# 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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### When is the closable composition of operators affiliated to a von Neumann algebra also an affiliated operator?

Let $S$ and $T$ be closed (unbounded) operators affiliated to a von Neumann algebra $M$. Can we say anything about when the closure of their composition $T \circ S$ is also affiliated to $M$ (assuming ...
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### Center of reduced von Neumann algebra

Consider the following fragment from the book "Lectures on von Neumann algebras" by Stratila and Zsido: Later, there is the following claim: Here, $z(e)$ is the central support of $e$, i.e....
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### Direct sum and isomorphism

Let $\mathcal{A}\subseteq \mathcal{B}(H)$ be a von Neumann algebra, $\,H$ be a separable Hilbert space and let $\xi\in H,\,\xi\neq 0$ such that $\overline{\mathcal{A}\xi}=H.$ We consider the ...
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### Let $B$ be a von Neumann subalgebra of $A$. Is it true that $L^2(B)\subseteq L^2(A)$?

Let $A\subseteq B$ be a unital normal inclusion of von Neumann algebras. Is it true that this induces an isometry of the standard Hilbert spaces $$L^2(A)\to L^2(B)?$$ I'm not sure if I can expect this ...
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### Question about the support of a normal weight on a von Neumann algebra

Consider the following fragment from Stratila's book "Modular theory in operator algebras": I understand everything in this fragment, except that equation $(3)$ holds. How can one show this ...
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### $s(e+f) = e\lor f$ for projections $p,q\in B(H)$

Let $H$ be a Hilbert space and $p,q\in B(H)$ orthogonal projections, i.e. $$p=p^*= p^2, \quad q=q^* = q^2.$$ I want to show that $s(p+q) = p\lor q$ where $s(p+q)$ is the support projection of $p+q$ ...
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### Doubt on exercise on Von Neumann factor

I am trying to solve the last point of the following exercise, but I don't know how to approach it.The exercise says: Let $M \subseteq B(\mathbb{H})$ be a type $\mathbb{II}_1$ factor Von Neumann ...
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### Doubt on exercise on finite Von Neumann Algebras

I am doing the following exercise: Let $M \subseteq B(\mathbb{H})$ be a von Neumann algebra, $\tau$ be a faithful, tracial, normal state $\tau: M \to \mathbb{C}$. Prove that M is finite. What happens ...
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### double commutant of unitary operators of a Von Neumann algebra.

Let $M$ be a Von Neumann algebra. I am having trouble showing that the double commutant of unitary operators for $M$ is equal to $M$. I have shown it with projections using borel functioncal calculus. ...
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### The GNS represenation for a Von neumann algebra is a Von neumann algebra.

Let $M$ be a Von Neumann algebra and $\omega$ be a faithful $\sigma$-weakly continuous positive tracial state on $M$. I know that we have an inner product space on $M$ because $\omega$ is faithful. ...
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### Multiplication of two unbounded operators and functional calculus

Let $A$ be a positive, self-adjoint unbounded operator defined on a Hilbert space $H$. Let $f,g: [0, \infty]\to \mathbb{R}$ be Borel measurable functions that are bounded on compact subsets. We can ...
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### relative commutant of von Neumann algebra

Let $M$ be a von Neumann algebra. If $N_1$ and $N_2$ are two subalgebras of $M$ such that $N_1$ is $*$-isomorphic to $N_2$. What is the relationship between $A_1:=N_1'\cap M$ and $A_2:=N_2'\cap M$? ...
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### Showing the double commutant of the image of GNS representation is a factor provided uniqueness of a tracial state.

Let $A$ be a $C^*$-algebra with an identity. If $A$ has a unique tracial state $\varphi:A \rightarrow \mathbb{C}$ i.e. $\varphi(ab) = \varphi(ba)$, $\varphi(x^*x) \geq 0$, and $\varphi(1) = 1$. I am ...
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### types of von Neumann subalgebras [closed]

Let $N$ be a von Neumann subalgebra of the von Neumann algebra $M$. Is the type of $N$ at most the type of $M$?
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### Showing that a von neumann algebra in the bounded operators of $l^2(S_\infty)$ is a factor

Consider $S_\infty$ i.e. the group of permutations with functions $\sigma:\mathbb{N} \rightarrow \mathbb{N}$ such that $\sigma(n) = n$ for all but finitely many. The left regular represenation defined ...
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### relative commutant of type III1 factor [closed]

Let $M$ be a type III$_1$ factor and $N$ be the non-trivial ($N\neq \Bbb C 1$)semi-finite von Neumann subalgebra of $M$. What is the type of relative commutant $N'\cap M$? Is it semi-finite?
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### Why does a projector being equal to its own central carrier imply that it is in the center of the commutant of its algebra?

In a passage in a proof, there is the following sequence of implications which I didn't quite understand: "The projector P in a representation $\pi$ of a Banach $*$-algebra $\mathscr{A}$, is its ...
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### If two representations have the scalars a its commutants, why does it imply that they are either unitarily equivalent or disjoint?

In Kadison's article he says that: "Since the commutant of an irreducible representation consists of scalars, two such are either unitarily equivalent or disjoint." Ref:”https://msp.org/pjm/...
### Why is the commutant of a unital $*$-representation a von Neumann algebra?
If $\pi$ is a $*$-representation (i.e. a $*$-homomorphism) of a unital Banach$*$-algebra, then $\pi(\mathscr{A})$, is trivially self-adjoint, but why is $\pi(\mathscr{A})'$ a von Neumann algebra? This ...