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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Type III factor examples?

How to prove the crossed product of $G$ and von Neumann algebra $M$, where $G$ is locally compact group acting on $M$ via free ergodic action and $M$ is type $II_{\infty}$ factor, is type $III$ factor,...
mathlover's user avatar
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On projection theory for inseparable Hilbert spaces

How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?
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What is spectral multiplicity for multiplication operators in general von Neumann algebra set up?

When two multiplication operators $M_{f}$ and $M_{g}$ acting on $L^2(X,\mu) $and $L^2(Y,\nu)$ are unitary equivalent? How multiplicity function look like here? What is the spectral multiplicity in ...
mathlover's user avatar
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Projections in properly infinite factor

Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let $e$ be a projection in a properly infinite factor. Is it always true that $e\sim 1$ or $1-e\...
Meisam Soleimani Malekan's user avatar
7 votes
1 answer
420 views

Open projections and Murray-von Neumann equivalence

Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
Masayoshi Kaneda's user avatar
3 votes
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Certain limits of normal completely positive idempotents

Suppose $L_n$ is a sequence of normal completely positive mappings of $B(H)$ into itself of norm strictly less than one with the property that $L_n(L_m(A)) \to L_m(A)$ in the strong operator topology ...
Bob Powers's user avatar
4 votes
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Algebraic sum of relative commutants in a finite von Neumann algebra

Let $(M,\tau)$ be a finite von Neumann algebra with faithful normal tracial state $\tau$. Suppose that $A \subset M$ is a finite-dimensional abelian unital subalgebra--say $A = W^*(p_1,\dots, p_n)$ ...
Scott Atkinson's user avatar
2 votes
1 answer
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an example of discrete factor group of exponential growth

I would like to understand if there is a discrete infinite group of exponential growth/intermediate growth such that its group von Neumann algebra is a $II_1$ factor. I would be happy to get an ...
Rauan Akylzhanov's user avatar
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85 views

Injectivity of Fermion algebras

Is the Fermion algebra or $-1$-Fock space as defined in https://arxiv.org/pdf/math/0303045.pdf hyperfinite? Any references?
Mathbuff's user avatar
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$e\precsim f$ and $1-e\precsim 1-f$ imply $e\sim f$?

Denote by $\precsim$ the order comes from "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let e and f be two projections in a properly infinite von Numann algebra ...
Meisam Soleimani Malekan's user avatar
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Minimal Tensor product of von Neumann algebras

Let $M$ and $N$ be two von Neumann algebras. A natural way to define $M\overline{\otimes}N$ is the von Neumann algebra generated by the algebraic tensor product $M\otimes N$ in $B(H\otimes K)$ where $...
Mathbuff's user avatar
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Tensor product of traces of von Neumann algebras

I am trying to understand how to define tensor product of normal semifinite faithful (in short n.s.f) traces between two von Neumann algebras $(M_1,\tau_1)$ and $(M_2,\tau_2),$ where $\tau_i$ is the n....
Mathbuff's user avatar
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1 answer
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Suppose $(A,H)$ is a vNa. When is an inner automorphism of $\mathcal{B}(H)$ an inner automorphism of $A$

Let $A$ be a von Neumann algebra acting on a Hilbert space $H$. Write $\mathcal{B}(H)$ for the bounded linear operators on $H$. Suppose that $\rho:\mathcal{B}(H) \rightarrow \mathcal{B}(H)$ is an ...
Peter's user avatar
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The space of $\mathcal A$-Hilbert Schmidt Operators.

Let $\mathcal A$ be a finite von Neumann algebra with finite trace $\tau$ and let $l^2(\mathcal A)$ be the Hilbert space completion of $\mathcal A$ with respect to the inner product induced by $\tau$, ...
H1ghfiv3's user avatar
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2 answers
249 views

Comparison-like lemma

Denote by $\precsim$ the order comes from "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let $e$ and $f$ be two projection in a von Numann algebra $\mathfrak M$. ...
Meisam Soleimani Malekan's user avatar
2 votes
0 answers
163 views

An operator valued Egoroff's theorem

The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
ABB's user avatar
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6 votes
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Relation between tracial norm and operator norm on a von Neumann algebra

First, let me preface this by saying that I am fairly new to the wide field of (finite) von Neumann algebras. In my studies of $L^2$-invariants, I am mostly concerned with Group von Neumann algebras, ...
H1ghfiv3's user avatar
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7 votes
2 answers
476 views

The von Neumann algebra generated by a non-closable operator

Let $H$ be a separable Hilbert space and let $M$ be a densely defined operator $\mathcal{D}(M) \subset H \to H$. It is closable iff its adjoint $M^{\star}$ is densely defined, and then its closure $\...
Sebastien Palcoux's user avatar
8 votes
1 answer
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Does every integer map generate a von Neumann algebra of type I?

