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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Existence of a projection in a strongly closed hereditary $C^*$-algebra of a von Neuman algebra

Let $A$ be a von Neumann algebra and $B$ is a strongly closed hereditary $C^*$-subalgebra of $A$. Prove that, there exists a unique projection $p$ in $B$ such that $B=pAp.$ $\underline{\text{...
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Confused on some parts in the proof of $p \in A'' \implies (A')_p=(A_p)'$

I was reading C-star Algebra and operator theory by Murphy and I stuck at the proof of the lemma $4.1.6$. I am attaching below: In the proof of the Lemma $4.1.6$ Murphy want to show that if $p \in A''...
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Decompose a positive contraction in a continuous masa in $L(H)$

Let $A$ be a continuous masa in $L(H)$ and $T$ be a positive contraction in $A$. Then we can assume that $0<\|Th\|<1$ for all unit vectors $h\in H$. Otherwise decompose $T$ as $P+T_0$ for some ...
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Commutant of corner of $*-$algebra on a hilbert space is corner of commutant

Let $A$ be a $*-$algebra on a Hilbert space $H$ and $p$ be a projection in $A'$, where $A'$ is the commutant of $A$, that is, $$A':=\{u \in B(H): ua=au~\text{ for all }~a \in A\}.$$ If also $p \in A''$...
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$\bigoplus_\lambda A_\lambda$ is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$.

Let $(H_{\lambda})_{\lambda\in \Lambda}$ is a family of Hilbert spaces and $A_{\lambda}$ is a von Neumann algebra on on $H_\lambda$ for each index $\lambda$. Then prove that the direct sum $\bigoplus_\...
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conditional expectation onto discrete MASA

In the above screnshot, $X$ is a MASA in $L(H)$ and $Z\in L(H)$. I tried to prove (3.1) and met with a question. When we take $Z =\begin{pmatrix}A & B\\C & D\end{pmatrix} $, where $A\in L(H_d),...
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Understanding Uhlmann Monotonicity Theorem on von Neumann Algebras

This is my first post, so apologies if this is a bad post. I'm reading "Quantum Entropy and its use' by M. Ohya and D. Petz. Theorem 5.3 states Let $M_1$ and $M_2$ be von Neumann algebras with ...
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Relatinships between Non-atomic von Neumann algebras and continuous von Neumann algebras

Does there exist relationships between non-atomic von Neumann algebras and continuous von Neumann algebras? A conditional expectation onto a continuous masa vansihes on compact operators. I found ...
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Von Neumann algebras are generated by its projections

In a lecture about operator theory we used the claim, that the set of projections in a von Neumann algebra $\mathcal M$ is dense in $\mathcal M$, with respect to the operator norm. Sadly that claim ...
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On double dual of C* algebra

Can anyone provide examples of double dual of $C^{*}$-algebras except $K(\mathcal{H})$, commutative cases? Thanks in advance!
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Normal states on a 2 by 2 complex matrix

Denote $S$ by the set of normal states on $M_2(\Bbb C)$. Suppose $p(p\neq 0,1)$ is a projection in $M_2(\Bbb C)$ and $\rho\in S$. Define $S_p:=\{w\in S:w(p)=0\}$ and $d(\rho,S_p):=\inf_{\omega\in ...
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Two von Neumann algebras not isomorphic as C*-algebras, can they be isomorphic as von Neumann algebras? [duplicate]

This might be a silly question. Consider two von Neumann algebras $M, N$, given that they are not isomorphic as C*-algebras to each other, is there a chance that they are isomorphic as von Neumann ...
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Will the continuity of functional calculus in another aspect still be ture?

It is a well known fact that if we have a fixed normal operator $T$ on $B(H)$, $f$ and $g$ are both continuous on $\sigma(T)$, then $||(f-g)(T)||=||f-g||_{\sigma(T)}$. It tells us that functional ...
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A question about the quotient space of von Neumann algebras

It is well known that in $C^*$ algebra category, If $\mathcal{A}$ is a $C^*$ algebra and $\mathcal{I}$ is a norm closed ideal of $\mathcal{A}$, then $\mathcal{I}$ is also a $C^*$ algebra, and $\...
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Extension of slice map to WOT closure

Given Hilbert spaces $H_1,H_2$ and a functional in the predual $\psi\in B(H_1)_*$ we may consider the slice map $S:B(H_1)\otimes B(H_2)\to B(H_2)$ defined on the spatial tensor product given by ...
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Compression map is an isomorphism from $pB(H)p$ to $B(K)$ via $u \to u_K$

I was reading a note on Von Neumann Algebra, and I am not able to understand this phrase as: Let $K$ be a closed vector subspace of a Hilbert space $H$ and let $p$ be the projection of $H$ onto $K$. ...
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Does any von Neumann algebra have $\sigma$-finite projections?

