Questions tagged [von-neumann-algebras]
A von Neumann algebra is a unital *-subalgebra of the algebra of bounded operators on a Hilbert space, closed in the weak operator topology. Also called a $W^*$-algebra, and may be regarded as a non-commutative generalization of $L^\infty$ space. These algebras are extensively used in knot theory, non-commutative geometry, and quantum field theory.
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Why are conditional expectations on a C*-algebra called so?
In what way does a conditional expectation on a C$^*$-algebra mimic the averaging property of conditional expectation in probability theory?
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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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Trace preserving isomorphism on von Neumann algebras
Is the condition on an isomorphism between von neumann algebras which says that the trace is conserved the same thing as the notion of spatial isomorphism?
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Subfactor of hyperfinite one
Is there a strict subalgebra of the hyperfinite $II_1$ factor that is separable and type $II_1$ factor?
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$J\Delta^{it}J = \Delta^{it}$ for modular operators associated to left Hilbert algebra
Let $\mathcal{A}$ be a (full) left Hilbert algebra with Hilbert space completion $H$. Let
$J: H \to H$ be the modular conjugation and $\Delta$ be the associated modular operator. Is it true that
$$J \...
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Free semicircular family: how to think of the elements?
Consider the following definition of a free semicircular family. Let $(A,\tau)$ be a C* probability space (so $\tau$ is a faithful state on $A$, which is a unital C* algebra). Let $NC_2(p)$ be the set ...
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$A$ is dense in $A''$ in the strong topology (Murphy's book)
In Lemma 4.1.4. Murphy is doing something strange that I don't seem to understand. for each $x\in H$ we find $v_n(x)$ such that it converges to $u(x)$. we wish to prove that is holds for EVERY $x$.
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Is any fixed element of $L^\infty(0, 1) \bar{\otimes} M$ contained in $L^\infty(0, 1) \bar{\otimes} M_0$ where $M_0$ is separable?
Suppose $M$ is a von Neumann algebra (not necessarily separable). Given any element $a \in L^\infty(0, 1) \bar{\otimes} M$, does there exist a separable subalgebra $M_0 \subseteq M$ s.t. $a \in L^\...
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Definitions of the support projection of a self adjoint
Let $T$ be a self-adjoint operator on $M$, then the Support $s(T)$ of $T$ is the smallest projection such that $s(T)T=T=Ts(T)$.
We also have the spectral projection of $T$ corresponding to $T$, take $...
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Centralize of hermitian normal linear functional
Let $\rho_1$ and $\rho_2$ be two normal states on a von Neumann algebra $M$. We denote by $M_{\rho_1-\rho_2}$ the centralizer of $\rho_1-\rho_2$ and $M_{\rho_i}$ the centralizer of $\rho_i(i=1,2)$.
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Why does $w_i^*xw_j\in e\mathscr{M}e$ here?
I'm reading Proposition IV.1.8 of Theory of Operator Algebras I. The statement is following:
At the last line of the proof the author said the following:
hence $(U^*xU)_{i,j}= w_i^*xw_j\in e\mathscr{...
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Multiplication of the support of the self-adjoint operators
How to prove that $s(x)s(A)=0$, which is marked red in the screenshot.
According to the definition, we know that $s(x)$ is the smallest projection such that $s(x)x=x=xs(x)$ and $s(A)$ is the smallest ...
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Support projection of a normal state
Let $M$ be a von Neumann algebra and $\rho$ a normal state on $M$.
Suppose that $a$ is in the centralizer of $\rho$ and $b$ is positive in $M$ such that $abs(\rho)=0$, where $s(\rho)$ is the support ...
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Jordan decomposition of a normal linear functional
Let $M$ be a von Neumann algebra. Suppose that $\rho$ is a normal linear functional on $M$.
We have the Jordan decomposition :$\rho=\omega_{+}-\omega_{-}$ and $\|\rho\|=\|\omega_{+}\|+\|\omega_{-}\|$, ...
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The subalgebras of algebras of $\tau$-measurable operators.
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite trace $\tau$, let $S(\mathcal{M},\tau)$ be the algebra of $\tau$-measurable operators.
Q1: If $\mathcal{B}$ is ...
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Find a projection in a type III$_1$ factor
Let $M$ be a type III$_1$ factor and $\omega$ a normal state on $M$. For any $\epsilon>0$, can we find a projection $p$ in $M$ such that $\omega(p)＜\epsilon$?
