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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Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
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$A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well

Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...
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Do all $x$ in Hilbert space $H$ equal $U T \eta$ for some $T \in R$ and unitary $U \in R'$, when $\eta$ is separating for $R$?

Let $H$ be a separable Hilbert space on which von Neumann algebra $R$ acts; let $\eta \in H$ be a separating vector for $R$ (i.e. the zero operator is the only $T \in R$ such that $T \eta = 0$); let $...
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Topology whose restriction to some sub-von-Neumann-algebra is its WOT?

Let $R \subset S$ be distinct von Neumann algebras having a separating vector in the separable Hilbert space $H$ on which they act. In what cases (if any) does there exist a topology $\tau$ on $S$ ...
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If $\|p-q\|<{1\over2}$ then $p$ is homotopy equivalent to $q$

Let $A$ be a $C^*$ algebra, $p,q \in A$ projections, such that $\|p-q\|< {1 \over 2}$. Show that $p$ homotopy equivalent to $q$. Proof. Let $a_t=(1-t)p+tq$, then $a_t$ is positive (self-adjoint ...
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If $\|p-q\|<1$ then they are Murray–von Neumann equivalent- proof

$\newcommand{\ran}{\operatorname{ran}}$ Let $p$, $q$ be projections in a $C^*$-algebra $A$. If $\|p−q\|<1$ then $p$ and $q$ are MvN equivalent. I'm tryind to understand the following proof: ...
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non abelian von Neumann algebras

I'm not familiar with von Neumann algebras, but I need the following fact (if it's true) for an other proof. Let $H$ be a Hilbert space, $A\subseteq L(H)$ a non abelian von Neumann algebra. Must $A$ ...
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Let $N$ be a normal operator. When $W^*(N)$ is a MASA?

I am trying to find conditions in which $W^*(N)$ is a MASA, where $N$ is a normal operator acting on some Hilbert space. I know that the multplication algebra $\{M_f| f\in L^{\infty}(X, \mu) \}$ is ...
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Von Neumann algebraic Quantum Object is direct sum of type I factors

I am looking at the non-standard quantum projective spaces $A:=\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. Now I want to show that if I take the von Neumann algebra ...
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projections in von Neuman algebra

Consider a semifinite von Neumann algebra $\mathcal{M}$ with a semifinite faithful normal trace $\tau$. If $Q, P$ are projections in $\mathcal{M}$ with $\tau(Q)< \tau(P)$, then does $\tau(P\wedge Q^...
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if $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$?

Suppose that $A_n,A$ are self-adjoint operators in $B(H)$. If $A_n$ weakly converges to $A$, does $|A_n| \rightarrow _{wo} |A|$? From Proposition. 2.3.2 of Pederson'book, I know the result holds in ...
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Trace and norm bounded sequence of positive elements has convergent subsequence in hyperfinite $II_1$ factor

Let $A$ is a hyperfinite $\operatorname{II_1}$ factor and $x_n \in A$ is some sequence of positive elements such that $||x_n||$ convergent and $\operatorname{Tr}(x_n^2) = 1$ (where $\operatorname{Tr}$ ...
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39 views

Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
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Tomita Theory: Involution

Given a Hilbert space $\mathcal{H}$. Consider a von Neumann algebra: $$M\subseteq\mathcal{B}(\mathcal{H}):\quad M=M''$$ Suppose a cyclic vector: $$\Omega\in\mathcal{H}:\quad\overline{\mathcal{M}\...
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Strong convergence of Spectral Projection

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded linear operators on $H$. Assume that $\{A_n\in B(H)\}_n$ strongly converges to $A$. $E^{|A|}(1,\infty)$ is a spectral projection of ...
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separable weakly dense subalgebra of a von Neumann algebra

This is an exercise of "A short course on spectral theory" by William Arveson. For any von Neumann algebra $\mathcal{M}$ on a Hilbert space $H$, I need to show there exists a unital $C^\ast$-...
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Characterization of noncommutative $L^2$-spaces as ordered vector spaces

If $M$ is a von Neumann algebra and $\tau\colon M_+\to[0,\infty]$ is a normal, semi-finite, faithful trace, the associated GNS Hilbert space is the completion of $\{x\in M\mid \tau(x^\ast x)<\infty\...
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predual of von Neumann algebra

I know that each separable finite von Neumann has separable predual. Can some one give me an example of a non-separable finite von Neumann algebra with separable predual?
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Automorphisms of group von Neumann algebras

I study group von Neumann algebras $L(G)$, and I extremely want to know about automorophism (groups) of these algebras. Is there any good reference about this? I appreciate of everybody help.
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Elements of $C(K)^{**}$, do they have a name?

