Questions tagged [von-neumann-algebras]
Subtag of the [oa.operator-algebras] tag for questions about von Neumann algebras, that is, weak operator topology closed, unital, *-subalgebras of bounded operators on a Hilbert space.
621 questions
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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 $\phi$ ...
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Connes Embedding Conjecture is false [closed]
This preprint from yesterday claims to prove that Connes Embedding Conjecture fails.
Since the paper is from outside Operator Algebras (Computer Science/Quantum Computing) and they actually work on ...
- 2,793
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1
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172
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Action of hyperbolic group on von Neumann algebra
Let $G$ be a hyperbolic group. Let $M$ be a vN algebra in standard form. Can there exist a faithful action of $G$ on $M$ such that
\begin{align*}
\sigma_{g_n} \rightarrow I
\end{align*}
for some ...
- 181
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0
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89
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On mixing and weak mixing subalgebras of finite von Neumann algebras
Let $M$ be a full $\mathrm{II}_1$ factor. Consider mixing and weak mixing subfactors $B$ and $C$ of $M$. Are $B$ and $C$ full?
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1
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0
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113
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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
2
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0
answers
135
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On crossed product of L^{P} spaces
Let $M$ be a von Neumann algebra with faithful normal state $\varphi$, and $G$ be a group action on $M$ preserving $\varphi$. Then is it true
\begin{align*}
L^{p}(M\rtimes G, \varphi^{M\rtimes G})\...
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2
votes
1
answer
564
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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.,...
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2
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Unusual crossed product constructions being factors
Let $A$ be an abelian von Neumann algebra and $G$ a countable group acting on $A$. In the literature we meet usually two kinds of crossed product $A \rtimes G$ being a factor:
if the action is (...
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0
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3k
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On prime factors
Let $M$ be a prime $\mathrm{II}_1$ factor. Let $N$ be a non hyperfinite finite index subfactor $N$, is $N$ prime factor?
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7
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0
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502
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Abstract characterization of group von Neumann algebra (II1 factor)
The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{...
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1
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310
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On conditional expectation from tensor products
Let $M$ be a $\mathrm{II}_{1}$ factor. Does there exist a conditional expectation from $M^{\otimes 2}$ to $M$ preserving the trace $\tau^{\otimes 2}$?
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3
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0
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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 $\mathrm{III}_{\lambda}$?
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8
votes
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224
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Is the invariant subalgebra of the von Neumann algebra $L(F_k)$ isomorphic to $L(F_k)$?
Let the symmetric group $G=S_{k}$ act on the von Neumann algebra of the free group $L(F_k)$ via permuting its generators. Is the fixed point algebra under the action isomorphic to the whole algebra, i....
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3
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158
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Given non-type-I subfactors $R \subset S$, must $S$ have a projection that meets no projection in $R$ except $1$?
Let $R \subset S$ be distinct non-type-I von Neumann factors; say two projections $P, Q \in S$ "meet" if they have a common non-null subprojection (i.e. if $P \wedge Q \neq 0$), and call $P$ "$R$-...
- 271
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0
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On an application dominated convergence theorem in vN algebras
$M$ be a $\mathrm{II}_{1}$ factor equipped with the faithful normal trace $\tau$ in the standard form. Let $\tau(Jx'J\eta)=0,\forall x' \in M'$ and fixed $\eta$ in $L^{1}(M,\tau)$. Is it true $\eta=0$?...
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On $L^{1}(M',\tau')$
Let $M$ be a $\mathrm{II}_{1}$ factor with the faithful trace $\tau$ in standard form. Suppose $\tau'(x')=\tau(Jx'^{*}J)$. If we complete $M$ with $\|\cdot\|_{1}$ norm, i.e., $\|x\|_{1}=\tau(|x|)$, ...
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4
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0
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120
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On existence of property gamma of C star simple group von Neumann algebra
We call a finite vN algebra $M$ with trace $\tau$ has property Gamma iff there exist unitary $u_{n}\rightarrow 0$ weakly and $\|xu_{n}-u_{n}x\|\rightarrow 0$. My question can we have C*-simple ...
- 181
2
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0
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100
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On relative commutant inside crossed product
Let $G$ be a discrete group acting on vN algebra $M$ in standard form. My question what is relative commutant of $M$ and $L(G)$ infact what is $M'\cap (M\rtimes G)$ and $L(G)'\cap(M\rtimes G)$?
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1
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1
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125
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On existence of certain operators in von Neumann algebra
Let $M\subset B(\mathcal{H})$ be an infinite dimensional vN algebra in standard form. Fix $\xi\neq 0 \in \mathcal{H}$, does there exist $M\ni x_{\xi}\neq I$ such that $x_{\xi}(\xi)=\xi$?
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Is this representation of Go (game) irreducible?
This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph.
Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
0
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0
answers
88
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On invertibility of ergodic averages
Let $x$ be invertible unbounded operator affiliated operator to the $\mathrm{II_{1}}$ factor $(M,\tau)$. Under which condition on $x$, the iterates also $1+\sigma(x)+\cdots+\sigma^{n}(x)$ are ...
- 181
3
votes
1
answer
148
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On analogue of ratio in operator algebras
For functions spaces we have ratio of two functions namley $\frac{f}{g}$, Can ratio of two operators in von Neumann algebra make sense? If at all it makes sense, what will be the proper definition of ...
