All Questions
Tagged with oa.operator-algebras von-neumann-algebras
504 questions
4
votes
0
answers
123
views
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....
- 2,830
1
vote
1
answer
234
views
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 ...
- 896
0
votes
1
answer
71
views
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 $\...
- 759
3
votes
0
answers
178
views
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 $\...
- 433
0
votes
1
answer
233
views
Compactly supported continuous functions as a Tomita algebra
Let $G$ be a locally compact group with modular function $\delta_G$ and consider $\mathcal{K}(G)$, the set of compactly supported continuous functions $G\to \mathbb{C}$, with the $*$-algebra structure ...
- 175
0
votes
1
answer
93
views
Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$?
Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that
$${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$
$${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...
- 617
3
votes
1
answer
106
views
Least upper bound of type I factors
Given two type I (von-Neumann algebra) factors $\mathcal M,\mathcal N$, is there a smallest type I factor containing both $\mathcal M,\mathcal N$?
Notes:
$\mathcal M,\mathcal N$ are over the same ...
- 896
3
votes
1
answer
244
views
Takesaki: question about lemma in section "Left Hilbert algebras and weights"
To make this question relatively self-contained, this post is quite long, but the question itself is rather short.
Consider the following fragments in Takesaki's second volume "Theory of operator ...
- 175
6
votes
1
answer
287
views
Sigma-weakly dense *-subalgebra of von Neumann algebra has increasing net of positive elements convergent to the identity
Let $M$ be a von Neumann algebra and $A\subseteq M$ a $\sigma$-weakly dense $*$-subalgebra of $M$. Does there exist an increasing net $\{a_i\}_{i\in I}\subseteq A\cap M^+$ such that $a_i\to 1$ in the $...
- 175
2
votes
1
answer
471
views
Takesaki II proposition 3.15 proof about modular automorphism groups: mistake in book?
Consider the following fragment from Takesaki's second volume "Theory of operator algebras" in chapter VIII "Modular automorphism groups" p122-123.
Why is it possible to choose an ...
- 175
3
votes
1
answer
244
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 ...
- 424
2
votes
2
answers
481
views
Takesaki II "Connes cocycle derivative"
Consider the following fragments from Takesaki's second volume "Theory of operator algebras" (chapter VIII $\oint 3$, Modular Automorphism groups, p107-108:
Why are the second and third ...
- 175
1
vote
0
answers
150
views
Intersection of finitely many type-I von-Neumann algebra factors
If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M\cap\mathcal N$ a type-I von-Neumann algebra factor?
Notes:
An elementary ...
- 896
0
votes
0
answers
104
views
Intersection of type-I von-Neumann algebra factors
Is the intersection of a (possibly infinite) family $\{\mathcal M_i\}$ of type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$) again a type-I von-Neumann algebra factor?
- 896
2
votes
1
answer
159
views
Question on tensor product of von Neumann algebras and subfactors
Let $M_1$ and $M_2$ be von Neumann algebras acting on Hilbert spaces $H_1,H_2$ and consider $M=M_1\overline\otimes M_2$ acting on $H_1\otimes H_2$. Let $K$ be an $M$-invariant subspace (so that $P_K\...
- 759
3
votes
1
answer
332
views
Takesaki II Lemma 1.13: stuck in proof
Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"):
Here, we associate with an ...
- 175
5
votes
1
answer
204
views
Continuity of the extension of a tracial state with respect to the strong operator topology
Problem: Let $M\subseteq B(H)$ be a finite von Neumann algebra with a faithful tracial state $\tau$. Let $\widetilde{M}$ be the $\tau$-measurable operators on $M$ (recalled below). Extend the trace $\...
- 85
3
votes
1
answer
225
views
$\tau$-measurable operator
Problem: Let $M$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. Let $m$ be a positive element in $M$ and let $e_{(0,\infty)}(m)$ be the spectral projection of $m$ ...
- 85
2
votes
1
answer
525
views
Difference in 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 ...
- 163
1
vote
0
answers
384
views
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 ...
- 85
5
votes
1
answer
239
views
Completely isometric coaction of discrete quantum group is multiplicative?
