A von Neumann algebra is a unital *-subalgebra of the algebra of bounded operators on a Hilbert space, closed in the weak operator topology. Also called a $W^*$-algebra, and may be regarded as a non-commutative generalization of $L^\infty$ space. These algebras are extensively used in knot theory, ...

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Support of a faithful representation

Let $A$ be a C*-algebra. Any representation $\pi:A\to B(H)$ is uniquely extended to a $w^*$-continuous representation $\tilde{\pi}:A^{**}\to B(H)$. Q: I am looking for an example of a faithful ...
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Is the bilinear map $M\times M^*\to M^*$ 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 ...
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Existence of uniform multiplicity projection in abelian Von Neumann algebras.

I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with ...
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20 views

A normal positive linear functional whose support is 1

Let $M$ be a W*-algebra with unit $1$. Q1) Does there exist a normal positive linear functional $\phi$ on $M$ whose support is 1? Q2) We know that there is a unique central projection $z$ in ...
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Support of a normal pure state is a rank one projection

Let $\phi$ be a normal pure state on a w*-algebra $M$ and $\{\pi, \xi, H\}$ its GNS representation associated to $\phi$. Suppose projection $e$ is the support of $\phi$. Show that $\pi(e)$ is a rank ...
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Form of the Polar decomposition for $M_{\varphi}$

Polar Decomposition:Let ‎$‎‎v$ ‎be a‎ ‎continuous ‎linear ‎operator ‎on a‎ ‎Hilbert ‎space ‎‎$‎‎H$.then ‎there ‎is a‎ ‎uniqe ‎partial ‎isometry ‎‎$‎‎u\in B(H)$ ‎such ‎‎$‎‎v=u‎‎\mid ‎v‎\mid‎‎‎$ ‎and ...
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Why the set of pure state ‎is ‎weak* ‎compact?

Let ‎$‎‎A$ ‎be a‎ ‎C*-algebra‎. ‎ ‎$‎S(A)‎$ ‎is ‎the ‎set ‎of ‎state ‎on ‎‎$‎‎A$ and $‎‎PS(A)$ ‎is ‎the ‎set ‎of ‎pure ‎state ‎on ‎‎$‎‎A$. ‎ ‎ I ‎know ‎that ‎if ‎‎$‎‎A$ ‎is ‎unital ‎then ‎‎$‎‎S(A)$ ...
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‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow ...
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left shift operator and compact operator [closed]

Let ‎$‎‎S$ ‎be the ‎left ‎shift ‎on ‎‎$‎‎\ell^2$ ‎i.e ‎‎$‎‎S(x_1,x_2,x_3,...)=(x_2,x_3,...)$‎‎. ‎ Assume that ‎$‎‎T$ ‎is a ‎compact ‎operator ‎such ‎that ‎‎$‎‎TS=ST$.‎ ‎ Q:‎$‎‎T$ ‎should ‎be ‎zero ...
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postliminal $C^*$-algebra

A ‎$‎‎C^*$-algebra ‎‎$‎‎A$ ‎is ‎said ‎to ‎be ‎postliminal ‎if ‎for ‎every ‎non-zero ‎irreducible ‎representation ‎‎$‎(H,‎\varphi‎)‎$ ‎we ‎have ‎‎$‎‎K(H)‎\subseteq‎ ‎‎\varphi‎(A)‎$‎ ‎ In ‎Murrphy's ...
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Equivalent non-degenerate representations of C*-algebras

For two non-degenerate representations $\pi_j:A\to B(H_{\pi_j})$ ($j=1,2$), we write $\pi_1\sim\pi_2$ if there exists a $w^*$-continuous isometrically isomorphism from $\pi_1(A)''$ onto ...
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Normal positive functional on Von Neumann algebras

Let $A$ be Von Neumann algebra. A positive linear functional ‎$‎‎‎\varphi‎$ on $A$ ‎is ‎said ‎to ‎be ‎normal ‎if ‎for ‎any ‎self‎adjoint and increasing nets such that ‎$‎‎u_{\alpha‎}\longrightarrow ...
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24 views

Second countable property for abelian von neumann algebras

I am looking at Murphy's book "$C^*$algebras and operator theory", in the section on abelian Von Neumann algebras (end of chapter 4). There, it is explained that any (unit containing) abelian Von ...
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11 views

Normal and singular part of a bounded linear map on von Neumann algebras

Assume $M$ and $N$ are von Neumann algebras. Every *-representation $\pi:M\to B(H)$ is a direct sum $\pi=\pi_n\oplus\pi_s$ where $\pi_n$ is normal and $\pi_s$ is singular *-representation. Based on ...
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28 views

Non-degenerate representations and central projections

Let $A$ be a C*-algebra and $\pi:A\to B(H)$ a non-degenerate *-representation. We also denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$ in $B(H)$. The predual of the von Neumann ...
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Polar decomposition for completely bounded linear maps

Let $M$ be a W*-algebra and $f$ be a norm one and normal functional on $M$. Polar decomposition says that, there is a unique positive linear functional, denoted by $|f|$, satisfying in: ...
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closed convex hull of projection