Let $M$ be a von Neumann algebra. Let $\Sigma$ be the set of $\sigma$-finite projections of $M$. In Takesaki's book "Theory of operator algebras II", chapter 7, p51, in the proof of theorem ...
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Convergence of vector states

Let $\mathfrak{A} \subset B(H)$ be a C*-algebra. For $x \in H$, $\|x\|=1$, define a vector state $\omega_x$ on $\mathfrak{A}$ by $\mathfrak{A} \ni A \mapsto \langle x, Ax \rangle$. Assume $(x_n)_{n\in ...
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When is a group von Neumann algebra a factor?

It is well-known that a von Neumann algebra (on a separable Hilbert space) can be written as a direct integrals of factors, i.e., von Neumann algebras with center $\mathbb C I$. As such, factors play ...
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Takesaki volume 2 lemma 1.9

Consider the following fragment from Takesaki's book "Theory of operator algebras II" chapter VI "Left Hilbert algebras" (p6): I am trying to understand how lemma 1.9 implies that ...
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Compactness of group of unitaries that commute with density operator

I study the paper "Doplicher and Longo: Standard and split inclusions of von Neumann algebras" and have a question about the proof of Lemma 3.2. Let $\omega$ be a faithful normal state on a ...
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SOT coincides with $||\cdot||_2$-topology on unit ball $L^{\infty}(X,\mu)$

Let $(X,\mu)$ be a probability space and $A=L^{\infty}(X,\mu)$. We view $A$ as a subspace of $L^2(X,\mu)$ and as a von Neumann algebra on $L^2(X,\mu)$. Show that on the unit ball $(A)_1$ of the von ...
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Showing that $B_0(H)^{**}= B(H)$

Let $H$ be a Hilbert space. Consider the $C^*$-algebra of compact operators $B_0(H)$ and its double enveloping von Neumann algebra $B_0(H)^{**}$. I want to show that $B_0(H)^{**}= B(H)$ as $C^*$-...
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Is every projection the support of a normal state?

Let $M$ be a von Neumann algebra and $p \in M$ a projection. Does there exist a normal state $\varphi \in M_*$ such that $s(\varphi)=p$? Here $s(\varphi)$ is the smallest projection $e$ with $\varphi =...
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When are the limits of (reduced) crossed product C*-algebras equal to (reduced) crossed product of limits?

Let $T$ be a directed set and let $\{G_t\}$ be a collection of discrete abelian groups and let $\{A_t\}$ be a collection of unital C*-algebras. Suppose we have some suitable action $\alpha_t$ for each ...
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Prove these two projections are equivalent

$\pmb Problem$: Let $e_1, e_2, f_1$ and $f_2$ be projections of $M$ which is a Von Neumann algebra, such that $e_1e_2=f_1f_2=0$. If $e_1+e_2=f_1+f_2$, $e_1 \sim e_2$ and $f_1 \sim f_2$, prove that $...
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$\sigma$-finite projection von Neumann algebra

Consider the following lemma from Takesaki's book "Theory of operator algebra II": Questions: (1) What is a $\sigma$-finite projection? I don't see a definition anywhere in either volume I ...
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Equivalence of projections in a Von Neumann algebra factor

$\pmb Problem:$ Let $M$ be a (properly) infinite factor and $e$ and $f$ be projections of $M$. Then $$ e\lor f \sim 1 \iff e\sim1\ \text{ or } f\sim1 .$$ $\pmb Idea:$ As $M$ is a factor, its center is ...
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Almost periodicity and approximation in tracial von Neumann algebra.

Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. Let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action of on $N$ which preserves the tracial state $\tau$ (i.e. $\...
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Polar decomposition of a functional in $C(\Gamma)?$

I'm reading Takesaki's first volume and in the proof of IV.7.3 (chapter 4, proposition 7.3) the following situation happens: $\Gamma$ is a compact Hausdorff space, and $\mu\in M(\Gamma)$ a complex ...
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If $A$ is a $W^*$-algebra, how to construct a nice predual for $M_n(A)$?

Let $A$ be a $W^*$-algebra. By concretely representing $A \subseteq B(H)$ as a von Neumann algebra and using that $M_n(A) \subseteq B(H^{n})$, one can check that $M_n(A)$ is again a $W^*$-algebra such ...
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4 votes
3 answers
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If $\varphi: A \to B$ is surjective, then is $\varphi^{**}: A^{**}\to B^{**}$ surjective?

