All Questions
Tagged with oa.operator-algebras von-neumann-algebras
504 questions
3
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0
answers
138
views
Is there a finite depth irreducible subfactor of prime index and not group-subgroup?
Let $N \subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N \subset P \...
3
votes
0
answers
154
views
"Non-group" ${\rm II}_1$ factors
Do we have existence/examples/criteria for a ${\rm II}_1$ factor that is not isomorphic to $L(G)$ for any group $G$?
- 121
3
votes
0
answers
111
views
Is there a non-irreducible maximal subfactor other than two-sided TLJ?
A subfactor $N \subseteq M$ is called:
irreducible if $N' \cap M = \mathbb{C}$,
maximal if for any intermediate subfactor $N \subseteq P \subseteq M$ then $P=\{N,M \}$.
The two-sided ...
3
votes
0
answers
86
views
Finite index subfactors of hyperfinite type $\mathrm{III}_{\lambda}$ factors
Let $M$ be a hyperfinite type $\mathrm{III}_\lambda$ factor. $N$ be a finite index type $\mathrm{III}$ subfactor, is it true $N$ is hyperfinite type $\mathrm{III}_{\lambda}$?
- 677
3
votes
0
answers
222
views
Odometer actions of groups
If a group $G$ acts on a Cantor set $(X,\mu)$ by odometers, my question is: what is the explicit automorphism $\alpha_{g}$ for the extended Koopman action on $L^{\infty}(X,\mu)$, for $g$ $\in$ $G$? I ...
- 181
3
votes
0
answers
227
views
Is there a noncommutative version of von Neumann's ergodic theorem? [closed]
The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem.
E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) ...
- 139
3
votes
1
answer
388
views
Fundamental group and group measure space construction
Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
3
votes
0
answers
109
views
Does this element belong to $\mathbb CG$?
Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...
- 2,118
3
votes
0
answers
97
views
An example of a particular vn-algebra
Let $A$ be a vn-algebra. Let us suppose $e$ is a finite projection in $A$ and $x$ is an isometry (meaning $x^*x=1$) in $A$ such that $e$ does not commute with $x$. Then $\{q_n=x^nex^{*n}\}$ forms a ...
- 4,058
3
votes
0
answers
137
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"Adding" a projection to a von Neumann algebra
This is a question about what happens when you "add" a new projection $p$ to a von Neumann algebra $\mathcal{R}$ to generate a larger v.N. algebra $(\mathcal{R} \cup \{p\})''$.
Suppose that $\mathcal{...
- 271
3
votes
0
answers
45
views
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 ...
- 365
3
votes
0
answers
200
views
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.
- 9,330
3
votes
0
answers
274
views
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: ...
- 241
3
votes
0
answers
141
views
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_{\...
- 31
3
votes
0
answers
152
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 ...
- 271
3
votes
0
answers
178
views
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 ...
- 4,058
3
votes
0
answers
211
views
Arveson spectrum for a unitary representation of a group on a Hilbert space
Although this is not research, I think the question is a little bit too specific for math.stackexchange
Let $G = \mathbb{R}$. By Stone's theorem, $U(t)\in\mathcal{B}(\mathcal{H})$ is generated by a ...
- 189
3
votes
0
answers
200
views
What are the first non-maximal non-group-subgroup simple irreducible subfactors?
Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$
is normal if the biprojections $e_{\...
3
votes
0
answers
156
views
A section from subfactors to transitive groups
A finite group-subgroup subfactor is a subfactor $(N \subset M)$ isomorphic to $(R^G \subset R^H)$ with $(H \subset G)$ an inclusion of finite groups acting as outer automorphism on the hyperfinite II$...
3
votes
0
answers
304
views
Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$?
The fundamental group $\mathcal{F}(N \subset M)$ of a unital inclusion of II$_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \...
3
votes
0
answers
455
views
Morphism of von Neumann Algebras
Hello,
Is there a counterexample to the following statement:
let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^* $-algebras is a $W^*$-homomorphism ?
( a $W^* $-...
- 663
3
votes
1
answer
335
views
What's the natural equivalence of subfactors in general?
Let $A$ be a factor and $\mathcal{C}_{A}$ be the category of all the subfactors $(M \subset N)$ such that $M$ and $N$ are isomorphic to $A$. The most famous of them is perhaps $\mathcal{C}_{R}$ with $...