- 181
3
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0
answers
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
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0
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127
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On examples of action of C-star simple group on von Neumann algebra
Can there exist a faithful action of a $C^{*}$-simple group $G$ on a von Neumann algebra $(M,\varphi)$ equipped with faithful normal state $\varphi$ such that action preserves the state $\varphi$ and ...
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0
votes
1
answer
204
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A certain class of representations
Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?
(The word "finite-dimensional" was ...
- 2,118
1
vote
0
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179
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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 $\mathcal ...
- 1,879
4
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type III$_1$ states
Given a von Neumann algebra that is a type III$_1$ factor with the state $\omega$ and any $\epsilon>0$ is it always possible to find a projection or a partial isometry in the algebra such that its ...
3
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1
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89
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Converegence of modulus in nocommutative $L_p$-spaces
Let $1\leq p<\infty.$ Let $\mathcal M$ be a von Neumann algebra equipped with a normal semifinite faithful trace $\tau.$ Let $L_p(\mathcal M,\tau)$ be the associated noncommutative $L_p$-space. Let ...
- 1,879
2
votes
1
answer
116
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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 $\...
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0
answers
108
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On standard form of corners
Let $(M, \tau)$ be a $\mathrm{II_{1}}$ factor equipped with the faithful normal tracial state $\tau$ acts on $L^{2}(M,\tau)$ is in standard form. The question is: if we consider vN algebras $PMP(P\in ...
- 677
2
votes
1
answer
182
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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
3
votes
2
answers
394
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Extension of trace on von Neumann subalgebra
Let $R$ be a finite von Neumann algebra and $S$ be a von Neumann subalgebra of $R$. Does every normal tracial state on $S$ extend to a normal tracial state on $R$?
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1
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1
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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
0
answers
111
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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
6
votes
1
answer
353
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Equivalence of $\sigma$-weak topology to another topology
Let $\mathcal H$ be a Hilbert space. Define a topology $\tau_1$ on $B(\mathcal H)$ by the family of seminorms $x\mapsto |Tr(xa)|,$ $a\in L^1(B(\mathcal H)).$ Here $B(\mathcal H)$ denotes the set of ...
- 1,879
5
votes
1
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499
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Variations on Kaplansky Density
Let $A$ be a $C^*$-algebra and $\pi:A\rightarrow B(H)$ a faithful $*$-representation, so $M=\pi(A)''$ is a von Neumann algebra and $A\rightarrow M$ is an inclusion. von Neumann's Bicommutant Theorem ...
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3
votes
0
answers
227
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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) ...
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2
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1
answer
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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, ...
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On crossed product subalgebra
For Compact pmp actions of a group $G$ on measure space $(X,\mu)$, if a subalgebra $B$ is such that $L(G)\subset B \subset L^{\infty}(X)\rtimes G$, is it always true that $B$ is also of the form ...
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116
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On relationship between cryptography and operator algebras [closed]
Does quantum cryptography connect two different areas of math operator algebras and Cryptography?
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0
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96
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Could we assume without loss of generality that all coefficients are positive?
Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...
- 2,118
1
vote
0
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64
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Being additive map using spectral decomposition theorem
Let $(\mathcal{A},\tau_\mathcal{A})$ and $(\mathcal{B},\tau_\mathcal{B})$ be semifinite von Neumann algebras with normal semifinite faithful traces $\tau_\mathcal{A}$ and $\tau_\mathcal{B}$. I defined ...
- 11
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0
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86
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Characterzing compact actions on von Neumann algebra
Suppose $G$ is a countable discrete group acting on vN algebra $M$, the action is compact. Can we have a topology on Aut$(M)$ such that $\{\sigma_{g}\in \text{Aut}(M):g \in G\}$ form a compact subset ...
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3
votes
0
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74
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Strong operator convergence of support in non-commutative $L^p$ spaces
Let $M$ be a von Neumann algebras with normal faithful semifinite trace $\tau.$ Let $L^p(M)$be the associated non-commutative $L^p$-space. Suppose, $x\geq 0$ be an element of $L^p(M)$ and $0\leq x_n\...
- 1,879
11
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0
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336
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Hausdorff dimension and von Neumann dimension
There are two subjects in which non-integral dimensions appear:
fractal geometry: consider the well-known Hausdorff dimension of fractals.
von Neumann algebra: consider a type ${\rm II_1}$ ...
7
votes
0
answers
268
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Enveloping von Neumann algebra of Clifford algebra
As explained in the book "Spinors in Hilbert Space" by Plymen and Robinson, if $V$ is a complex (separable) Hilbert space with a real structure, and $\mathrm{Cl}(V)$ the corresponding Clifford algebra,...
- 9,853
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0
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54
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On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
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0
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68
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On existence of sequence of unitaries in $II_{1}$ factor $M$
Let $M$ be a $\mathrm{II}_{1}$ factor acting on $L^{2}(M, \tau)$ in standard form, let $\{e_{n}:n \in \mathbb{N}\}$ be fixed orthonormal basis of $L^{2}(M, \tau)$, does there exist sequence of ...
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vote
0
answers
118
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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 ...
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2
votes
1
answer
83
views
On existence of fixed point operator
Let $M$ be an infinite dimensional non-type $I$ factor, given $\xi$ in $\mathcal{H}$, does there exist a not identify operator $x$ in $M$ such that $x\xi=\xi$, I have tried with taking projection $P_{\...
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