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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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 ...
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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 $\...
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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 $E_i:M\to M_i$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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 ...
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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 =\|...
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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 ...
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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$ ...
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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 ...
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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?
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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
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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$ ...
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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 ...
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$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}...
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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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2
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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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2
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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 $\...