Let $K$ be a compact (Hausdorff) space, and let $C(K)$ be the Banach algebra of contunous functions on $K$ (with the usual $\sup$-norm). The enveloping von Neumann algebra of $C(K)$ is its second dual ...
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Types of W*- algebras

Searching to find a reference for "homogeneous type $I_{{\aleph}_0}$ W*-algebra", I was not successful. Please guide me. Thanks in advance.
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What is the motivation for Murray-von Neumann equivalence

Definition: If $p,q$ are projections in a $C^*$-algebra $A$, we say that they are Murray-von Neumann equivalent, and we write $p\sim q$, if there exist $u\in A$ such that $p=u^*u$ and $q=uu^*$. I ...
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Is there an example of a non von Neumann algebra with this property?

What is an example of a $C^{*}$ subalgebra $A$ of $B(H)$ such that $A$ contains the identity $I_{H}$ and satisfies the following properties: 1) For every $T\in A$, The orthogonal projection $\...
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When set of projections is closed under multiplication

Let $\mathcal A\subset B(H)$ be an unital $C^*$ algebra of operators on a Hilbert space $H$. Let's denote by $\mathcal P$ the set of projections in $\mathcal A$, that is $\mathcal P:=\{a\in \mathcal A ...
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Is the algebra of universally integrable functions a von Neumann algebra?

I would like to continue this discussion. Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel ...
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Are Baer Rings worth studying for significant modern progress in vNa's?

I am wondering if Kaplansky's "Ring of Operators" is worth studying if I'm interested in functional analysis (more specifically von Neumann algebras). Yes any vNa is a B. ring but my question is more ...
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Do a von Neumann Algebra's subalgebras have unique “complements”?

Let $A \subset B \subset C$ be von Neumann algebras (and more specifically factors, if it helps) either all type II or all type III, acting on the same separable Hilbert space. Call $A' \subset B$ ...
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Representations of $SU_q(n)$

I am searching for a classification of all irreducibel representations of the quantum group $SU_q(n)$ for general $n$. Can someone give referenced or some statements about this? Moreover does one has ...
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Atomic W*- algebra

Let $A$ be a C*-algebra. Put $z: = \sup\{ e\in A^{**} ;\text{ e is a minimal projection}\}$. Easily can see $z$ is a central projection. Set $M:= A^{**}z$. 1) Is $M$ an atomic W*-algebra in general? ...
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Approximate unit for a certain C*-algebra

Let $A$ be a C*-algebra and $p$ a projection in $A^{**}$. To prove $p$ is the smallest unit for $B: = \{a\in A; pap=a\}$, suppose $\{u_i\}$ is an approximate unit for $B$. It's easy to see $q: = w^*-\...
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Comparing two positive linear functionals when support$(\psi)\leq$ support$(\phi)$

The following point has been addressed in the significant paper "order ideal in C*-algebras and its dual (By E. Effros - Lemma 4.1). The reference is Dixmier's book (1957 French). Unfortunately, I ...
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Atomic W*-algebras

I am looking for any information concerning atomic W*-algebras. Def. A W*-algebra $M$ is called atomic if for any projection $p$ in $M$ there is a minimal projection $e\in M$ with $e\leq p$. Q. ...
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A certain *-isomorphism

let $A$ be a C*-algebra and $z\in A^{**}$ the supremum of all the minimal projections in $A^{**}$. How can show $* -$ homomorphism $A\to zA\subset zA^{**}$ is injective?
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Two approaches on the support of functionals

I am writing concerning support of functionals on C*-algebras. I feel there are two different definitions for this notion. Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. ...
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A review of an example given by C. A. Akemann

I am witting concerning the example II.6 given in the paper "The General Stone-Weierstrass Problem" by C. A. Akemann: Let me review this example. Let $H_i$ be the two dimensional Hilbert space $\...
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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 $\tau \leq 1$ $e_ie_j=...
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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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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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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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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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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 $M^{**}...
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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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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 ‎...