Let $\mathbb{G}$ be a compact quantum group (in the sense of Woronowicz) with discrete dual $\widehat{\mathbb{G}}$ which we view as a von Neumann algebraic locally compact quantum group (in the sense ...
- 525
6
votes
1
answer
166
views
Every element of a $W^*$-algebra is a linear combination of exponential unitaries?
I am trying to understand a proof in this paper (specifically theorem 5.4). In it, a fact is used that every element of the $W^*$-algebra $A$ is a linear combination of exponential unitaries.
I've ...
- 63
2
votes
1
answer
250
views
Norm continuity of the predual of a von Neumann algebra
Let $M$ be a von Neumann algebra and let $(p_i)$ be a net of projections in $M$ decreasing to $0$. Let $f\in M_{\ast} $, the predual of $M$.
It is well known that
$\| p_i f \|_{M_\ast}\to_{i} 0$
for ...
- 617
4
votes
1
answer
332
views
Support projection vs closed support projection of a normal state in enveloping von Neumann algebra
I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may ...
- 135
4
votes
1
answer
226
views
Definition of Radon measure on Takesaki's first volume
Consider the following theorem from Takesaki's first volume "Theory of operator algebras":
In $(i)$, it is claimed that $L^\infty(\Gamma,\mu)$ is an abelian von Neumann algebra. How does ...
- 175
0
votes
0
answers
144
views
Type III von Neumann algebra
Let $\mathcal M$ be a type $\mathrm{III}$ von Neumann algebra. Is it true that for all $n\geq 1,$ $\mathcal M$ contains a copy of $M_n$ as a von Neumann subalgebra? By Theorem 9.24 of these lecture ...
- 1,879
1
vote
0
answers
87
views
irreducible subfactor inclusion and commutativity of induced projections
Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$.
Let $\Omega\in H$.
Does it follow that the projections onto $[N\Omega]$ and $[M'\...
- 759
1
vote
1
answer
129
views
Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?
Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product
$$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \...
- 525
5
votes
1
answer
428
views
Separable C* algebras and type I states
Let $A$ be a separable $C^*$-algebra and let $\omega$ be a state on $A$.
Then there is an "orthogonal" probability measure $\mu$ on the pure state space $P(A)$ of $A$ such that $\omega(x) = \...
- 759
8
votes
1
answer
643
views
CCR vs. CAR vs. Clifford algebras, infinite tensor products and type of the corresponding von Neumann algebra
$\newcommand\CAR{\mathit{CAR}}\newcommand\Cl{\mathbb C\mathit l}$This question will be rather long and it will be my attempt to finally clarify many issues concerning CCR, CAR and Clifford algebras ...
- 9,330
1
vote
1
answer
114
views
Continuous surjection between spectra of commutative von Neumann algebras
Suppose that $V_1,V_2$ are two commutative von Neumann algebras and $V_1 \subset V_2$. Being in particular commutative $C^*$-algebras we have that $V_1 \cong C(X_1), V_2 \cong C(X_2)$ for some ...
- 9,330
1
vote
1
answer
177
views
Commuting and generating subfactors of $ B(H)$
I have a question on subfactors of $B(H)$ (the von Neumann algebra of bounded operators on a complex Hilbert space).
Say I have a subfactor $M$ of $B(H)$. Is it true that another subfactor $N \subset ...
- 759
0
votes
1
answer
101
views
"Project" an operator outside of a von Neumann Algebra into it
Suppose $W$ is a proper von Neumann Algebra contained in $B(H)$ and the identity in $W$ is the identity mapping of $H$ (namely, $W$ does not have non-trivial null space).
Given a self-adjoint $T\in W$...
- 307
4
votes
0
answers
110
views
non centrally free actions of ameanable groups on the hyperfinite III_1 factor
Let $R$ be a hyperfinite $\mathit{III}_1$ factor, and let $Out(R)$ be its set of automorphisms modulo inner automorphisms. There is a canonical and important homomorphism $\phi:\mathbb R\to Z(Out(R))$ ...
- 43.2k
0
votes
0
answers
390
views
Monotone convergence theorem for increasing net of positive functions
Suppose that we have $(\Omega,\mu)$ a $\sigma$-finite measure space. I have the following question.
