Questions tagged [von-neumann-algebras]

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, non-commutative geometry, and quantum field theory.

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conditional expectations on $C^*$-algebras

I am trying to get a feel for conditional expectations on arbitrary $C^*$-algebras but I am not able to find many examples. Obviously I can find conditional expectations from an $C^*$-algebra into $\...
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If $B\subset B(H)$ is a C*-subalgebra and $T\colon B''\to B''$ is linear, bounded and weakly continuous, then $\|T\|=\|T|_{B}\|$

Let $H$ be a Hilbert space and let $B\subset B(H)$ be a C*-subalgebra. Suppose that $T\colon M\to M$ is linear, bounded and operator-weakly continuous, then I want to prove that $\|T\|=\|T|_{B}\|$. ...
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If $(H,\pi)$ is a finite dimensional irreducible representation of a C*-algebra $A$, then $\pi(A)=B(H)$

Let $A$ be a C*-algebra and $(H,\pi)$ an irreducible representation such that $\dim(H)<\infty$. Then how do I show that $\pi(A)=B(H)$? Here is what I tried: Since $(H,\pi)$ is irreducible, we ...
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On Direct integral decomposition of von Neumann algebras

I have a question. We know by theory that any von Neumann algebra is direct integral of factors. Then how to get the decomposition in practical situation. Basically what is the decomposition examples ...
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Commuting elements from Enveloping von Neumann algebra.

Let $A$ be a $C^*$-algebra. We can regard its double dual as the universal enveloping von Neumann algebra of $A$ as follows: $A$ sits inside of $A^{**}$ via $i: A \hookrightarrow A^{**}$, where $i(a)(\...
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What are some examples of C*-algebras that do not admit a bounded tracial linear functional?

Realizing a C*-algebra, say $A$, as a subalgebra of some $B(H)$ we could equip it with the trace it inherits from $B(H)$. But this is unbounded and not even defined on most elements. To my knowledge (...
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Conditional expectations on von Neumann algebras-change of state

Let $M$ be a finite von Neumann algebra equipped with a faithful finite normal trace $\tau$. Let $N$ be a von Neumann sub-algebra of $M$. Let $E_\tau: M \to M$ be the faithful normal conditional ...
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Relation of spectral multiplicity and commutant

As we can see for, self adjoint operator $x$ in discrete spectrum if eigenvalues repeat $m$ times then a copy of $M_{m}(\mathbb{C})$ sits inside commutant of the vN algebra generated by $x$. So my ...
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Non-triviality of “Weak closures of $*$-subalgebras are von Neumann algebras”

I suspect there is a slight error in Murphy's C*-algebras and Operator Theory: Murphy defines a von Neumann algebra on a Hilbert space $H$ as a $*$-subalgebra of $B(H)$ that is strongly closed. I ...
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When are crossed product von Neumann algebras injective?

Let $\alpha$ be a continuous action of a discrete group $\Gamma$ on a von Neumann algebra $\mathcal{M}$. We can build the corresponding crossed product von Neumann algebra $\mathcal{N}:=\mathcal{M} \...
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$C^\ast$-algebra theory associates a compact Hausdorff space to every measure space. What is this compact Hausdorff space?

Commutative von Neumann algebras are "the same" as measure spaces, and unital commutative $C^\ast$-algebras are the same as compact Hausdorff spaces. Moreover, every commutative von Neumann algebra is ...
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Isomorphisms of bimodules associated to group representations

Let $G$ be a discrete group and $M = L G \subset B(\ell^2 G)$ its left regular von Neumann algebra. According to [Section 13.1.3, AP] it is possible to associate to any given (unitary) group ...
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On one property of infimum of two projections

Suppose $M$ is von Neumann algebra and $p,q\in M$ are projections. One can show that $r=p\wedge q + p\wedge (1-q)$ is also projection, where $p\wedge q$ means the $\inf\{p,q\}$. Indeed, $ (p\wedge q)\...
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Visualizing projections in type $II_1$ AFD von Neumann algebras

I'm having a lot of trouble to picture the elements of the AFD (hyperfinite) $II_1$ von Neumann algebra. I would like to see concrete examples of operators and projections belonging to the hyperfinite ...
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Von Neumann algebra is generated by its projections

I know von Neumann algebras are generated by their projections. The proofs I've seen use spectral measures (this is the case in Murphy's and Conway's texts). I was wondering if there's a more ...
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Von Neumann Correspondence

In Popa's preprint https://www.math.ucla.edu/~popa/popa-correspondences.pdf his initial definition of an $N-M$ correspondence between two von Neumann algebras is a Hilbert space $\mathcal{H}$ and a ...
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Left Ideals in Finite Dimensional $\mathrm{C}^\ast$-Algebras

