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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A remark on projections in von Neumann algebras

Let $M$ be a von Neumann algebra and $e,f$ be projections in $M$. For a given central projection $z\in M$, is the following true? $$z(e\vee f)=ze\vee zf$$
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Jones construction projections

Let M be a von Neumann algebra with faithful normal normalized trace tr. Let $\{ e_i | i=1,2,\dots \}$ be projections in M such that: $e_ie_{i \pm 1}e_i=\tau e_i $ for some $\tau \leq 1$ ...
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The left kernel of a positive linear functional and its $w^*$-extention

Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. We let $\tilde{\phi}$ be its unique $w^*$-continuous extension on $A^{**}$. It is supposed to focus on the left kernel of ...
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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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38 views

The point-wise closure of the space of continuous functions

Let $X$ be a locally compact space and consider $C_0(X)$. We denote $b(X)$, by the set of all bounded functions on $X$. It is easy to be checked that $b(X)$ may be considered as a C*-sub algebra of ...
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Multiplicity free representation contain irreducible representation (for type I representation)?

While looking at Arveson's "An invitation to C* algebras", at the moment of defining type I representations (p. 47), he says that a (non degenerate) representation is type I if every central ...
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Which open projections are support of closed left ideals in a C*-algebra

Let $A$ be a C*-algebra. Let $\phi$ be a positive linear functional on $A$. The support of $\phi$ is defined by the smallest projection $q\in A^{**}$ with $\phi(q)=||\phi||$ and in the literature ...
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28 views

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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25 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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20 views

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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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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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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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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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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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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Understanding minimal projections

This might be very easy but it is not quite clear for me. Detailed explanation appreciated! I went through the commutative case but beyond that I lack intuition. Let $A$ be a C*-algebra and let ...
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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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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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Minimal projections vs maximal left ideals

I've seen in some papers a statement (which is referred to a very old book of Dixmier in French which I have no access to / can't read anyway) saying that maximal left ideals of a (unital) C*-algebra ...
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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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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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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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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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Is an ultrapower of the hyperfinite factor still hyperfinite?

Let $\mathcal{R}$ be the hyperfinite type $II_{1}$ factor and let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$. Is it true that $\mathcal{R}^{\mathcal{U}}$ is never hyperfinite ? How can I see ...
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Ultraweak continuity of power maps on $W^*$-algebras

Let $\mathcal{A}$ be a $W^*$-algebra. Is the map $a \mapsto a^2$, or more generally the map $a \mapsto a^k$, ultraweakly continuous? (Of course, products are not jointly ultraweakly continuous in ...
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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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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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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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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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29 views

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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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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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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21 views

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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36 views

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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26 views

$\omega$ 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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35 views

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 ...