All Questions
Tagged with oa.operator-algebras von-neumann-algebras
504 questions
3
votes
1
answer
254
views
Reference request for c*-algebra embedding into the atomic part of its double dual
Let $A$ be a C*-algebra. According to operator algebraists, it is well known that $A$ embeds into the atomic part of its double dual in the following sense: if $z$ is the central projection in $A^{**}$...
- 388
18
votes
0
answers
756
views
An "exercise" on von Neumann algebra tensor product
The following problem appears to be an easy exercise on von Neumann algebra tensor products, but since I've been failing to find a rigorous proof, I'd like to make sure it's not that trivial. Suppose $...
- 10.1k
7
votes
0
answers
1k
views
Books on von Neumann algebras
I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
- 455
3
votes
2
answers
540
views
Von Neumann Algebra isomorphism extension
I have Von Neumann algebra $\mathfrak{U}$ and a weakly dense *-subalgebra $A$. I have another Von Neumann Algebra $\mathfrak{V}$ and an injective *-homorphism $$\phi: A\longrightarrow \mathfrak{V}$$ ...
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
4
votes
0
answers
118
views
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
8
votes
2
answers
812
views
Weak*-norm continuous operators on von Neumann algebras
Let $M$ be a von Neumann algebra with predual $M_*$, and let $T\colon M\to M$ be a bounded, linear map. Let us say that $T$ is (sequentially) weak*-norm continuous if for every net (sequence) $(a_j)_j$...
- 3,497
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
1
vote
1
answer
164
views
A Baer *-ring which is not embedded into $B(H)$
Assume $A$ is a complex $*$-algebra which is also a Baer*-ring.
Q. Can we concluded that there exists a Hilbert space $H$ such that $A$ is embedded in $B(H)$ as a Baer*-ring? What about when $A$ ...
- 4,058
5
votes
1
answer
158
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Does every non-type-I factor's projection lattice admit a dense embedding of the standard continuum-collapsing poset?
Let $R$ be a non-type-I factor acting on a separable Hilbert space.
Let $P(R)$ be the set of $R$'s projections with the usual ordering ($x \leq y \iff$ range$(x) \subseteq$ range$(y)$) under which it ...
- 271
3
votes
1
answer
238
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How much of a factor's structure is determined by the order-type of its projection lattice?
H. A. Dye showed that a type II or III factor $R$ is determined, up to *-algebraic isomorphism or anti-isomorphism, by the ortholattice-isomorphism type of its projection lattice ("ortholattice-...
- 271
3
votes
1
answer
212
views
Embedding of von Neumann algebras
Let $A$ and $B$ be two purely infinite von Neumann algebras. Under which conditions $A$ can be embedded in $B$? For example is it true that every infinite factor can be embedded into the another ...
- 9,330
0
votes
1
answer
122
views
On index 2 and square of subfactors without extra intermediate
Let $N \subsetneq K_i \subsetneq M$, $i=1,2$, be a square of irreducible finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, such that there is no extra intermediate, with $K_1 \not \...
1
vote
1
answer
181
views
Existence of a third intermediate if there are two intermediate subfactors of index 2
Let $(N \subset M)$ be an irreducible finite index unital inclusion of hyperfinite ${\rm II}_1$ factors.
Let $K_1$ and $K_2$ be two distinct intermediate subfactors $N \subset K_i \subset M$, such ...
1
vote
1
answer
187
views
On the intersection of index 2 subfactors
Let $H_1$ and $H_2$ be two distinct index $2$ subgroups of a finite group $G$.
We can deduce several properties about the intersection $H_1 \cap H_2$:
$H_1$ and $H_2$ are normal subgroups of $G$. ...
15
votes
2
answers
1k
views
Is a C*-algebra with an isomorphic predual a von Neumann algebra?
It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...
- 3,497
14
votes
0
answers
647
views
Countably decomposable von Neumann algebras
A von Neumann algebra is countably decomposable if every family of mutually orthogonal nonzero projections is countable. Even a singly-generated von Neumann algebra need not be countably decomposable; ...
- 42.8k
6
votes
1
answer
486
views
Dye's Theorem for real von Neumann algebras
Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically ...
- 388
15
votes
1
answer
540
views
On spatial tensor products of von Neumann algebras
Let $H$ be a Hilbert space, and let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras.
We write $\odot$ for the algebraic tensor product,
and $\bar\otimes$ for the spatial tensor ...
- 43.2k
4
votes
1
answer
250
views
Can we solve the FGF problem by finding an appropriate action?
If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
1
vote
0
answers
80
views
weak convergence in operator space structure
Let $M$ be von Neumann algebra and $B(H)$ be it's universal representation. Let $(e_i)$ be a Hilbert basis of $H$ and $\zeta_n\xrightarrow{w}\zeta $ in $H$. I know that $[w_{\zeta_n ,e_i}]_{1\times I}\...
- 11
13
votes
2
answers
349
views
Is the domain of an operator valued weight closed under Hahn-Jordan decomposition?
Let $N\subseteq M$ be an inclusion of semi-finite factors with normal faithful semi-finite traces $\operatorname{Tr}_N$ and $\operatorname{Tr}_M$ respectively. Let $T: M^+\to \widehat{N^+}$ be the ...
- 5,425
13
votes
3
answers
1k
views
Separable von Neumann algebra
What is the simplest argument which shows that each infinite dimensional von Neumann algebra is not separable (in the norm topology)? It seems that this is a kind of folklore: at least I never saw the ...
