}_{t\in \mathbb{R}}$ its associated modular automorphism group. We also consider the GNS-triplet $(\pi_\varphi, \...

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### Why do von Neumann algebras possess identity?

My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...

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### Unitary group of a von Neumann algebra: is it a retract of $U(H)$?

Let $M\subset B(H)$ be a properly infinite von Neumann algebra (the case I care about is $M=$ hyperfinite $\mathrm{III}_1$).
Consider the unitary groups $U(M)$ and $U(H)$ in their strong operator ...

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### A question regarding certain sequences in hyperfinite type $II_1$ factor

Let us consider the hyperfine type $II_1$ factor $\mathcal M$ arising from the inclusion $M_2\subseteq M_{2^2}\subseteq \dots M_{2^k}\subseteq\dots$ of matrix algebras with the normalised trace $\tau$....

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### Inclusion of finite dimensional C*-algebras and relative commutants of subfactors

Given a subfactor $N\subset M$ with finite Jones index, the inclusion of relative commutants $N^{\prime}\cap M\subset N^{\prime}\cap M_1$ (here, $M_1$ is the basic construction of $N\subset M$) is a ...

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### Directed sets of positive elements in noncommutative $\mathrm L^p$ spaces

Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra $\mathcal M$.
If $1<p<\infty$ and $E$ is a nonempty subset of $\mathrm L^p(\mathcal M,\tau)_+$ such that
for every $x\...

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### Relation between factor condition on von Neumann algebras and modularity condition on ribbon fusion categories

A modular tensor category is defined to be a ribbon fusion category $\mathcal{C}$ in which the only objects which commute with all other objects are multiples of the identity. That is, if we denote by ...

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### Depth of the reduced subfactor

Suppose $N\subset M$ is a finite depth subfactor with $[M:N]<\infty$. Consider the reduced subfactor $pNp\subset pMp$ for some projection $p\in N$. How to calculate the depth of $pNp\subset pMp$ in ...

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### Two more topologies on unitary groups

Let $H$ be a separable Hilbert space and let $\operatorname{U}(H)$ be the group of unitary transformations of $H$. It is well known that the weak, strong and compact-open topologies on $\operatorname{...

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### Approximation from below of positive elements in tensor product of von Neumann algebras

Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras.
If $x$ is a positive element of $\mathcal M$ and $y$ is a positive element of $\mathcal N$, it is known that $x\otimes y$ is a positive ...

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### Noncommutative maximal weak $L_1$ norms with respect to sub algebra

Let $(\mathcal M,\tau)$ be a von Neumann algebra with normal finite faithful trace $\tau.$ For any sequence $(x_n)_{n\geq 1}\in \mathcal M$ define $\|(x_n)\|_{\Lambda_{1,\infty}(\mathcal M;\ell_\infty)...

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### Choosing a net of projections from a given collection

Let $\mathcal M$ be a von Neumann algebra. Let $S$ be a nonempty set. Consider $\Omega$ to be the collection of all finite subsets of $S$. Suppose for all $\omega\in\Omega$ we have $I_\omega$ a ...

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### Hyperexpectations from injective subfactors of a type $II_1$ factor

Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $...

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### Ultralimit of $w^*$-continuous maps

Let $\omega$ be a free ultrafilter on $\mathbb N.$ Let $(\mathcal M_n)$ be a sequence of finite von Neumann algebras. Let $\mathcal N$ be another finite von Neumann algebra and we have maps $\phi_n:\...

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### Backwards stable factors

A factor $R$ is called stable if $M_n(R)\cong R$ for all $n>0$. For the sake of this question, we call a factor backwards stable if $R\cong M_n(S)$ implies $S\cong R$ where $S$ is allowed to be any ...

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### von Neumann algebra of canonical commutation relations

In quantum mechanics we have position and momentum operators $P$ and $Q$ acting on $L^2(\mathbb{R})$ in the usual way. I'm wondering what the von Neumann algebra generated by the bounded functions of $...

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### Examples of discrete quantum group actions on commutative von Neumann algebras

Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ ...

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### Inner product on Standard form of von Neumann algebra

Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that
$$\langle x,yz\rangle=\langle zx,y\rangle$...

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### Von Neumann algebras as complemented subspaces

Question: Does there exist a non-injective von Neumann algebra $M\subseteq B(H)$, which is a complemented Banach subspace of $B(H)$?
According to an MO post, this problem was still open as of 2013. I'...

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### Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions

Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...

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### Representing measurable map to compact space as a continuous map

Let $\Omega$ be a measurable space equipped with a $\sigma$-ideal $\mathcal{N}$ (though of as the "null sets"). Define the compact Hausdorff space
$$ \tilde{\Omega} := \mathrm{Spec}(L^\infty(...

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### Defining states on von Neumann algebras from filters on the projection lattices

Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for ...

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### Conditioning a $\mathrm{C}^*$-algebra state with infinite precision

This question (and a second part) have been asked at MSE and gone through two bounties without an answer. I have been beating my head at it for a while without success.
Let $\mathcal{A}$ be a unital $\...

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### Banach isomorphisms between von Neumann algebras

It seems that most people are talking about $*$-isomorphisms of von Neumann algebras. However, I can not find any references for the Banach isomorphisms, i.e., let $A,B$ be two different von Neumann ...