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If $0\le a\le p$ in a von Neumann algebra then $a \in pMp$ [duplicate]
Let $M$ be a von Neumann algebra over a Hilbert space $H$ and $a \in M$ such that $a \ge 0$. Let $p\in M$ be a projection such that $a \le p$. I want to show that $a \in pMp$.
I know that, if $q\le p$ ...
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The inequality for the conditional expectation in a von Neumann algebra
Suppose there exists a conditional expectation $E$ from the von Neumann algebra $M$ to its subalgebra $N$, the following inequality holds:
$E(a)^*E(a)\leq E(a^*a)$ for all $a\in M$.
Under what ...
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Why does $\omega((I-E)T)=0$ here?
I am reading Kadison-Ringrose-Vol2. The authors defined the completely additive states and their supports in Definition 7.1.1. And just after the definition they remarked the following:
If $E$ is a ...
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Is integration with respect to a Radon measure a normal weight on $L^\infty(X)$?
This might be trivial to experts in von Neumann algebra theory, but here goes.
Let $\mu$ be a Radon measure on a locally compact Hausdorff space. Consider the weight
$$\varphi: L^\infty(X, \mu) \to [0,...
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Group von Neumann algebras and the $\|\cdot\|_{2}$ norm
Let $G$ be a discrete group and $LG$ the corresponding group von Neumann algebra. For a set $F\subset G$, let $P_{F}:\ell^{2}(G)\rightarrow\ell^{2}(F)$ denote the orthogonal projection onto $\ell^{2}(...
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Relationship between two partial isometries
Let $M$ be a von Neumann algebra. Suppose $v,w$ are partial isometries in $M$ such that $v^*v=w^*w$ and $vv^*=ww^*$. We know that there exist partial isomeries $c_1,c_2$ such that $v=c_1w$ and $v^*=...
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Amplification commutes with double commutant – problem understanding proof
In Takesaki, Theory of Operator Algebras I, we find the following theorem and proof (page 184):
I do not understand the last step (starting from "hence"). I manage to see that $\pi(\mathcal ...
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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)
where we must think $\mathcal{N}$ and $\mathcal{M}$ as concrete von Neumann algebras over the ...
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If $\varphi$ is a normal faithful semifinite weight, is $\eta_\varphi(\mathfrak{n}_\varphi\cap\mathfrak{n}_\varphi^*)$ dense in $\mathfrak{H}_\varphi$
Let $M$ be a von Neumann algebra and $\varphi: M_+\to [0, \infty]$ be a normal, faithful semifinite weight. Consider its associated semi-cyclic representation
$$\pi_\varphi: M\to B(\mathfrak{H}_\...
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Intermediate von Neumann subalgebra of a $\sigma$-finite von Neumann algebra
I was confused the existence of the intermediate von Neumann subalgebra $P$ in the above Lemma.
We know that $M^{\varphi}$ and $M$ must be globally invariant under $\sigma^{\varphi}$. Can we construct ...
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How to show that an operator in $B(L^2(N))$ is actaully in $N$?
I am reading a note written by Claire Anantharaman and Sorin Popa recently. Here is a link to the notes.
https://www.math.ucla.edu/~popa/Books/IIun.pdf
I am reading Chapter 13 right now, in section 13....
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Non-unitary isometry and a norm equality
I am looking at a paper which asserts the following equality relating a non-unitary isometry. There is no explanation given for this, and I cannot figure out why this is true:
Here is the proposition: ...
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Type transmutation of von Neumann factors
The crossed product of a Type $III_{1}$ von Neumann factor with its modular automorphism group is generically Type $II$. Does there exist a similar construction turning a Type $II$ (or a Type $III_{1}$...
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Can we have some examples about i.c.c groups?
In the following discussion, we always assume $G$ is a countable discrete group.
I am learning about the group von Neumann algebra recently, We know that $L(G)$ is a factor if and only if $G$ is an i....
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A question about standard representation for type $II_1$ factors?
We suppose that $M$ is a type $II_1$ factors, then we have a standard representation, we know that every element in $L^2(M)$ can be approximated by elements in $M$ by 2-norm, in another word, for ...
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Proposition 3 from Chapter 2 of Dixmier's Von Neumann Algebras
Please help me understand this argument from the proof of the above Proposition.