Let $A$ and $B$ be $C^*$-algebras and consider a $*$-homomorphism $\varphi: A \to B$. Then the biduals $A^{**}$ and $B^{**}$ carry natural $C^*$-structures (coming from the enveloping von Neumann ...
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Hilbert algebra definition clarification

Consider the following fragment from Takesaki's second volume: In (c), we read that the involution $\sharp: \mathfrak{A}\to \mathfrak{A}$ is preclosed. What does this mean?
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Is a positive map $\varphi$ between von Neumann algebras normal if $x_i \nearrow x$ implies $\varphi(x_i)\nearrow \varphi(x).$

Let $\varphi: M \to N$ be a positive map between von Neumann algebras. Assume that for any increasing net of positive elements $\{x_i\}_{i \in I}$ we have $$\varphi\left(\sup_{i \in I} x_i\right) = \...
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2 votes
2 answers
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Perturbing a partial isometry to part of a unitary

In Narutaka Ozawa's solution over at Mathoverflow, the following result is implicitly used: Let $M\subseteq B (H)$ be a von Neumann algebra, and let $\xi\in H$. If $u\in M$ is a partial isometry, ...
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2 votes
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Takesaki II: lemma 1.2

Consider the following fragment from Takesaki's book "Theory of operator algebra II": Questions: (1) How does the proof show that every element of $\mathfrak{m}$ is a linear combination of ...
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Hahn-Banach extension of positive functional is positive

Consider the following lemma from Takesaki's book "Theory of operator algebra I": Why is the sentence "Any Hahn-Banach extension of a positive linear functional on a $C^*$-subalgebra ...
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$\omega_1(p) \le \omega_2(p)$ for all projections $p$ implies $\omega_1 \le \omega_2$.

Let $M\subseteq B(H)$ be a von Neumann algebra. Let $\omega_1, \omega_2$ be functionals on $M$ satisfying $$\omega_1(p) \le \omega_2(p) \quad (*)$$ for all projections $p \in M$. Is it true that $\...
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3 votes
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The annihilators in dual operator algebras.

It is well known that in every von Neumann algebra $\mathcal{M}$, the left annihilator of a given subset $S\subseteq \mathcal{M}$ is in the form $\mathcal{M}p$ for some projection $p$ in $\mathcal{M}...
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3 votes
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Is $\|\varphi\| = \|\varphi p\| + \|\varphi(1-p)\|$?.

Let $M$ be a von Neumann algebra and $p \in M$ a central projection. Let $\varphi \in M$ be a normal functional and define new normal functionals $\varphi p $ and $\varphi(1-p)$ by $$(\varphi p)(m) = \...
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Spectral value of reduced von Neumann algebra

Let $M$ a von Neumann algebra and $x$ positive and $a$ a projector, $N_x$ is the von Neumann algebra generated by $x$. Do you know if $min Spec_{aN_xa}(axa)\in Spec_M(x)$ please? [Attempt added from ...
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3 votes
2 answers
147 views

Takesaki lemma 4.5

Consider the following fragment from Takesaki's book "Theory of operator algebra I" (p82 and previous pages): The notation $\mathscr{L}_G$ means all normal operators with spectrum contained ...
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Takesaki corollary 3.6

Consider the following fragment from Takesaki's book "Theory of operator algebra I": I can't see why the marked equality is true. In particular, I don't see why $\pi(\mathscr{M}'')\subseteq \...
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Minimum of Spectrum

Let $M$ be a von Neumann algebra and $e$ a projector. Is it true that $min Spec_M(x)\leq min Spec_{eMe}(exe)$ for $x$ positive in $M$? Thank you.
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1 vote
1 answer
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Spectrum of reduced von Neumann algebra

Let $M$ be a von Neumann algebra and $e$ a projector. Do you know if $Spec_{eMe}(exe)\subset Spec_M(x)$ for $x\in M$? Thank you
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6 votes
1 answer
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What exactly is "halving" a projection in a von Neumann algebra?

In Kadison-Ringrose Vol II (Lemma 6.3.3) the authors introduce the concept of "halving" a projection inside a von Neumann algebra. What I understood is that: "halving" a projection ...
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2 votes
1 answer
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Proposition 1.4 in Takesaki's book "Theory of operator algebra": about the spectral integral

Consider the following fragment from Takesaki's book "Theory of operator algebra I": Can someone explain why the restriction $x\vert_{e_n(\mathscr{H})}$ is invertible? Intuitively, I would ...
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2 votes
1 answer
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When is the sum of two partial isometries again a isometry?

Suppose $V$ and $W$ be two partial isometries in a von Neumann algebra $\mathscr{R}$ such that $V^*V = E_1;~VV^*=F_1$ and $W^*W=E_2; ~ WW^*=F_2$. I am thinking about When is $V + W$ a partial ...
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0 votes
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If $V$ is a partial isometry with initial projection $E$ then is $EVE=V$?

Let $\mathscr{R}$ be a finite von Neumann algebra and for an $n\in \Bbb{N}$ consider the von Neumann algebra $M_n(\mathscr{R})$. If $E_1$ denotes the projection (in $M_n(\mathscr{R})$) whose $(1, 1)$-...
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2 votes
1 answer
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Is every normal state on $A''$ a vector state?

Perhaps this is a simple fact that I'm missing. In a paper I'm reading ("Nuclear C*-Algebras and the Approximation Property" by M. D. Choi and E. G. Effros) they claim that if $A$ is a C*-...
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