- 9,330
7
votes
1
answer
378
views
The quantum group SUq(n) as von Neumann algebra
i have a question about a "presentation" of the quantum $SU(n)$. Here presentation means the following. Let $(M,\Delta)$ be a quantum group in the sense of Kustermanns and Vaes. One can show that the ...
4
votes
0
answers
187
views
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
1
answer
135
views
Is the module action $M\times M^*\to M^*$ jointly continuous?
Let $M$ be a W*-algebra and consider the following map:
$$\gamma: M\times M^*\to M^*: (a,f)\to af$$
where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
- 4,058
1
vote
1
answer
123
views
A relation among projections of a von Neumann algebra
This is a follow-up question on this. Let $A$ be a von Neumann algebra and $P$ be its projection lattice.
For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = 1-...
- 1,731
2
votes
0
answers
157
views
Support vectors and relative modular operator
I'm studying the relative modular operator and I'm looking for o good text to do it. Until now I'm using Araki's papers but I don't know how to deal with the support of a vector, $s^M(\xi)$, which is ...
8
votes
1
answer
422
views
The positive cone of the standard representation of a Von Neumann algebra
Let $A$ be a von Neumann algebra, let $L^2(A)$ be the underlying Hilbert space of the standard form of $A$, and $P \subset L^2(A)$ the canonical positive cone (see for example this paper by Haagerup). ...
- 42.4k
5
votes
0
answers
151
views
Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra
While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove:
Let $H$ be a Hilbert space, $A$ and $B$ trace von Neumann subalgebras ...
- 743
0
votes
0
answers
134
views
semifinite projection
Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$.
( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
- 101
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
5
votes
1
answer
239
views
second dual of minimal tensor products of $C^*$-algebras
Let $A$ be a unital $C^*$-algebras and $K(H)$ is $C^*$-algebras of compact operators on separable Hilbert space $H$. Is it true that $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?
- 85
6
votes
0
answers
441
views
Infinite number of non-isomorphic von Neumann algebras with property Gamma?
A II$_1$ factor $\mathcal M$ with trace $\tau$ has property Gamma if for every $\epsilon > 0$ and finite set $\{x_1,\cdots, x_n\} \subset \mathcal M$ there exists a trace 0 unitary element $y\in\...
- 3,984
5
votes
1
answer
203
views
What are the applications of the depth 2 reduction to the subfactors theory?
Let $(N \subset M)$ be an irreducible finite depth ($>2$) finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, then for $n$ sufficiently large the subfactor $(N \subset M_n)$ is depth ...
2
votes
0
answers
165
views
Rank–nullity theorem for finite von Neumann algebras
The rank-nullity theorem states that for $U, V$ finite dimensional vector spaces and $T:U \to V$ a linear map $$\dim(U) = \dim(im(T)) + \dim(ker(T)) $$
Let $M \subset B(H) $ be a finite von Neumann ...
3
votes
1
answer
196
views
A relative property gamma and $L(\mathbb F_2)$
Given any unital non-commutative subalgebra $\mathcal M$ of $L(\mathbb F_2)$ is it true that $\mathcal M' \bigcap L(\mathbb F_2)^\mathcal U = \mathbb C I$ for any free ultrafilter $\mathcal U$?
- 3,984
6
votes
2
answers
1k
views
Commutative von Neumann algebras and localizable measure spaces
This is not my subject so I apologize if my question is too obvious or understood from other pages.
I read some pages such as
Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
- 1,344
4
votes
1
answer
328
views
Tomita Takesaki theory and boundeness of $S$
Let $M$ be a von Neumann algebra, $\xi$-separating and cyclic vector for $M$. Let $S$ be antilinear operator acting as $x \xi \mapsto x^* \xi$ where $x \in M$. Then one can show that $S$ is closable ...
- 9,330
6
votes
1
answer
257
views
The (Hecke) double coset von Neumann algebra
It it well-known in the von Neumann algebra theory that for $\Gamma$ a non-trivial countable group, the von Neumann algebra $L(\Gamma)$ generating by $\Gamma$ acting by left multiplication on $l^2(\...
1
vote
0
answers
222
views
Examples of non-extremal subfactors
Every subfactor $(N \subset M)$ in this post are supposed to be finite index (unital) inclusion of ${\rm II}_1$ factors.
Definition (here p64): Such a subfactor is called extremal if $tr_{N'} = tr_M$ ...
0
votes
0
answers
76
views
The completely reducible bimodules coming from subfactors
This post is a sequel of: Are all the R-R-bimodules completely reducible?
Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...
5
votes
0
answers
428
views
Koopman representation, weakly compact action, Ozawa Popa
Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
- 193
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
1
vote
0
answers
81
views
Why does one only consider one-parameter groups in Borchers-Arveson theorem?
(question from math.stackexchange)
The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter ...
- 189
2
votes
0
answers
101
views
Are there infinitely many amenable Hadamard-Petrescu subfactors?
The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by W....
5
votes
2
answers
934
views
Question about projections in von Neumann algebras
Let $M$ be a von Neumann algebra, and let $\mathcal{P}$ be the set of nontrivial (not equal to $0$ or $e$) projections of $M$. Define $p,q \in \mathcal{P}$ to be equivalent if there exist projections ...
- 388
7
votes
0
answers
269
views
Approximation in the tensor square of a weakly exact von Neumann algebra
Background. I think I can prove something about a certain construction definition for Fourier algebras of discrete groups, under the assumption that the group is exact (well, really I use Yu's ...
- 25.8k
6
votes
1
answer
245
views
Comparing cardinalities of the spectrum of two masas in $B(H)$
If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
- 5,407
0
votes
1
answer
122
views
Is there an irreducible subfactor with an infinite homogeneous single chain lattice?
We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here).
Now ...