Let $A$ be a finite dimensional $\mathrm{C}^\ast$-algebra. Let $I$ be a left ideal in $A$. I believe there is a projection $p\in A$ (an element such that $p=p^2=p^*$) such that: $$I=Ap.\qquad(1)$$ ...
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KMS States on a $C^*$- dynamical system

Let $\omega$ be a $(\tau, \beta)$- KMS state of the $C^*$- dynamical system $(\mathcal{U}, \tau)$ with $\beta \in \mathbb{R}-\{0\}$. What is mean by the normal extension of $\omega$ to the weak ...
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Question on the commutant of a von Neumann algebra

Let $\mathfrak{R}$ be a von Neumann algebra acting on a complex Hilbert space $\mathcal{H}$. A projection $P\in\mathfrak{R}$ is said to be cyclic if there exists $x\in\mathcal{H}$ such that $\text{Ran}...
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$\sigma-$ weakly continuous functionals on Von Neumann Algebra.

Why and in which sense a Von Neumann algebra $\cal{M}$ can be considered as the dual space of $\sigma-$ weakly continuous linear functionals on $\cal{M}$?
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Cone of positive Hilbert-Schmidt operators

Let $H,K$ be complex Hilbert spaces and $\operatorname{HS}(H),\operatorname{HS}(K)$ the spaces of Hilbert-Schmidt operators. I will identify $\operatorname{HS}(H)$ with $H\otimes \overline{H}$ (where $...
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Atomic von Neumann algebras

Let $M$ is atomic $\sigma-$finite von Neumann algebra. Atomic means that every projection has minimal subprojection. $\sigma-$finite means that the cardinality of any set consisting of mutually ...
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Why is a completely additive isomorphism of Von-Neumann Algebras ultraweakly continuous?

Suppose $M$ and $N$ are Von-Neumann algebras on say $B(H)$ and $B(K)$ and $\theta: M \rightarrow N$ is an isomorphism if $M$ and $N$ are considered $C^*$-algebras. If also $\theta(\sum\limits_{i\in I}...
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Equality of sum and direct sum of projection operators

I am reading the book "An introduction to Operator Algebras" by Kehe Zhu and I am confused by the proof of the following lemma: $\textbf{Lemma 26.2}$ Let $\mathcal A$ be a von Neumann Algebra. The ...
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Morita-invertible von Neumann algebras

I am familiar with the Morita theory of rings, and the hermitian Morita theory of rings with involution, and I am trying to understand some parallels and differences with the Morita theory of von ...
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How to see a matrix algebra is the closed linear span of its projections

A von Neumann algebra is the norm closed linear span of its projections. If we restrict to the case of matrix algebras which are themselves von Neumann algebras why is this the case? I think it should ...
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Von Neumann Bicomutant theorem detail

at some point in the proof of the thoerem of double commutant of Von Neumann, you have to show that $$\pi(\mathcal{M})'' = \pi(\mathcal{M}'') $$ I was looking the proof on Theory of Operator ...
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weak* topology is just relative $\sigma$_weak topology in von Neumann algebras

I know that if $A$ be a von Neumann algebra on a Hilbert space $H$ then $A$ is isometric linear isomorphism with $(\frac{L^1(H)}{A\bot})^*$ where $A\bot = \{ v \in L^1(H) : \operatorname{tr}(uv) = 0 (...
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Properties of image of a projection under *-homomorphism into Tensor Product

Let $A$ be a finite dimensional $\mathrm{C}^*$-algebra. Suppose that $q$ is a projection and $$T:A\rightarrow A\otimes A$$ is a *-homomorphism. Suppose we write: $$T(q)=\sum_{j=1}^n q_j\otimes p_j.$...
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Linear Combination of Linearly Independent Small Projections

The conditions on the $p_j$ and $q_j$ that I believe hold do not in fact hold. See the edit for the linked question showing this. Let $A$ be a finite dimensional $\mathrm{C}^*$-algebra. Suppose that ...
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On conditional expectation on vN subalgebras

Let $M$ be tracial vN algebra with trace $\tau$, $E$ be a conditional expectationonto a subalgebra $B$ preserving the trace $\tau$, if a unitary $u$ in $M$, is it true $E(u)$ is again unitary in some ...
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Abelian/Finite Projections in Von Neumann Algebras

I know that provided a von Neumann Algebra acting on Hilbert $(\mathcal{M},\mathcal{H})$ a projection $e \in P(\mathcal{M})$ is said to be abelian if $e\mathcal{M} e$ is $\textbf{abelian}$ and the ...
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On Hyperfinite ness

If $M\subset B(\mathcal{H})$ is hyperfinite type $\mathrm{II}_1$ factor, does it imply $PM$ is again hyperfinite type $\mathrm{II}_1$, where $P$ is a projection in $M'$.
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Isomorphisms of reduced group $C^*$-algebras with unique trace lift to the group von Neumann algebras.

