All Questions
Tagged with oa.operator-algebras von-neumann-algebras
504 questions
4
votes
0
answers
120
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Projection behavior under the automorphism $\Phi$ on a diffuse semi-finite von Neumann algebra
Let $\mathcal{M}$ be a diffuse semi-finite von Neumann algebra acting on a Hilbert space $\mathcal{H}$, equipped with a faithful normal semi-finite trace $\varphi$. Let $\Phi : \mathcal{M} \to \...
- 113
1
vote
0
answers
69
views
Compressions and (ir)rational trace
Let $\mathcal{M}$ be a type $II_1$ factor with tracial state $\tau$ and $P$ be a projection in $\mathcal{M}$ such that $\tau(P)=1/n$ for some natural number $n.$ It is known (Ananatharaman-Popa "...
1
vote
0
answers
86
views
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
0
votes
1
answer
119
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Reference request: hyperfinite cross product
Given a countable essentially free ergodic non-singular group action $G \curvearrowright (X, \mu)$ on a standard measure space, suppose $\mu$ is a non-atomic probability measure and $\alpha: G \...
0
votes
0
answers
123
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crossed product by compact groups
Do we need the ambient measure on G to be a Haar measure in order to form the crossed product by a compact group of a von Neumann algebra M? If the measure is indeed Haar, then we can obtain the ...
- 15
6
votes
0
answers
110
views
Standard form of fiber product of von Neumann algebras
Let $Z$ be an abelian von Neumann algebra, and let $A$ and $B$ be two von Neumann algebras that receive central maps $Z \to Z(A)$ and $Z \to Z(B)$.
We may then construct the fiber product of $A$ and $...
- 43.2k
3
votes
0
answers
109
views
Faithful traces on reduced $C^*$-algebra of a measured groupoid
Let $G$ be a measured étale groupoid with quasi-invariant measure $\mu$ (that induces the reduced $C^* $-algebra, meaning it has full support) with associated equivalent measures $\nu,\nu^{-1}$.
Is ...
- 261
4
votes
1
answer
251
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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
$$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
- 175
3
votes
0
answers
97
views
Is a localised "restricted symmetry" automorphism implementable as a unitary operator on the GNS Hilbert space?
I have a pure state $\omega$ on a quasilocal algebra $\mathcal{A}$ on a 2d lattice $\Gamma = \mathbb{Z}^2$ with a $\mathbb{C}^d$ vector space on each site. Let there be a unitary symmetry action $U_g(...
- 569
5
votes
1
answer
183
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Question about modular group (Modular theory in operator algebras, section 2.14)
Consider the following fragment from Stratila's book "Modular theory in operator algebras", section 2.14, p20:
I'm trying to understand the claim $(3)$ (see the red box). The main strategy ...
- 175
1
vote
0
answers
91
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Subfactors with integer Jones index
Is there any integer (Jones) index subfactor which is not extremal?
- 393
1
vote
1
answer
186
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Takesaki II "Bimodule" question
Consider the following fragment from Takesaki's book "Theory of operator algebras", chapter IX Non-commutative integration, Section 3 on p187-188:
I have trouble understanding the equality
$...
- 175
0
votes
0
answers
111
views
Integral decomposition
Let $\mathcal{A}$ be a separably acting von Neumann algebra and let $$\mathcal{A}\cong \int_{\Gamma}^{\oplus} \mathcal{A}_{\gamma}\,d\mu(\gamma)$$ be its direct integral decomposition into factors $\...
3
votes
0
answers
117
views
Is every strongly singly generated type $II_1$ factor generated by a pair of hyperfinite $II_1$ subfactors with a common MASA?
A type $II_1$ factor $M$ is strongly singly generated (SSG) if every amplification $M^{t}$ of $M$ is singly generated as a von Neumann algebra. Interesting characterizations of SSG type $II_1$ factors,...
- 7,047
2
votes
0
answers
93
views
Amenability and the unitary group of an operator algebra
Let $M$ be a von Neumann algebra and $U(M)=\{x\in M: x^*=x^{-1}\}$ be its unitary group. In this post, we equip $U(M)$ only with the relative weak$^*$ topology $\sigma(M,M_*)$. Then, $U(M)$ is a ...