(Assume that $(f_i)_{i\in I}$ be an increasing net of positive measurable functions such that $f_i\...
- 1,879
0
votes
1
answer
214
views
Semi-commutative von Neumann algebras
Suppose $\Omega$ is a $\sigma$-finite measure space with measure $\mu.$ Let $\mathcal M\subseteq B(H)$ be a von Neumann algebra.
Can an element of $L_\infty(\Omega)\overline{\otimes}\mathcal M$ be ...
- 1,879
4
votes
1
answer
246
views
Takesaki lemma 1.16 (volume II, chapter VII)
I am trying to understand the proof of the implication $(i)\implies (ii)$ in Takesaki's book "Theory of operator algebras II", chapter VII, which says the following:
The relevant setting ...
- 175
2
votes
0
answers
68
views
What about the structure theory in Baer *-rings?
In the literature, Baer *-rings are called as the algebraic analogue of von Neumann alegars.
It is well-known that
Theorem. Every von-Neumann algebra is decomposed into a direct sum of the algebras of ...
- 4,058
3
votes
1
answer
121
views
Impact of annihilators in C*-algebras
Let $A$ be a unital C*-algebra. Let $S\subseteq A$. We put $$\operatorname{Ann}_r(S)=\{a\in A : \forall s\in S,~ ~as=0\}$$
Suppose that $A$ satisfies the following property:
For every $S\subseteq ...
- 4,058
2
votes
0
answers
176
views
Projections in von Neumann algebra tensor product
Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra with normal faithful semifinite trace $\tau.$ Consider the von Neumann algebra $\mathcal N:=L_\infty([0,1])\overline{\otimes}\mathcal M$...
- 1,879
4
votes
2
answers
298
views
Takesaki volume II chapter VII lemma 1.15
Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...
- 175
9
votes
3
answers
568
views
Defining the abstract tensor product of W*-algebras via a universal property
I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt:
It is easy to show that such an object ...
- 175
2
votes
2
answers
158
views
Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki)
I originally asked this on MSE, but did not get an answer there.
Let $M$ be a von Neumann algebra. Let $\varphi: M_+ \to [0, \infty]$ be a weight on $M$. Consider
\begin{align*}&\mathfrak{p}_\...
- 175
2
votes
0
answers
192
views
Almost periodicity and approximation in tracial von Neumann algebra
Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. For a countable group $G$, let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action on $N$ which preserves the tracial ...
- 73
2
votes
1
answer
218
views
Extending a $\sigma$-weakly continuous map: Takesaki IV.5.13
Consider the following fragment from chapter IV in Takesaki's book "Theory of operator algebra I":
Why is the boxed line true? Takesaki argues that
$$\theta_0: \mathscr{M}_1\otimes_{\...
- 175
2
votes
1
answer
297
views
Predual theorem proof in Takesaki's volume I
Consider the following fragment from Takesaki's book "Theory of operator algebra I" (Section III.3 ,p133-134).
Why is the boxed line true? I can see that $\epsilon: \widetilde{A}\to A$ is ...
- 175
3
votes
1
answer
255
views
Takesaki: Lemma about enveloping von Neumann algebra
Consider the following lemma with proof from Takesaki's book "Theory of operator algebra I" (p121):
It appears to me that Takesaki claims at the end of the proof that $\pi(A)_1$ is $\sigma$-...
- 175
5
votes
1
answer
321
views
Takesaki's proof of the Kaplansky density theorem
Consider the following fragment from Takesaki's book "Theory of operator algebra I":
Why is the boxed sentence true? It looks like they replace $A$ by its strong$^*$-closure. Is this ...
- 175
6
votes
0
answers
241
views
Tomita–Takesaki theory and subfactors
Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $\Omega$ be a cyclic and separating vector in $H$. Let $J$ and $\Delta$ be the corresponding modular conjugation and modular ...
4
votes
1
answer
303
views
Matrix units in von Neumann algebras, and $K_0$ groups
This question arises from trying to understand the proof of Lemma 3.1.4 in De Commer, Martos, and Nest - Projective representation theory for compact quantum groups and the quantum Baum–Connes ...
- 18.7k