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.
621 questions
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Reference for the Gelfand duality theorem for commutative von Neumann algebras
The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann ...
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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 ...
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What does it mean for a category to admit direct integrals?
Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals.
Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...
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Is this representation of Go (game) irreducible?
This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph.
Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
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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 claim?...
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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 $B \subset A$. ...
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About the category of von neumann algebras
I am looking for one (or more) reference about properties of the category of von Neumann algebra.
More precisely, in an answer of a previous question, Dmitri Pavlov mentions
that the $W^*$ category ...
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Is the opposite category of commutative von Neumann algebras a topos?
By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict ...
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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 $\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost ...
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Relative commutants of abelian von Neumann algebras
This question arose from the discussion over at the question Centralizers in $C^*$-algebras.
Which von Neumann algebras $N$ satisfy the property that $A' \cap N = B' \cap N \implies A = B$, for all ...
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What's a noncommutative set?
This issue is for logicians and operator algebraists (but also for anyone who is interested).
Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]):
Let $H$ ...
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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 $\...
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Is Dependent Choice all we really need?
http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...
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On complemented von Neumann algebras
Edit: according to Narutaka Ozawa, question 3) is still open in the type $\mathrm{II}_1$ case. In other terms, it is not known whether every topologically complemented type $\mathrm{II}_1$ factor in $...
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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 ...
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What is the relationship between algebraic geometry and quantum mechanics?
The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...
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In which sense the GNS-construction is a functor?
I asked this at mathstackexchange a week ago, without success.
I think the Gelfand–Naimark–Segal construction must be a functor in some sense, but I can't find an explicit statement anywhere. Can ...
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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 ...
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An "exercise" on von Neumann algebra tensor product
The following problem appears to be an easy exercise on von Neumann algebra tensor products, but since I've been failing to find a rigorous proof, I'd like to make sure it's not that trivial. Suppose $...
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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 ...
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If two projections are close, then they are unitarily equivalent
Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$.
The proof that immediately occurs to me uses comparison of ...
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Which sigma-ideals in a sigma-algebra are ideals of null sets?
My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...
17
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reference request : constructive measure theory
As the title said, I would like to know if constructive measure theory has been developed somewhere ?
I am more precisely interested in the (constructive) theory of completely continuous valuation on ...
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answer
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The cyclic subfactors theory: a quantum arithmetic?
Context: First recall some results:
Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980).
A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-Popa ...
16
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4
answers
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Von Neumann algebra associated to the infinite Cuntz algebra
The Cuntz algebra $\mathcal{O}_{\infty}$ is the universal $C^*$-algebra generated by countably infinitely many isometries $s_i$ satisfying the relations $s_i^*s_j = \delta_{ij}$ (there is no condition ...
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Is the group von Neumann algebra construction functorial?
Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...
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Non-commutative geometry from von Neumann algebras?
The Gelfand transform gives an equivalence of categories from the category of unital, commutative $C^*$-algebras with unital $*$-homomorphisms to the category of compact Hausdorff spaces with ...
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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 ...
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Can one associate a "nice" topos to a von Neumann algebra?
The question here inspires my present question.
Reyes proves here that the contravariant functor Spec from the category of commutative rings to the category of sets cannot be extended to the category ...
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Is a C*-algebra with an isomorphic predual a von Neumann algebra?
It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...
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Denseness of inner automorphisms inside automorphisms of hyperfinite type III_1 factor
Let $R$ be the hyperfinite type $III_1$ factor,
and let $Aut(R)$ be its group of automorphisms, equipped with the $u$-topology
(topology of pointwise convergence on the predual).
An automorphism $\...
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On spatial tensor products of von Neumann algebras
Let $H$ be a Hilbert space, and let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras.
We write $\odot$ for the algebraic tensor product,
and $\bar\otimes$ for the spatial tensor ...
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Categorifying the Reals via von Neumann Algebras?
So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to ...
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Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?
I recently gave a talk, where I talked about the tensor category
of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor.
Vaughan Jones, who was in the audience, later told me ...
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answers
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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 ...
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votes
2
answers
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Do torsion-free groups give projectionless group ($C^\ast$) algebras?
One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
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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 ...
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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 ...
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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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Countably decomposable von Neumann algebras
A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...
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Separable von Neumann algebra
What is the simplest argument which shows that each infinite dimensional von Neumann algebra is not separable (in the norm topology)? It seems that this is a kind of folklore: at least I never saw the ...
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Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent?
Regarding the hyperfinite $II_{1}$ factor $R$ as $C^{*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of ...
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Ideals in Factors
One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I_n$, type $II_1$, and type $...
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Do subgroups have "two sided bases"?
Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of
$$
E(g)=\begin{cases}
g &\text{if } g\in H\\\
0 &...
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Do Baumslag-Solitar Group von Neumann algebras have Property $\Gamma$?
A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
- 7,057
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Factor states on C*-algebras
Which C$^*$-algebras admit factor states for which the von Neumann algebra it generates in the corresponding GNS representation is a type III$_1$ factor? For example, do all purely infinite algebras ...
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answer
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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 $Ind(E)$ ...
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Is the domain of an operator valued weight closed under Hahn-Jordan decomposition?
Let $N\subseteq M$ be an inclusion of semi-finite factors with normal faithful semi-finite traces $\operatorname{Tr}_N$ and $\operatorname{Tr}_M$ respectively. Let $T: M^+\to \widehat{N^+}$ be the ...
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A left inverse for the comultiplication on a Hopf von Neumann algebra
Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments.
$\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following ...
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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 ...
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