Consider a map $m: \mathbb{N} \to \mathbb{N}$ (we call it an integer map). Let $E_r$ be the set $m^{-1}(\{r\})$. Let $H$ be the Hilbert space $\ell^2(\mathbb{N})$ and consider the densely defined ...
Sebastien Palcoux's user avatar
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Shift on trivalent directed tree, operator and von Neumann algebra

Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\...
Sebastien Palcoux's user avatar
4 votes
1 answer
104 views

Does the inclusion of the discretized group into itself lift to the group von Neumann algebras?

Let $G$ be a locally compact, Hausdorff and $2^{nd}$-countable group and let $G_{disc}$ be the same group with the discrete topology. We have a continuous (and bijective) homomorphism given by $$ ...
Adrián González Pérez's user avatar
3 votes
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298 views

Connes' fusion product

Let $B$ be a von-Neumann algebra and let $M$ be a right-Hilbert-$B$-module and let $N$ be a left-Hilbert-$B$-module. In this situation, Connes' fusion product $M \boxtimes_B N$ of $M$ and $N$ over $B$ ...
Matthias Ludewig's user avatar
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shifts in Baer*-rings

Let $A$ be a Baer*-ring with the unit $1$. An important point that holds in every Baer*-ring is this: for every element $y\in A$ there is an smallest projection $e_y\in A$, called the left projection ...
ABB's user avatar
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relative amenability of von Neumann algebra

Let $\cal{M}$ be a finite von Neumann algebra and $\cal{N}$ be a von Neumann subalgebra of $\cal{M}$. The von Neumann algebra $\cal{M}$ is is amenable relative to $\cal{N}$ if there exists a norm ...
Albert harold's user avatar
3 votes
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Type III von Neumann algebra generated by one operator

Is it possible to explicitly construct the Hilbert space $H$ and operator $T \in B(H)$ such that the von Neumann it generates is type $III$ factor? I would like to see an example.
truebaran's user avatar
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Index of a subfactor of a full $II_1$ factor

On pg. 151 of "Coxeter Graphs and Towers of Algebras" by F.M. Goodman, P. de la Harpe, and V.F.R. Jones (1989), it is stated that there is no known example of a full $II_1$ factor having a subfactor ...
L.C. Ruth's user avatar
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$C^*$-algebra which is not von Neumann, but satisfies the property that ever self-adjoint element is

Consider the following theorem by Aupetit. Let $A$ and $B$ be two von-Neumann algebras and let $\phi$ be a spectrum-preserving linear mapping from $A$ onto $B$. Then $\phi$ is a Jordan isomorphism. ...
user860374's user avatar
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254 views

Can we extend c.p. normal maps on a finite von Neumann algebra $M$ to $L_0(M)_+$?

Suppose that $M$ is a von Neumann algebra with a finite, normal, faithful trace $\tau$. Let $T\colon M\to M$ be a completely positive, normal map. Can $T$ be extended to a `positively linear map' ...
Tomasz Kania's user avatar
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10 votes
1 answer
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A small corner w.r.t. a masa in a von Neumann algebra

Let $A \cong L^\infty[0,1]$ be a non-atomic maximal abelian *-subalgebra in $M \cong B(L^2[0,1])$ (or any von Neumann algebra $M$). Is the following true? For every $T \in M$ and $\epsilon>0$, ...
Narutaka OZAWA's user avatar
2 votes
1 answer
314 views

The fundamental group of the von Neumann algebra of a free group of infinite rank

It is well-known that that the fundamental group (in the sense of Murray and von Neumann) of the factor $L(F_{\mathbb{N}})$ is $\mathbb{R} \smallsetminus \{0\}$. I think that by the cutting and ...
heller's user avatar
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0 answers
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Decomposition of a von Neumann Algebra into Factors

I know that a von Neumann algebra on a separable Hilbert space can be (uniquely) decomposed into factors. But is there any non-trivial example showing how we can explicitly compute it? Non-trivial: ...
Fan's user avatar
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0 answers
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Approximately inner conditional expectations of $II_{1}$ factors

In many contexts it is helpful to think of conditional expectations as averages of unitary conjugates, a standpoint vindicated by many standard techniques in the theory of finite von Neumann algebras. ...
Jon Bannon's user avatar
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5 votes
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Examples of non-proper conditional expectations onto von Neumann subalgebras of $II_{1}$ factors

Denote by $M$ be a type $II_{1}$ factor with trace $\tau$, and $1\in N\subseteq M$ a von Neumann subalgebra. What are some examples of such inclusions for which the normal conditional expectation $\...
Jon Bannon's user avatar
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3 votes
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Existence of a unique cyclic and separating vector in a *-representation