Anyone have a nice proof or reference of the following fact. Suppose, G,H are infinite discrete groups and $\theta: C^*_r(G) \to C^*_r(H)$ is a $*$-isomorphism and $C^*_r(G)$ has a unique trace then $\...
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On kernel of an unbounded operator

If spectral measure of $A$ is nonatomic, for which class of measurable (with respect to spectral measure) functions we have ker$f(A)=0.$
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Resolution of identity of self adjoint belongs to von Neumann Algebra

If $x \in M \subset B(H)$ where $M$ is a Von Neumann Algebra and \begin{equation} x = \int \lambda \, dE(\lambda) \end{equation} Is selfadjoint, how can I show that the resolution of Identity \begin{...
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On conditional expectation of tracial C star algebra

Let $A$ be a C*-algebra with a trace $\tau$. $B$ be a C*-subalgebra of $A$. Do always exist trace preserving conditional expectation exist like in von Neumann algebra?
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In a von Neumann factor, trace of nonzero projection is strictly positive

I'm following Popa and Anantharaman's "An introduction to $II_1$ factors" and in Proposition 4.1.3 they prove that: A von Neumann factor $M$ has at most one tracial state, and then such a tracial ...
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Comparison of projections in $B(H)$

Suppose $P$, $Q$ are two non-trivial projections in $B(H)$, can we deduce that $P\leq Q$ or $Q\leq P$?
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An injective $C^*$-algebra is generated by its projections

Let $A$ be a unital, injective $C^*$-algebra. Recall that $A$ is called injective if whenever $S\subseteq T$ is an embedding of operator systems, and $\phi: S\to A$ is a unital completely positive ...
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Let $\tau,\phi:A \rightarrow \mathbb{C}$ be faithful states on a C*-algebra A, with $\tau \geqslant \phi$. How do the GNS representations compare?

Let $\tau,\phi:A \rightarrow \mathbb{C}$ be faithful states on a C* algebra A, with $\tau \geqslant \phi$. Write $H_{\tau}, H_{\phi}$ for the GNS representation with respect to $\tau,\phi$ ...
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Image of a normal $*$-homomorphism

Let $\mathcal M$ be a von Neumann algebra. Let $\pi:\mathcal M\to\mathcal M$ be a normal $*$-homomorphism Is $\pi(\mathcal M)$ again a von Neuman algebra? By [J. Dixmier, Les algebres d’operateurs ...
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Definition of von Neumann subalgebra

Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra. Then, what should be natural definition of a von Neumann subalgebra?
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equivalent subprojection in $\Bbb M_m(A)$ [closed]

Let $A$ be a $C^*$ algebra,$p$ is a projection in $\Bbb M_n(A)$,$q$ is a projection in $\Bbb M_m(A)$.If there exists a projection $p_0$ in $\Bbb M_k(A)$ such that $q$ is equivalent to $p\oplus p_0$,...
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Is every ideal of a unit regular ring unit regular?

A ring $R$ with unity $1_R$ is unit regular if for any $a\in R$, $a=aua$ for some unit element $u\in R$. An ideal $I$ of a ring $R$ is a unit regular ideal if for any $x\in I$, there exists $u\in R$ ...
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the $K_0$ group of von Neumann algebra factor of type $II_{\infty}$ and type $III$

Suppose $M$ is a is a von Neumann algebra factor of type II$_{\infty}$, and $N$ is a is a von Neumann algebra factor of type III. I have no idea how to prove that $K_0(M)=K_0(N)=0$. What are the ...
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matrix algebras of von Neumann algebra factor of type $II1$

Suppose $M$ is a von Neumann algebra factor of type $II_1$ ,Is $\Bbb M_n(M),n\in \Bbb N$ also a von Neumann algebra factor of type $II_1$ ?
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the range of trace on projections of $II_1$ factor

Suppose $M$ is a von Neumann algebra factor of type $II_1$ factor,let $P(M)$ be the set of projections in $M$,how to prove that $\{tr (p),p\in P(M)\}=[0,1]$,where tr is a tracial state on $M$?
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Projection operators and Analytic Elements

Consider the following element: $P_{n}=\sqrt{\frac{n}{π}}\int_{-\infty}^{+\infty}e^{-nt^{2}}τ_{t}(P)dt.$ where $P$ is a projection and $τ_{t}(P)=U(t)PU(t)^*$ for some group of unitaries in a von ...
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Analytic Elements for a group of Isometries in a von Neumann algebra

Why can an analytic element with respect to translations along a timelike direction not belong to a local algebra in a Haag Araki theory? I know the claim is true, but is there any concise ...

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