$1$:I know that if ‎$‎‎F$ is a ‎locally convex ‎compact ‎space ‎then ‎‎$‎‎‎\overline{co}(‎Ext (F))=F$‎ ($Ext$: means extreme point) $2$:I ‎know ‎that ‎if ‎‎$‎‎M$ ‎is a ‎Von ‎Neumann ‎algebra ‎then ...
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projecton and positive element in $C^*$algebras

Let ‎$‎‎A$ ‎be ‎a‎ ‎$‎‎C^*$-algebra.‎$‎‎p\in A$ ‎is a ‎‎projections. ‎‎‎ Assume ‎that ‎‎$‎‎a$ ‎is a element in‎$‎‎ Ball(A_+)$ ‎such ‎that ‎‎$‎‎a‎\leq p‎$‎ Q:May I‎ ‎say ‎‎$‎‎ap=pa$?why?‎ ‎
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weakly convergent sequence of operator on $B(H)$

Let $H$ be a Hilbert space. Assume that $\{u_n\} \subseteq B(H)$ is W.O.T convergent.(‎ ‎$‎u_n‎\rightarrow u‎$‎‎ ‎in ‎W.O.T ‎topology ‎iff ‎‎$‎‎‎‎<‎u_n(x),y>\rightarrow‎ ‎‎‎<‎u_n(x),y> ‎ ...
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44 views

which von Neumann algebras have many sufficiently normal irreducible representations?

Let $M$ be a von Neumann algebra. We say that $M$ has many sufficiently normal irreducible representations, namely $\{\pi_i\}$, if $||a||=\sup ||\pi_i(s)||$. For example the second dual of a ...
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28 views

Equivalent projections and 2-sided ideals in von Neumann algebras

Let $M$ be a von Neumann algebra. For given $x$ in $M$, we put $I_x=MxM$ (the algebraic ideal generated by $x$). We know that if two projections $p$ and $q$ are equivalent (that is $\exists u\in M$ ...
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Minimal projections on von Neumann Algebras

A projection $p \neq 0$ in a von Neumann Algebra $A$ is called minimal, if for every projection $0\neq q\in A$ with $q \leq p$ already $q=p$. I want to prove the following theorem: For a minimal ...
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Von Neumann algebras with uncountable sets of incompatible projections

Which von Neumann algebras acting on separable Hilbert space $H$ have uncountable antichains of projections? ("Antichain" meaning a set of projections any pair of which has no shared nonzero ...
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an isomorphism from $L^\infty(\mathbb{T})$ to $L^\infty([-1,1],\displaystyle\frac{2}{\pi } \sqrt{1-t^2}\mathrm{d}t)$

$\mathbb{T}$ is the boundary of unit ball.Consier $\phi:[-1,1]\rightarrow\mathbb{T},\phi(t)=exp^{2i(arcsint+t\sqrt{1-t^2})},t\in[-1,1]$. It is easy to check that $L^2(\mathbb{T})\ni f\mapsto f\phi\in ...
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Is a unital $*$-homomorphism preserving a state is one-to-one?

Let $M$ be a von Neumann algebra and let $\varphi$ be a faithful normal state on $M$. Suppose that $T \colon M \to M$ is a normal unital $*$-homomorphism preserving $\varphi$, i.e. $\varphi \circ T ...
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the isomorphism of $L^\infty$ spaces

If we have an unitary operator from $L^2(\mathbb{T})$ to $L^2(X,d\mu)$ ,$\mathbb{T}$ is boundary of the unit ball ,$d\mu$is the Borel probability measure.is there an isomorphism between ...
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33 views

union of group von neumann

If we have an increasing chain of group von Neumann algebras such as $L(G_1)\subseteq L(G_2)\subseteq\ldots$ what can we say about the weak closure of their union? Is it a group von Neumann algebra? ...
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If$ p \in B(H)$ is a projection, then $r \in A'$ if and only if the closed vector subspace $p(H)$ of $H$ is invariant for $A$.

In the proof of the theorem $4.1.12$ on the page $120$ in Murphy, he uses a central remark that: If $p$ is a projection in $B(H)$ , then $p$ belong to $A'$ if and only if the closed vector subspace ...
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direct sum of a collection of Von-Neuman algebras is still Von-Neuman

If {Aα} be a collection of some Von Neuman algebras then their direct sums is still Von Neuman ? I can prove that if Aα are unital then (⊕ Aα)= (⊕ Aα)" that is because of the fact that (⊕ Aα)'= ...
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29 views

Why is a von Neumann algebra is closed with respect to weak * topology?