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### Questions related to a paper of Cowling-Haagerup and uniform lattices of $\mathrm{Sp}(1,n)$

I am reading the following paper of Cowling and Haagerup for my master’s thesis. I am new to this area so I am not very conversant. So I do apologize if the questions are silly.
Question 1. In the ...

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### Invariant weights associated to algebraic quantum groups

Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$.
...

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### *-homomorphisms from $L^\infty(0, 1)$ to itself that acts as the identity on continuous functions

Let $\pi: L^\infty([0, 1], \lambda) \rightarrow L^\infty([0, 1], \lambda)$ be a *-homomorphism (where $\lambda$ is the Lebesgue measure on $[0, 1]$), not necessarily normal (otherwise the question is ...

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### Tensor product of operator values weights (in the theory of locally compact quantum groups)

Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object
$$\iota \otimes \psi: (M\overline{\otimes} M)_+ \to \widehat{M_+}.$$
...

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### Extreme points in the space of ucp maps

Suppose $M$ and $N$ are $\mathrm{II}_1$ factors. Let $\tau\mathrm{UCP}(M,N)$ be the convex space of trace-preserving UCP maps from $M$ to $N$, equipped with the topology of pointwise weak* convergence....

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### Intersection of von-Neumann algebra factors

Given two von-Neumann algebra factors $\mathcal M,\mathcal N$, is $\mathcal M\cap\mathcal N$ a factor?
And how about the intersection of infinitely many factors?
Notes:
I know that the intersection ...

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### Cyclic vectors and subfactor inlcusion

Let $N\subset M$ be a be factors acting on a Hilbert space $H$.
Denote by $\mathrm{Cyc}(A)\subset H$ the set of cyclic vectors for $A = M,N$.
I am interested in the equality case of the inclusion $\...

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### What's the purpose of the operator $(\Delta^{-1}+\lambda)^{-1}$ in Tomita-Takesaki modular theory?

I was reading Tomita-Takesaki modular theory (from all the books, and articles), the goal is to relate a von Neumann algebra $\mathcal{A}$ with its commutant $\mathcal{A}'$ on a Hilbert space $\...

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### Compactly supported continuous functions as a Tomita algebra

Let $G$ be a locally compact group with modular function $\delta_G$ and consider $\mathcal{K}(G)$, the set of compactly supported continuous functions $G\to \mathbb{C}$, with the $*$-algebra structure ...

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### Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$?

Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that
$${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$
$${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...

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### Every locally compact group gives rise to a locally compact quantum group

A locally compact quantum group (in the sense of Vaes-Kustermans) consists of the data $(M, \Delta, \varphi, \psi)$ with $M$ a von Neumann algebra, $\Delta: M \to M \overline{\otimes} M$ a normal ...

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### Least upper bound of type I factors

Given two type I (von-Neumann algebra) factors $\mathcal M,\mathcal N$, is there a smallest type I factor containing both $\mathcal M,\mathcal N$?
Notes:
$\mathcal M,\mathcal N$ are over the same ...

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### Takesaki: question about lemma in section "Left Hilbert algebras and weights"

To make this question relatively self-contained, this post is quite long, but the question itself is rather short.
Consider the following fragments in Takesaki's second volume "Theory of operator ...

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### Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity

Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...

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### Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?

Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123.
Why is it possible to choose an ...

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### inclusion of von Neumann algebras implies reversing inequality of its modular operators

I'm working with von Neumann algebras and I stumbled with this statement in a work of Borchers (1999)
Since $\mathcal N \subseteq \mathcal M$, it follows by standard arguments that $\Delta_{\mathcal ...

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### Takesaki II "Connes cocycle derivative"

Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108:
Why are the second and third ...

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### Intersection of finitely many type-I von-Neumann algebra factors

If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M\cap\mathcal N$ a type-I von-Neumann algebra factor?
Notes:
An elementary ...

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### Intersection of type-I von-Neumann algebra factors

Is the intersection of a (possibly infinite) family $\{\mathcal M_i\}$ of type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$) again a type-I von-Neumann algebra factor?

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### Question on tensor product of von Neumann algebras and subfactors

Let $M_1$ and $M_2$ be von Neumann algebras acting on Hilbert spaces $H_1,H_2$ and consider $M=M_1\overline\otimes M_2$ acting on $H_1\otimes H_2$. Let $K$ be an $M$-invariant subspace (so that $P_K\...

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### Takesaki II Lemma 1.13: stuck in proof

Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"):
Here, we associate with an ...

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### subfactors and trace

For every factor $M$ there is an (essentially unique) dimension function (or trace) $\tau_M : M_p \to [0,\infty]$ (where $M_p$ is the lattice of orthogonal projections in $M$).
My question is: Under ...

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### Continuity of the extension of a tracial state with respect to the strong operator topology

Problem: Let $M\subseteq B(H)$ be a finite von Neumann algebra with a faithful tracial state $\tau$. Let $\widetilde{M}$ be the $\tau$-measurable operators on $M$ (recalled below). Extend the trace $\...

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### $\tau$-measurable operator

Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...

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### Difference in tracial and finite von Neumann algebras

A tracial von Neumann algebra $(M,\tau)$ is a von Neumann algebra with a faithful normal tracial state $\tau$ on $M$. That is, $\tau$ is a function from $M \to \mathbb{C}$ such that it is a faithful ...

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### Existence of conditional expectations map onto subalgebras

Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existence of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any ...