2
votes
1
answer
182
views
Normal $*$-homomorphism
Let $\pi:\mathscr M\to\mathscr M$ be a normal $*$-homomorphism between a von Neumann algebra $\mathscr M.$ Assume $\mathscr M$ has a normal semifinite faithful trace. Does $\pi$ extend as a bounded ...
- 1,879
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
1
answer
111
views
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
2
votes
1
answer
369
views
On diagonal part of tensor product of $C^*$-algebras
Suppose we have a $C^*$-algebra $\mathcal{U}$, Consider the $C^*$-subalgebra generated by elements of the form $a\otimes a$, what is it isomorphic to? Is it isomorphic to $\mathcal{U}$ itself?
- 677
2
votes
1
answer
208
views
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 ...
- 361
2
votes
2
answers
252
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$. ...
- 2,118
2
votes
1
answer
233
views
${\rm II}_1$-factors with finite commutant: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \Rightarrow \mathcal{A}' \cap \mathcal{B}'$ hyperfinite?
Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}', \mathcal{B}' $ are also a ${\rm II}_1$-factors.
Question: $\mathcal{A} \cap \mathcal{B} = \mathbb{...
2
votes
1
answer
429
views
Proof of uniqueness of predual of von Neumann algebra
I am currently reading Jesse Peterson's lecture notes on von Neumann algebras. I'm confused by lemma 4.4.2. In particular, it seems to me that Hahn Banach theorem here can only conclude the map $\phi$ ...
- 53
2
votes
2
answers
867
views
Decomposition of an abelian von Neumann algebra
Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance.
"Let $\pi$ be a non-degenerate ...
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
2
votes
1
answer
527
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
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
272
views
A question on quantum tori
Let $\mathbb T_\theta^2$ be quantum tori generated by two unitary operators $u,v$. can $u,v$ be finite dimensional?
- 1,879
2
votes
1
answer
709
views
Topology of state space in von Neumann algebras
What are the sufficient conditions for a von Neumann algebra to have a first countable set of states with respect to the weak * operator topology?
- 123
2
votes
1
answer
757
views
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 $...
- 455
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 $...
- 7,057
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
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
2
votes
1
answer
67
views
Analogue of spectral values of automorphisms in vN algebra
Is there any analog of studying spectral properties of automorphisms of von Neumann algebra? Does it make sense, if anybody knows please give a reference.
- 677
2
votes
1
answer
160
views
Approximation of the central support
Let $(M,\tau)$ be a tracial von Neumann algebra, i.e.
a unital subalgebra $M=M''\subset \mathbb{B}(H)$;
a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ implies ...
- 743
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
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
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
2
votes
1
answer
462
views
Direct integral decomposition relative to a given measure space
It is well known that a separable Hilbert space $H$ decomposes as a direct integral in the presence of an abelian von Neumann algebra $\mathscr A\subseteq B(H)$.
More precisely, and quoting from ...
- 483
2
votes
2
answers
302
views
Is $x \mapsto x \otimes 1$ $\sigma$-weakly continuous?
Let $M\subseteq B(H)$ be a von Neumann algebra. Is it true that the mapping
$$\psi: M \to B(H \otimes H): m \mapsto m \otimes \text{id}_H$$
is $\sigma$-weakly continuous? Here the $\sigma$-weak ...
user167952
2
votes
1
answer
564
views
Ultrapower of an ultrapower of von Neumann algebras
Let $M$ be a $\mathrm{II}_1$ factor, fix a ultrafilter $\omega$, we know the ultrapower $M^{\omega}$, is again $\mathrm{II}_1$ factor. The question is that what is the ultrapower of $M^{\omega}$, i.e.,...
- 181
2
votes
1
answer
116
views
Extending $C^*$-norms from $*$-subalgebras
Let $A$ be a unital $*$-algebra, and $B$ a unital $*$-subalgebra of $A$. In addition, assume that there exists a $B$-$B$-sub-bimodule $C \subset A$, such that
$$
A \simeq B \oplus C,
$$
where $\...
- 969
2
votes
1
answer
114
views
Order isomorphic order intervals
Let $M$ be a von Neumann algebra. If $x$ is positive, then Lemma 2.1(3) of the paper "Order Isomorphisms of Operator Intervals in von Neumann Algebras" (Mori, Integral Equations and Operator Theory, ...
- 388