I was trying to prove that the identity map between a von Neumann algebra $(A,\mbox{ultra weak topology})$ with respect to ultra weak topology and the von Neumann algebra $A$ with respect to weak* ...
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40 views

Ultra weakly continuous linear functional on von neuman algebras

I started to learn von Neumann algebras and I wonder about the proof of the following. Can anyone outline it? this is theorem 4.2.10 in Murphy's book: A linear functional $\tau$ on von Neuman ...
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Ultra weakly continuous linear functional on von neuman algebras

I started to learn von Neumann algebras and I wonder about the proof of the following. Can anyone outline it? every linear functional on von neuman algebra is ultra-weakly continious iff there exist a ...
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Weak* topology on a von Neuman algebra $A$ is just the relative ultraweak topology on $A$

The following is a remark of a theorem in Murphy's C*-algebras and operator theory that every von Neuman algebra is a dual of a C*-algebra. But how does it implies that these topologies coincide in ...
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Completeness of the lattice of projectors of a von Neumann algebra

Consider a von Neumann algebra of operators $R$ in a complex generally non-separable Hilbert space $H$ and let $L\subset R$ be the lattice of orthogonal projectors included in $R$. Is $L$ complete? ...
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Construct a projection satisfying a certain property

Let $\cal G$ be a group of finite order $n$. For every prime divisor $p$ of $n$, construct a projection $P\in \cal N(G)$ such that $\operatorname{tr}_{\cal N(G)}(P)=1/p$. Here $\cal N(G)$ denotes ...
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semifinite projections

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 ...
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finite and properly infinite projections

Let $M$ be a von Neumann algebras and $p$ be a projection in $M$. $Q1:$I want to prove that there is a central projection $z \in M$ such that $pz$ is finite and $P(1-z)$ is properly infinite. $Q2:$ ...
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$\omega$ is cyclic for $M\subset B(H)$ if and only if $\omega$ is separating for $M'$

Let $H$ be a Hilbert space, $M\subset B(H)$ a von Neumann algebra and $\omega \in H$ a vector. Then $\omega$ is cyclic for $M$ if and only if $\omega$ is separating for $M'$. I proved ...
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Trace-class, Hilbert Schmidt operators, $L^p(H)$: duality theorems

Let $H$ be a Hilbert space, separable if necessary, and let $tr$ be the usual trace on $L^1(H)$. It is classical theory that $K(H)^*=L^1(H)$, and $L^1(H)^*=B(H)$, via the canonical application ...
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Relative weak-star topology on pure states

Let $A$ be a (unital) C*-algebra and consider $PS(A)$, the set of all pure states on $A$ with the relative weak-star topology. I would like to check (a weaker form of) Uryshon's lemma on $PS(A)$ in ...
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23 views

Kaplansky density theorem

Let $H$ be a Hilbert space and $A$ a C*-subalgebra of $B(H)$, and $1_H\in A$. Show that the unitaries of $A$ are strongly dense in the unitaries of $\overline{A}^{sot}$. Suppose $U(A)$ be unitaries ...
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A formula for representation

Let $A$ be a C*-algebra. Do you confirm the following discussion? Let us consider a representation $\pi:A\to B(H)$. We denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$. If we denote ...
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Minimal projections II

Let $M_1$ and $M_2$ be two W*-algebras. Let $A$ be a C*-algebra and $\pi_j:A\to M_j$ be two faithful representations with $M_j=\overline{\pi_j(A)}^{w^*}$. Assume that $$\textrm{The unit of}~ ...
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Is $B(H)$ sot separable

To prove that the unit ball of $B(H)$ is separable in strong operatior topology using the fact that $K(H)$ is separable and also is sot- dense in $B(H)$. I think we can conclude that $B(H)$ is also ...
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30 views

projection in a factor von Neumann algebra.

We know that center of a factor von Neumann algebra $\mathcal{A} $ is trivial. Let $P_1$ be a projection in $\mathcal{A} $ such that $P_1\neq I,0$ . undoubtedly there exist another projection like ...
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26 views

Von neumann contains the range projections of all of its elements

The following is a theorem of Murphy's C*-algebra and operator theory: I think it can prove easier, while I'm not sure about my proof : Let $a\in A$ be positive. Consider $C^*(a)$, and let ...
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45 views

Generator of an abelian von Neumann Algebra

I want to show that $L^{\infty}(S^1)$(where $S^1$ is equipped with its Haar Measure) as a von Neumann algebra is generated by the multiplication operators $M_{e^{in\theta}}$ where $n \in Z$ and ...
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29 views

Subordinate projections: Transitivity

Let $A$ be a $C^*$-algebra. If necessary, let us assume that $A$ is a von Neumann algebra. For projections $p,q \in A$ one writes $p \prec q$ if $p$ is Murray-von Neumann-equivalent to a subprojection ...
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1answer
20 views

Computing complex or irrational powers of the modular operator

Let $M\subset B(H)$ be a von Neumann algebra, with cyclic separating vector $\xi$. Then the modular conjugation operator $S$ is defined to be the closure of the operator $$S_{0}:M\xi\to M\xi\text{ ...
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Definition of Modular Automorphism Group

Is there a way to concisely define the modular automorphism group of a von Neumann algebra $M$ for a "lay" person? I have little more than a vague fuzzy overview of Tomita Takesaki Theory and I'm ...