# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...