All Questions
Tagged with oa.operator-algebras von-neumann-algebras
504 questions
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
1
vote
1
answer
491
views
Is this result of Spain correct?
Let us have a look on the proof of Theorem 2 in [P. G. Spain, Boolean algebras of projections, Proceedings of the Edinburgh Mathematical Society (Series 2) 19, 03, March 1975, 287-289]
The author ...
- 203
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
1
vote
1
answer
234
views
Intersection of von-Neumann algebra factors
Given two von-Neumann algebra factors $\mathcal M,\mathcal N$, is $\mathcal M\cap\mathcal N$ a factor?
And how about the intersection of infinitely many factors?
Notes:
I know that the intersection ...
- 896
1
vote
1
answer
189
views
Normal states on a type III$_1$ factor
Let $M$ be a type III$_1$ factor. Suppose $\rho$ is a normal state on $M$, given any $c\in [0,2]$, can we find a normal state $\rho'$ on $M$ such that $\|\rho-\rho'\|=c$? Or can we find a sequence of ...
- 427
1
vote
1
answer
408
views
Separability of von Neumann algebra
In a lecture note, from where I am studying theory of von Neumann algebras, the author has commented that the following are equivalent. Let $A$ be a von Neumann algebra.
$A$ is SOT separable.
$A$ is ...
- 1,879
1
vote
1
answer
126
views
Subalgebras of $II_{1}$ factor
Let $M$ be a type $II_{1}$ factor, Let $B$ is an infinite dimensional nonabelian subalgebra. Is it true that $B$ always type $II_{1}$ ?
- 677
1
vote
1
answer
86
views
Subprojections of the sum of mutually orthogonal Abelian projections
Let $f_i$, $i=1,\dotsc,n$, be mutually orthogonal Abelian projections in a von Numann algebra, and let $e\leq\sum f_i$. Is it true that there exist mutually orthogonal Abelian projections $e_j$, $j=1,\...
- 2,118
1
vote
1
answer
108
views
Clarification on predual on existence of separating vector
We know predual of a von Neumann algebra $M$ as a Banach space is independent of Hilbert space where the $M$ is represented. Now the question is if we represent $M$ in $B(\mathcal{H})$, where $M$ has ...
- 227
1
vote
1
answer
174
views
On predual of von Neumann algebra
Suppose $T_{n}$ converging $T$ in vN algebra $M$ in weak operator topology, can we conclude $||T_{n}||$ is uniformly bounded? Another question if a linear functional $\varphi$ is continuous in unit ...
- 227
1
vote
1
answer
190
views
Infinite amenable group subfactors
Let amenable groups $\Gamma$ and $\Gamma'$. They act outerly of only one manner on the hyperfinite ${\rm II}_1$-factor $\mathcal{R}$.
Question: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) ...
1
vote
1
answer
251
views
On the relation between the set of extreme points of the unit ball of $M(X)$ and $M(X)^{**}$
Suppose that $X$ is a locally compact topological space. Let $M(X)$ denote the Banach space of regular Borel measures on $X$. It is known that the bidual of $C_0(X)$ is a commutative $C^*-$algebra. ...
- 306
1
vote
1
answer
186
views
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
1
vote
1
answer
129
views
Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?
Let $X\subseteq B(H)$ be an operator system and let $M\subseteq B(K)$ be a von Neumann algebra. We form the Fubini-tensor product
$$X \otimes_\mathcal{F} M := \{z \in B(H\otimes K): (\sigma\otimes \...
- 525
1
vote
1
answer
114
views
Continuous surjection between spectra of commutative von Neumann algebras
Suppose that $V_1,V_2$ are two commutative von Neumann algebras and $V_1 \subset V_2$. Being in particular commutative $C^*$-algebras we have that $V_1 \cong C(X_1), V_2 \cong C(X_2)$ for some ...
- 9,330
1
vote
1
answer
96
views
On boundedness of sequence of operators in vN algebra
Let $x_{n}$ be a sequence of operators in vN algebra $M$, $\Omega$ is a cyclic vector for $M$, if $x_{n}\Omega$ converges in $\mathcal{H}$, can we say there exist a subsequence $\{y_{n}\}$ of $\{x_{n}\...
- 677
1
vote
1
answer
172
views
Examples of isomorphic W* algebra with non-homeomorphic weak topology
Due to the uniqueness of the predual, a W* algebra, when realized as a von Neumann algebra in any way, always has a unique, well-defined ultraweak (or $\sigma$-weak) topology. The same can be said ...
- 1,586
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
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
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
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
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
1
vote
1
answer
177
views
Commuting and generating subfactors of $ B(H)$
I have a question on subfactors of $B(H)$ (the von Neumann algebra of bounded operators on a complex Hilbert space).
Say I have a subfactor $M$ of $B(H)$. Is it true that another subfactor $N \subset ...
- 759
1
vote
1
answer
125
views
On commutant of $II_{1}$ factors
Suppose $M$ is $II_{1}$ factor but need not be in standard form. Under what condition (on $M$ or Hilbert space) is the commutant $M'$ of $M$ again $II_{1}$ factor on the Hilbert space acted by $M$?
- 677
1
vote
1
answer
170
views
The range projection of product of projections
Let $A$ be a von Neumann algebra. Let $p$ be a projection in $A$. Suppose that $e$ is a finite projection. Can we determine all types of vn-algebras in which $p-p\wedge(1-e)$ is a finite projection?...
