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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2 votes
0 answers
71 views

About normal states in abstract von Neumann algebras

In the book "Fundamental of the theory of operator algebras" (KAdisong and Ringrose, Vol 2) we have the Corollary 7.1.16 but this was state only for concrete von Neumann algebras (because ...
4 votes
0 answers
121 views

Is every pointwise-weakly continuous one-parameter group of automorphisms of B(H) given by a Hamiltonian?

Let $\mathcal H$ be a Hilbert space, $\mathscr B(\mathcal H)$ be the von Neumann algebra of all bounded operators on $\mathcal H$, and let $\sigma $ be a one-parameter group of automorphisms of $\...
2 votes
1 answer
206 views

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 ...
7 votes
1 answer
233 views

Approximately semifinite factors

For the sake of this question, lets call a factor $M$ approximately semifinite if there exists an increasing net of semifinite subfactors $M_i$, $i\in J$, with conditional expectations $E_i:M\to M_i$ ...
8 votes
0 answers
229 views

Question about the homogeneity of the state space of a type $\rm{III}_1$ factor

I'm reading the paper Homogeneity of the State Space of Factors of Type $\rm{III}_1$ by Connes and Størmer. Homogeneity of the state space means that all normal states are approximately unitarily ...
7 votes
1 answer
361 views

Positive cone in Haagerup L²-space: how much information does it contain?

Given a von Neumann algebra $A$, its Haagerup $L^2$-space $H:=L^2A$ (also known as the standard form of the Neumann algebra) comes equipped with a positive cone $P\subset H$. Question:    How much ...
4 votes
1 answer
215 views

Strengthening the direct integral decomposition of von Neumann algebas

Let $M$ be a von Neumann with separable predual. It well known that one can write $M$ as a direct sum $M=M_I\oplus M_{II} \oplus M_{III}$ of von Neumann algebras of types $I$, $II$ and $III$. It is ...
0 votes
0 answers
161 views

How to show that every Von Neumann algebra is unital?

I was reading the book on operator algebra by Kehe Zhu. The proof of theorem 17.7 (page 107) goes like this : He first considered the set of all non-empty finite subsets of the set of all projections ...
4 votes
1 answer
197 views

Ergodic actions and deviation from invariance

Let $M$ be a von Neumann algebra and let $(\phi_t)$ be an ergodic point-$\sigma$-weakly continuous one-parameter group of automorphisms $\phi_t\in \mathrm{Aut}(M)$, i.e., $\Vert\omega-\omega\circ\...
1 vote
0 answers
264 views

Using the von Neumann crossed product to introduce a measure on the orbit space?

Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space). Question: is there a natural way of using the ...
1 vote
1 answer
242 views

Intersection of two intermediate subalgebras

Suppose $B\subset A$ is an inclusion of simple $C^*$-algebras with a conditional expectation of (Watatani) index-finite type and $B^{\prime}\cap A=\mathbb{C}$. Then we know $B^{\prime}\cap A_1$ is ...
4 votes
1 answer
180 views

Does $N \mathbin{\bar{\otimes}} N^{\mathrm{op}}$ act on $L^2(N)$?

Let $N$ be a von Neumann algebra and $N^{\mathrm{op}}$ its opposite. The standard form $L^2(N)$ is an $N$-$N$-bimodule, or equivalently a module over $N \otimes_{\mathrm{alg}} N^{\mathrm{op}}$. Does ...
5 votes
2 answers
292 views

Projections in atomless von Neumann algebras

Let $\varepsilon>0$. If we consider a sequence $\{f_n\}$ in $L_\infty(0,1)$, then there exists a very small subset $A$ of $(0,1)$ with $m(A)<\varepsilon$ such that $$\|f_n \chi_A\|_\infty =\|...
5 votes
1 answer
217 views

Function algebra of Furstenberg boundary $\partial_F \Gamma$: when is it a $W^*$-algebra?

Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely ...
1 vote
1 answer
122 views

Compare the weight of $p\vee q$ and that of $p+q$

Let $M$ be a von Neumann algebra. If it has a semifinite faithful normal trace $\tau$, then we have $\tau(p\vee q)\le \tau(p)+\tau(q)$. However, for the weight (even a faithful normal state) $\omega$ ...
1 vote
1 answer
154 views

Tensor product of faithful normal states is faithful

I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful. I also ...
5 votes
1 answer
390 views

von Neumann subalgebra having separable predual

Let $M$ be a von Neumann algebra. Let $x,y$ be two self-adjoint operators in $M$. Are there any von Neumann subalgebra $A$ of $M$ containing $x,y$ such that the predual of $A$ is separable?
4 votes
1 answer
167 views

weights of projections and norms of operators in a von Neumann algebra

Let $M$ be an atomless von Neumann algebra equipped with a (semifinite faithful normal) weight $w$. Let $x\in M$ and let $\varepsilon>0$. Can we find a constant $\delta>0$ such that whenever a ...
5 votes
1 answer
182 views

Hyperfinite factors and increasing fatorization of states

If a factor $R$ contains a matrix algebra $M\subset R$ (i.e., a $M$ is a type $I_n$ factor), then $R \cong M \otimes M^c$ where $M^c=R\cap M'$ is the relative commutant. Each state $\omega$ on $R$ ...
4 votes
1 answer
217 views

How to classify von Neumann algebra bundles?

If we consider algebra bundles over X where the fiber is an algebra of bounded operators in a separable Hilbert space H over the complex numbers. I learn from "Isomorphism Classification of Operator ...
4 votes
1 answer
343 views

Families of representations of von Neumann algebras

Let $A$ be a von Neumann algebra and let $H$ be a (separable) Hilbert space. It is known (see e.g., Section IV, Thm. 5.5 of Takesaki I) that there exists a Hilbert space $K$ such that $A \subset \...
2 votes
1 answer
118 views

$K_0$ group of an infinite factor

The following question was already posted in this link but I could not understand hints given in this post. Let $\mathcal{M}$ be an infinite factor and my question is how to prove that $K_0(\mathcal{M}...
2 votes
0 answers
138 views

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 ...
13 votes
0 answers
204 views

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 ...
2 votes
1 answer
148 views

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:\...
3 votes
1 answer
86 views

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 ...
0 votes
0 answers
64 views

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 $\tau$....
1 vote
0 answers
104 views

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 ...
2 votes
2 answers
129 views

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\...
3 votes
0 answers
230 views

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{...
3 votes
0 answers
199 views

What are the first non-maximal non-group-subgroup simple irreducible subfactors?

Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$ is normal if the biprojections $e_{\...
2 votes
0 answers
113 views

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 ...
4 votes
1 answer
154 views

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 ...
3 votes
0 answers
55 views

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)...
4 votes
1 answer
697 views

Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant. Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
5 votes
1 answer
163 views

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 $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
7 votes
3 answers
689 views

Jordan-Hölder theorem for subfactors?

All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors. First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the ...
1 vote
1 answer
90 views

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 ...
2 votes
1 answer
89 views

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 $...
3 votes
0 answers
302 views

Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$?

The fundamental group $\mathcal{F}(N \subset M)$ of a unital inclusion of II$_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \...
4 votes
1 answer
148 views

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 ...
5 votes
1 answer
240 views

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 $...
4 votes
1 answer
135 views

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 $M$ ...
1 vote
0 answers
76 views

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$...
3 votes
0 answers
251 views

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'...
3 votes
2 answers
248 views

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(...
2 votes
1 answer
158 views

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 ...
1 vote
1 answer
164 views

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 $\...
2 votes
0 answers
85 views

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 $\psi$. ...
4 votes
0 answers
105 views

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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