I'm interested in knowing the requirements for a $*$-representation, $\pi_{\omega}$, of a C*-algebra, $\mathbb{C}(\mathcal{G})$, (or equivalently the requirements for the unitary representation, $U_{\...
B. T.'s user avatar
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3 votes
1 answer
233 views

Reference request for c*-algebra embedding into the atomic part of its double dual

Let $A$ be a C*-algebra. According to operator algebraists, it is well known that $A$ embeds into the atomic part of its double dual in the following sense: if $z$ is the central projection in $A^{**}$...
Marten Wortel's user avatar
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Foliations, von Neumann algebras and measurability

In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...
truebaran's user avatar
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1 vote
1 answer
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Example of an amenable enveloping von-Neumann algebra

I am looking for an example of an infinite dimensional $C^{*}$-algebra whose second dual is amenable. Can anyone supply a suggested reference? Many thanks in advance. Edit: If this is ...
roo's user avatar
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18 votes
0 answers
694 views

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 $...
Narutaka OZAWA's user avatar
7 votes
0 answers
1k views

Books on von Neumann algebras

I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
Mathbuff's user avatar
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3 votes
2 answers
507 views

Von Neumann Algebra isomorphism extension

I have Von Neumann algebra $\mathfrak{U}$ and a weakly dense *-subalgebra $A$. I have another Von Neumann Algebra $\mathfrak{V}$ and an injective *-homorphism $$\phi: A\longrightarrow \mathfrak{V}$$ ...
Carlos De la Mora's user avatar
4 votes
1 answer
617 views

Von Neumann Algebras and Measure Theory

We know that a abelian von Neumann algebras $\mathcal{A}$ is isomorphic to $\mathscr{L}^{\infty}(X)$ for some special measure space $(X,\Sigma,\mu)$. For that reasone one can see an arbitrary von ...
Alexander Yates's user avatar
3 votes
0 answers
138 views

Whether a projection can "overlap" certain projections yet not commute with them

Question here about how the projections of a von Neumann algebra $\mathcal{R}$ might be arranged, relative to a projection that is not in $\mathcal{R}$. Stipulate the following:     $H$ is ...
Doug McLellan's user avatar
4 votes
0 answers
117 views

Smoothness in von Neumann algebra of measurable functions

Let $A=L^{\infty}(M)$ be an algebra of essentially bounded measurable function on manifold $M$. Let $D$ be a first order elliptic differential operator acting on some hermitian bundle $S$ over $M$ (...
truebaran's user avatar
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8 votes
2 answers
775 views

Weak*-norm continuous operators on von Neumann algebras

Let $M$ be a von Neumann algebra with predual $M_*$, and let $T\colon M\to M$ be a bounded, linear map. Let us say that $T$ is (sequentially) weak*-norm continuous if for every net (sequence) $(a_j)_j$...
Hannes Thiel's user avatar
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3 votes
0 answers
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A point concerning absolute value of functionals

Let $M$ be a von Neumann sub-algebra in $B(H)$. Let $\phi$ be a normal functional on $M$. Assume $\psi$ is a normal functional on $B(H)$ with $\psi_{|_M}=\phi$ (note that $\phi$ and $\psi$ may have ...
ABB's user avatar
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1 vote
1 answer
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A Baer *-ring which is not embedded into $B(H)$

Assume $A$ is a complex $*$-algebra which is also a Baer*-ring. Q. Can we concluded that there exists a Hilbert space $H$ such that $A$ is embedded in $B(H)$ as a Baer*-ring? What about when $A$ ...
ABB's user avatar
  • 3,898
5 votes
1 answer
157 views

Does every non-type-I factor's projection lattice admit a dense embedding of the standard continuum-collapsing poset?

Let $R$ be a non-type-I factor acting on a separable Hilbert space. Let $P(R)$ be the set of $R$'s projections with the usual ordering ($x \leq y \iff$ range$(x) \subseteq$ range$(y)$) under which it ...
Doug McLellan's user avatar
3 votes
1 answer
231 views

How much of a factor's structure is determined by the order-type of its projection lattice?

H. A. Dye showed that a type II or III factor $R$ is determined, up to *-algebraic isomorphism or anti-isomorphism, by the ortholattice-isomorphism type of its projection lattice ("ortholattice-...
Doug McLellan's user avatar
6 votes
2 answers
657 views

The topology of pointwise convergence with the adjoint operator on a von Neumann algebra

Let $H$ be a Hilbert space and ${\mathcal A}$ a von Neumann algebra in $B(H)$. Let us endow ${\mathcal A}$ with the topology of pointwise convergence with the Hermitian adjoint operator, i.e. a net $...
Sergei Akbarov's user avatar
3 votes
1 answer
201 views

Embedding of von Neumann algebras

Let $A$ and $B$ be two purely infinite von Neumann algebras. Under which conditions $A$ can be embedded in $B$? For example is it true that every infinite factor can be embedded into the another ...
truebaran's user avatar
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