- 4,058
1
vote
1
answer
171
views
On projection theory for inseparable Hilbert spaces
How can one see that $I$ is an infinite projection in $B(\mathcal{H})$, where $\mathcal{H}$ is an inseparable Hilbert space?
- 227
1
vote
1
answer
127
views
Suppose $(A,H)$ is a vNa. When is an inner automorphism of $\mathcal{B}(H)$ an inner automorphism of $A$
Let $A$ be a von Neumann algebra acting on a Hilbert space $H$. Write $\mathcal{B}(H)$ for the bounded linear operators on $H$. Suppose that $\rho:\mathcal{B}(H) \rightarrow \mathcal{B}(H)$ is an ...
- 556
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$. ...
1
vote
1
answer
287
views
How coarse is the coarse correspondence?
Let $M$ denote a finite von Neumann algebra with trace $\tau$, and $L^{2}(M)$ denote the standard (trivial) M-M correspondence (binormal bimodule). The coarse correspondence is $L^{2}(M) \overline{\...
- 7,057
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
1
vote
0
answers
91
views
Subfactors with integer Jones index
Is there any integer (Jones) index subfactor which is not extremal?
- 393
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
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
1
vote
0
answers
150
views
Intersection of finitely many type-I von-Neumann algebra factors
If $\mathcal M,\mathcal N$ are type-I von-Neumann algebra factors (over the same Hilbert space $\mathcal H$), is $\mathcal M\cap\mathcal N$ a type-I von-Neumann algebra factor?
Notes:
An elementary ...
- 896
1
vote
0
answers
385
views
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
1
vote
0
answers
87
views
irreducible subfactor inclusion and commutativity of induced projections
Let $N\subset M$ be an irreducible subfactor inclusion, i.e., $N'\cap M =\mathbb C1$, acting on a Hilbert space $H$.
Let $\Omega\in H$.
Does it follow that the projections onto $[N\Omega]$ and $[M'\...
- 759
1
vote
0
answers
101
views
Sequences in von Neumann algebras
Let $(x_n)$ be a sequence in a von Neumann algebra $M$ or its predual $M_*$.
Is there a hyperfinite von Neumann subalgebra $N$ of $M$ such that $(x_n)\subset N$ or $N_*$?
- 617
1
vote
0
answers
87
views
Showing the existence of a right-inverse in a von Neumann probability space
Disclaimer: This is my first post here on Overflow as opposed to the "normal" forum, so if this question is too elementary for this forum, I'd appreciate y'all letting me know. I posted it ...
- 172
1
vote
0
answers
399
views
Pairs of subfactors
Suppose we have two subfactors $P\subset M$ and $Q\subset M$ with finite Jones indices (here $P,Q$ and $M$ all are $II_1$ factors). Under what condition the von Neumann algebra $L$ generated by $M,e_P$...
- 393
1
vote
0
answers
113
views
Property gamma for type III factors
I am struggling to find a definition which uses centralizing sequences for property gamma in type III factors?
If $M$ is type $\mathrm{III}$ factor, what is the exact definition of property gamma ...
- 181
1
vote
0
answers
179
views
Noncommutative analogue of Radon-Nikodym derivative
Let $\mathcal M$ be a noncommutative probability space i.e. there is a trace $\tau$ on $\mathcal M$ such that it is normal, faithful and $\tau(1)=1.$ Let $\tau_1$ be another normal trace on $\mathcal ...
- 1,879
1
vote
0
answers
111
views
On a doubt on spectral measure on Gelfand spectrum
In the lecture notes of Peterson https://math.vanderbilt.edu/peters10/teaching/spring2013/vonNeumannAlgebras.pdf he proves the following theorem
THEOREM 2.7.5: Let $\mathcal H$ be a Hilbert space and $...
- 1,879
1
vote
0
answers
118
views
Some doubt on crossed product von Neumann algebras
There are two definitions in different books. Let $G \curvearrowright M$, then there is the definition of forming Group ring $M[G]$, define product and addition then make its algebra. represent the ...
- 677
1
vote
0
answers
160
views
Projections in tensor product of vN algebras
Can we write any projection in the tensor product vN algebra $M\otimes N$ in terms of limits of projections $p\otimes q$, where $p$ and $q$ are projections in M, N or somewhat relate the projections ...
- 677
1
vote
0
answers
68
views
Studying fixed point algebra under group action
If $M$ is in standard form, consider the action of a finite group on $M$, does the fixed point subalgebra under the action is in standard form? What we can say if $M$ is hyperfinite $\mathrm{II}_{1}$ ...
- 677
1
vote
0
answers
83
views
Sequence of unitaries in type III von Neumann algebra
Consider a type III von Neumann algebra $\mathcal{M}$ and an isometry $w$. How does one show that there exists a sequence of unitaries $u_n\in\mathcal{M}$ that converge strongly to $w$?
For instance,...
1
vote
0
answers
83
views
Are these kinds of "crossed product" studied?
Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...
- 2,118
1
vote
0
answers
74
views
About crossed product of the group von Neumann algebra
Let $G$ be a group with a normal subgroup $N$. If $K$ is a field, then it is well known in ring theory that $K[G]\cong K[N]*(G/N)$ ($*$ stands for the crossed product). Do we have such an isomorphism ...
- 2,118
1
vote
0
answers
185
views
Unitary element of the group algebra
Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...
- 2,118