All Questions
Tagged with von-neumann-algebras fa.functional-analysis
162 questions
1
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0
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86
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Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group
I am posting my question of mathstack exchange here. (see: My post on MSE)
Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, ...
- 261
4
votes
1
answer
252
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Show that $\Lambda_\varphi(x_n)\to \Lambda_\varphi(x)$ for an nsf weight $\varphi$ on a von Neumann algebra
Let $\varphi$ be an nsf weight on a von Neumann algebra $M$. Fix a square-integrable element $x\in \mathscr{N}_\varphi$. Put
$$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
- 175
5
votes
1
answer
183
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Question about modular group (Modular theory in operator algebras, section 2.14)
Consider the following fragment from Stratila's book "Modular theory in operator algebras", section 2.14, p20:
I'm trying to understand the claim $(3)$ (see the red box). The main strategy ...
- 175
3
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0
answers
96
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Excising the trace of a $II_1$-factor
Recall that a state $\varphi$ on a $C^*$-algebra $A$ is said to be excised by projections if there exists a net of projections $e_i \in A$ such that $\| e_i a e_i - \varphi(a) e_i\| \to_{i} 0$ for all ...
- 297
4
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0
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148
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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 $\...
- 2,263
1
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0
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210
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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 ...
- 173
5
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2
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342
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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 =\|...
- 617
1
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1
answer
128
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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) $\omega$ ...
- 617
1
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1
answer
211
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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 ...
- 439
5
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1
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453
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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?
- 617
4
votes
1
answer
203
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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 ...
- 617
2
votes
0
answers
157
views
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 ...
- 433
2
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2
answers
135
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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\...
- 407
3
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0
answers
253
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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{...
- 9,853
5
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1
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324
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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 $...
- 439
1
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1
answer
180
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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 $\...
- 1,027
2
votes
0
answers
177
views
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 ...
- 617
4
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0
answers
115
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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 ...
- 131
0
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1
answer
93
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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
244
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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
288
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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
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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
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
226
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
1
vote
0
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385
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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 ...
- 85
2
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1
answer
250
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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
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334
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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
0
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0
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144
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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
0
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1
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101
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"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
2
votes
1
answer
282
views
Measurability of a net
Let $(f_\epsilon)_{\epsilon>0}$ be a family of positive measurable functions on $L_p(\mathbb R)$ where $1<p<\infty.$ Assume that the pointwise supremum $f^*(x)=\sup_{\epsilon>0}|f_\epsilon(...
- 1,879
4
votes
1
answer
246
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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
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
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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
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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
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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
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
4
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0
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132
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$L_1$-subspace of the predual of a von Neumann algebra
If $M$ is a type $II$ von Neumann algebra, then the predual has a complemented subspace isometric to $L_1(0,1)$. It follows from the existence of expectation. However, I don't know whether such a ...
- 617
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
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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
10
votes
3
answers
860
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Takesaki theorem 2.6
I originally posted this question on MSE and didn't get a satisfactory answer, even after putting a bounty on it. Hence, I thought I should ask here:
Consider the following theorem in Takesaki's book &...
- 175
2
votes
0
answers
105
views
Comparing two quantities related to the norm of an inner derivation
Let $M$ be a von Neumann algebra sitting in $B(H)$.
Let $U(M)$ denote the unitary group of $M$.
Let $I(M):=\{\tau\in M\,|\,\tau=\tau^*=\tau^{-1}\}$ the set of involutions in $M$.
Let $SAC(M):=\{h\in M\...
- 71
1
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1
answer
188
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Uniqueness of the predual of a W*-algebra
Consider the following fragments in Takesaki's "Theory of operator algebras" (volume I):
Question: So, we have an abstract Banach space $F$ with $A \cong F^*$. In Lemma 3.6, one considers ...
- 175
2
votes
1
answer
111
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About $\sigma$ strong$^*$-functionals and seminorms
I'm reading the book "Theory of operator algebras" by Takesaki. In this book, the $\sigma$-strong$^*$ topology on the space $B(H)$ (bounded operators on the Hilbert space $H$) is defined (...
- 175
1
vote
2
answers
148
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Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra
Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions:
B is a von Neumann algebra with $A'' = B$.
The inclusion $A \...
- 175
3
votes
1
answer
306
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Opposite $C^*$ algebras
$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
- 1,879
0
votes
0
answers
144
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Dual operator space
Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|...
- 1,879
12
votes
1
answer
2k
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Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"?
I've listened to many interviews and lectures of Alain Connes, in which he says something which goes roughly as follows
"Every non-commutative algebra has its own time (evolution of), by which I ...
- 6,853
4
votes
3
answers
801
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Quick derivation of classical probability theory from von Neumann algebraic framework
Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko https://www.youtube.com/watch?v=F8Urtr39jM0, I'm led to consider the following question
Question. How can one ...
- 6,853
3
votes
1
answer
221
views
Coincidence of two topology on a bounded subset of a finite von Neumann algebra
Let $M\subset B(H)$ be a finite von Neumann algebra with a faithful normal trace $\tau$. There is a norm $\|.\|_\tau$ on $M$ given by $\sqrt{\tau(xx^*)}$. How to show the $\|.\|_\tau$-topology ...
- 391
0
votes
2
answers
123
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$(ST)_{[13]}= S_{[13]}T_{[13]}$ for $S,T \in B(\mathcal{H}\otimes \mathcal{H}).$
Let $T\in B(\mathcal{H} \otimes \mathcal{H})$ where $\mathcal{H}$ is a Hilbert space. We can define operators
$$T_{[12]}= T \otimes 1;\quad T_{[23]}= 1 \otimes T$$
and if $\Sigma: \mathcal{H} \otimes \...
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