- 2,605
3
votes
2
answers
132
views
$w^*$-limit of projections in von Neumann algebra
Let $\mathcal M$ be a semi finite von Neumann algebra with a normal faithful semi finite trace $\tau$. Let $(e_i)_{I\in I}$ be a net of projections in the von Neumann algebra which converges to an ...
- 1,879
3
votes
0
answers
96
views
Excising the trace of a $II_1$-factor
Recall that a state $\varphi$ on a $C^*$-algebra $A$ is said to be excised by projections if there exists a net of projections $e_i \in A$ such that $\| e_i a e_i - \varphi(a) e_i\| \to_{i} 0$ for all ...
- 297
4
votes
1
answer
137
views
Fixed point algebra of a non-amenable factor
Let $M$ be a non-amenable factor. Suppose $\alpha$ is an action of a group $\Bbb R$ on $M$.
Define $$M^{\alpha}:=\{x\in M: \alpha_t(x)=x \text{ for any} \; t\in \Bbb R\}.$$
If we know that there exits ...
- 427
7
votes
1
answer
244
views
Approximately semifinite factors
For the sake of this question, lets call a factor $M$ approximately semifinite if there exists an increasing net of semifinite subfactors $M_i$, $i\in J$, with conditional expectations $E_i:M\to M_i$ ...
- 759
8
votes
0
answers
252
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Question about the homogeneity of the state space of a type $\rm{III}_1$ factor
I'm reading the paper Homogeneity
of the State Space of Factors of Type $\rm{III}_1$ by Connes and Størmer.
Homogeneity of the state space means that all normal states are approximately unitarily ...
- 759
7
votes
1
answer
391
views
Positive cone in Haagerup L²-space: how much information does it contain?
Given a von Neumann algebra $A$, its Haagerup $L^2$-space $H:=L^2A$ (also known as the standard form of the Neumann algebra) comes equipped with a positive cone $P\subset H$.
Question: How much ...
- 43.2k
4
votes
1
answer
291
views
Strengthening the direct integral decomposition of von Neumann algebas
Let $M$ be a von Neumann with separable predual. It well known that one can write $M$ as a direct sum $M=M_I\oplus M_{II} \oplus M_{III}$ of von Neumann algebras of types $I$, $II$ and $III$.
It is ...
- 759
4
votes
1
answer
211
views
Ergodic actions and deviation from invariance
Let $M$ be a von Neumann algebra and let $(\phi_t)$ be an ergodic point-$\sigma$-weakly continuous one-parameter group of automorphisms $\phi_t\in \mathrm{Aut}(M)$, i.e., $\Vert\omega-\omega\circ\...
- 759
4
votes
1
answer
201
views
Does $N \mathbin{\bar{\otimes}} N^{\mathrm{op}}$ act on $L^2(N)$?
Let $N$ be a von Neumann algebra and $N^{\mathrm{op}}$ its opposite. The standard form $L^2(N)$ is an $N$-$N$-bimodule, or equivalently a module over $N \otimes_{\mathrm{alg}} N^{\mathrm{op}}$.
Does ...
- 6,406
5
votes
2
answers
342
views
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
vote
1
answer
128
views
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
vote
1
answer
211
views
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
4
votes
1
answer
203
views
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
5
votes
1
answer
208
views
Hyperfinite factors and increasing fatorization of states
If a factor $R$ contains a matrix algebra $M\subset R$ (i.e., a $M$ is a type $I_n$ factor), then $R \cong M \otimes M^c$ where $M^c=R\cap M'$ is the relative commutant.
Each state $\omega$ on $R$ ...
- 759
1
vote
1
answer
256
views
Intersection of two intermediate subalgebras
Suppose $B\subset A$ is an inclusion of simple $C^*$-algebras with a conditional expectation of (Watatani) index-finite type and $B^{\prime}\cap A=\mathbb{C}$. Then we know $B^{\prime}\cap A_1$ is ...
