All Questions
Tagged with oa.operator-algebras von-neumann-algebras
172 questions with no upvoted or accepted answers
4
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118
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Smoothness in von Neumann algebra of measurable functions
Let $A=L^{\infty}(M)$ be an algebra of essentially bounded measurable function on manifold $M$. Let $D$ be a first order elliptic differential operator acting on some hermitian bundle $S$ over $M$ (...
- 9,330
4
votes
0
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187
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The weak-star closure of closed left ideals corresponding to pure states
I asked this question at math.stackexchange and received no comment.
Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous ...
- 4,058
4
votes
0
answers
195
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The groupoid VN algebra of the transversal to a uniquely ergodic action
I have a uniquely ergodic dynamical system preserving a finite ergodic measure (specifically, I have a nice aperiodic tiling space with an action of $\mathbb{R}^d$). Thus the transformation group von ...
- 1,010
4
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0
answers
151
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Is there a maximal finite depth infinite index irreducible subfactor?
A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...
4
votes
0
answers
282
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Extensions of completely positive mappings
I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-...
- 619
4
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417
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Connes fusion and the composition of completely positive maps
Let $N$ be a type $II_{1}$-factor with trace $\tau$.
An $N-N$ correspondence is a Hilbert $N$-bimodule $H$ where the left and right actions are both ultraweakly continuous. Equivalently, a ...
- 7,047
4
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answers
282
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If the flip automorphism of a finite factor can be connected to the identity is it approximately inner?
This intriguing question is due to Sorin Popa, so I'm marking it community wiki. I'd like to know the status of the question, and if it is still open, perhaps gather some new ideas for attacking it.
...
3
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109
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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
3
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0
answers
97
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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
3
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0
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117
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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
3
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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
3
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answers
60
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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
3
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258
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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
3
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0
answers
178
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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
3
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106
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When can a state on a C*-algebra descend to a quotient?
Has anything been written on the following question:
Let $(\mathfrak{A}, \phi)$ consist of a C*-algebra $\mathfrak{A}$ equipped with a tracial state $\phi$. Let $\pi : \mathfrak{A} \twoheadrightarrow ...
- 172
3
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138
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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
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answers
154
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"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
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111
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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
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0
answers
86
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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
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222
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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
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answers
109
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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
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0
answers
97
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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
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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
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0
answers
45
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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
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200
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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
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274
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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
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0
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141
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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
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152
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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
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0
answers
178
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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
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211
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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
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0
answers
200
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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
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0
answers
156
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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
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0
answers
304
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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
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0
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455
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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
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
2
votes
0
answers
157
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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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0
answers
118
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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
2
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0
answers
177
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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
2
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0
answers
121
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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
2
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answers
68
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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
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
2
votes
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192
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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
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136
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McDuff-to-hyperfinite step in Connes' Injectivity $\Rightarrow$ Hyperfiniteness
In Connes' "Classification of Injective Factors" (1976) the last step in Injectivity $\Rightarrow$ Hyperfiniteness (Thm. 5.1) is the implication 2. $\Rightarrow$ 1., where
$N \cong R$,
a) $...
- 121
2
votes
0
answers
105
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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
2
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0
answers
139
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Fixed point subalgebra
Suppose that $M$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^{G}$ the fixed point subalgebra and suppose that $M^{G}=\mathbb{C}$ (i.e., we have an ergodic ...
- 59
2
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0
answers
112
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Anticommutation of convolution products on trace class operators of quantum groups
This question was originally posted to MathStackExchange.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
- 59
2
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0
answers
113
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Induction of group von neumann algebra for group homomorphism with amenable kernel
Let $\alpha:H\to G$ be a group homomorphism betwenn discrete countable groups, and assume that the kernel of $\alpha$ is an amenable group, denoted by $K$. I would like to know ...
- 2,630
2
votes
0
answers
99
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Convergence of Brown measures
For each $n\in \mathbb N$, let $\mathcal M_n$ be a finite von Neumann algebra with a faithful trace $\tau_n$. Fix a non-principal ultrafilter $\omega$ on $\mathbb N$. Let $\mathcal M^\omega$ be the ...
- 1,418
2
votes
0
answers
156
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Extension of a theorem of Bisch to cyclotomic integers of fixed degree
Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such ...