Proposition 3: Let $\mathcal{A} \subset \mathcal{L}(H)$ be a Von Neumann algebra (VNA) of operators on a Hilbert space $...
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Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$
Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
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The reduction of a Von Neumann algebra's center is the center of its own reduction
I'm trying to get understand this basic fact about Von Neumann algebras (VNA). I cannot for the life of me understand what's going on in the Corollary to Proposition 2, Chapter 2 of Dixmier's book on ...
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What are your favorite books or textbooks dealing with operator algebras? [closed]
The title has it all. I am looking for some books with lot of exercises, and not very hard to read.
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Irreducible subfactor inclusion and von Neumann type
Let $M$ and $N$ be factors and $N\subset M$ be an irreducible subfactor inclusion, i.e., $N$ has trivial relative commutant in $M$.
Does it follow that $M$ and $N$ have the same type?
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How to use Spectral theorem in this proof?
In the book Kadison-Ringrose II in proposition 8.5.2 (image attached below) it seems that the authors used the fact:
Given a non-zero positive operator $T$ in a von Neumann algebra there exists a ...
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Commuting group matrices
Let $G$ be a group and form the associated Hilbert space $\ell^2(G)$ with canonical orthonormal basis $(e_g)_{g\in G}$. For $g \in G$, define the operators $u_g, v_g ∈ B(\ell^2(G))$ by
$$u_g(e_h) := ...
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Definition of normal state on von Neumann algebra
I have a question about the notion of normal state on a von Neumann algebra and its relation to a particular representation of the algebra. Let me take from the book by Bratteli and Robinson:
...
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Tracial and finite von Neumann algebras
A tracial von Neumann algebra $(M,\tau)$ is a von Neumann algebra with a faithful normal tracial state $\tau$ on $M$. That is, $\tau$ is a function from $M \to \mathbb{C}$ such that it is a faithful ...
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Existance of conditional expectations map onto subalgebras
Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existance of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any ...
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Densely defined and closed operator
Question: Let $N\subseteq B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$ and let $n\in N$. Let $B$ be a von Neumann subalgebra of $N$. Let $\mathbb{E}_B: N\rightarrow B$ be ...
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continuous decomposition of type III von Neumann algebras
The above statement is from Takesaki's book 'Theory of Operator Algebras vol II'(See page 375).
I was struggled with verifying the conclusion $N=M_{\varphi}$.
How to check that for any $x\in N$, we ...
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For positive semidefinite unit trace $\Gamma$, what is $\{ x \in \mathbb R^N: x^T A x = \text{tr}(A \Gamma) \ \forall A \in\mathbb R^{N \times N} \}$?
My question.
Let $N > 1$ and $\Gamma \in \mathbb R^{N \times N}$ be a positive semidefinite matrix with unit trace.
Can we describe in a simple manner the following set? $$S(\Gamma) := \{ x \in \...
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Inclusion of factors with separable predual and with expectation
Let $N\subset M$ be any irreducible (i.e.$ N'\cap M=\Bbb C 1$) inclusions of factors with separable predual and with expectation. Popa proved that if $N$ is semifinite, then there exists an AD ...
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commutant of a finite and semifinite von Neumann algebra
When $M$ is a type I von Neumann algebra on a Hilbert space, then the commutant $M'$ is also type I.
The conclusion is also true for type II and type III von Neumann algebras.
My question is as ...
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Comparing minimal projections in two finite type $I$ subfactors
Let $R$ be a von Neumann algebra containing two finite type $I_{n}$ subfactors $M$ and $N$ with matrix units $\{E_{rs}\}$ and $\{F_{rs}\}$, respectively. My question is: is there necessarily a partial ...
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$\sigma$-Weakly Continuous Bounded Linear Functional on a von Neumann Algebra is Normal
I have been working on Exercise 4 in Chapter 4 of Murphy's "$C^*$-Algebras and Operator Theory", which is as follows:
Let $A$ be a von Neumann algebra on $H$, and suppose that $\tau$ is a ...
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the faithful normal conditional expectation is uniquely determined
I was puzzled by the proof of $(i)\rightarrow (ii)$.
How to check that $E:M\rightarrow N$ is uniquely determined by the condition $\varphi\circ E=\varphi$?
To my mind, we need to verify the following ...
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A nice operator topology on bidual
I have a proof of something but it is very much provisional on some operator topology questions that I am looking for help with. I will state the problem generally but in fact what I am working with ...
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