- 393
3
votes
1
answer
142
views
$K_0$ group of an infinite factor
The following question was already posted in this link but I could not understand hints given in this post.
Let $\mathcal{M}$ be an infinite factor and my question is how to prove that $K_0(\mathcal{M}...
- 307
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
14
votes
0
answers
220
views
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 ...
- 43.2k
1
vote
0
answers
111
views
Inclusion of finite dimensional C*-algebras and relative commutants of subfactors
Given a subfactor $N\subset M$ with finite Jones index, the inclusion of relative commutants $N^{\prime}\cap M\subset N^{\prime}\cap M_1$ (here, $M_1$ is the basic construction of $N\subset M$) is a ...
- 393
2
votes
0
answers
118
views
Depth of the reduced subfactor
Suppose $N\subset M$ is a finite depth subfactor with $[M:N]<\infty$. Consider the reduced subfactor $pNp\subset pMp$ for some projection $p\in N$. How to calculate the depth of $pNp\subset pMp$ in ...
- 393
5
votes
1
answer
165
views
Approximation from below of positive elements in tensor product of von Neumann algebras
Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras.
If $x$ is a positive element of $\mathcal M$ and $y$ is a positive element of $\mathcal N$, it is known that $x\otimes y$ is a positive ...
- 407
3
votes
0
answers
60
views
Noncommutative maximal weak $L_1$ norms with respect to sub algebra
Let $(\mathcal M,\tau)$ be a von Neumann algebra with normal finite faithful trace $\tau.$ For any sequence $(x_n)_{n\geq 1}\in \mathcal M$ define $\|(x_n)\|_{\Lambda_{1,\infty}(\mathcal M;\ell_\infty)...
- 1,879
1
vote
1
answer
97
views
Choosing a net of projections from a given collection
Let $\mathcal M$ be a von Neumann algebra. Let $S$ be a nonempty set. Consider $\Omega$ to be the collection of all finite subsets of $S$. Suppose for all $\omega\in\Omega$ we have $I_\omega$ a ...
- 1,879
2
votes
1
answer
101
views
Hyperexpectations from injective subfactors of a type $II_1$ factor
Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $...
- 7,047
2
votes
1
answer
170
views
Ultralimit of $w^*$-continuous maps
Let $\omega$ be a free ultrafilter on $\mathbb N.$ Let $(\mathcal M_n)$ be a sequence of finite von Neumann algebras. Let $\mathcal N$ be another finite von Neumann algebra and we have maps $\phi_n:\...
- 1,879
5
votes
1
answer
158
views
Backwards stable factors
A factor $R$ is called stable if $M_n(R)\cong R$ for all $n>0$. For the sake of this question, we call a factor backwards stable if $R\cong M_n(S)$ implies $S\cong R$ where $S$ is allowed to be any ...
- 759
1
vote
0
answers
86
views
Inner product on Standard form of von Neumann algebra
Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that
$$\langle x,yz\rangle=\langle zx,y\rangle$...
- 131
3
votes
0
answers
258
views
Von Neumann algebras as complemented subspaces
Question: Does there exist a non-injective von Neumann algebra $M\subseteq B(H)$, which is a complemented Banach subspace of $B(H)$?
According to an MO post, this problem was still open as of 2013. I'...
- 2,605
6
votes
1
answer
175
views
Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions
Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
- 407
2
votes
1
answer
170
views
Defining states on von Neumann algebras from filters on the projection lattices
Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for ...
- 2,830
1
vote
1
answer
180
views
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
votes
0
answers
115
views
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
2
votes
0
answers
121
views
Invariant weights associated to algebraic quantum groups
Consider an algebraic quantum group $(A, \Delta)$ in the sense of Van Daele, i.e. a regular multiplier Hopf $^*$-algebra with a positive left integral $\varphi$ and a positive right integral $\psi$.
...
- 175
0
votes
2
answers
149
views
Tensor product of operator values weights (in the theory of locally compact quantum groups)
Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object
$$\iota \otimes \psi: (M\overline{\otimes} M)_+ \to \widehat{M_+}.$$
...
- 175