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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, ...

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Projections correspond to double duals of $C(X)$-algebras fibers

Let $A$ be a $C(X)$-algebra ($X$ compact). For $x\neq y$ in $X$, we have the Glimm ideals in $A$: $I=C_0(X\setminus \{x\})A$ and $J=C_0(X\setminus \{y\})A$. The fibers are denoted by $A_x=A/I$ and $...
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$p$ and $q$ orthogonal projections in $B(H)$, then the orthogonal projection onto the intersection space is the strong operator limit of $(pq)^n$ [duplicate]

$p$ and $q$ orthogonal projections why is it true that the orthogonal projection onto $p(H) \cap q(H)$ is the strong operator limit of $(pq)^n$ ?
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nuclear $C^*$ algebra

If $(A_i)$ is a sequence of nuclear $C^*$ algebras,Is $\oplus_{c_0}A_i$ ($c_0$ direct sum)and $\prod A_i$($\ell ^\infty $ direct sum) also nuclear?
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functional calculus under the $*$ homomorphism

If $A,B$ are two $C^*$ algebras,$\psi:A \rightarrow B$ is a non $*$ homomorphism.Suppose $b$ is a nonzero normal element in $B$,we have a $*$ isometric isomorphism $\phi:C(\sigma_B(b))\to C^*(b,b^*),\;...
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functional calculus for non-unital $C^*$ algebras

If $A$ is a non-unital $C^*$ algebra,$a$ is a normal element of $A$, there is a $*$ isometric isomorphism from $C_0(\sigma_{A}(a))$ to $C^*(a)$. I have a question:what is the set of $C_0(\sigma_{A}(a)...
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Extension of a von Neumann algebra by a von Neumann algebra

Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras: $$0\to A\to C\to B\to 0$$ Assume that $A,B$ are generated by their ...
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compute the multiplier algebra

If $A$ is a $C^*$ algebra, $\oplus_{c_0} M_{k(n)} (\mathbb{C} )$ is the essential ideal of $A$ ,then we have $\oplus_{c_0} M_{k(n)} (\mathbb{C} ) \subset A \subset \prod M_{k(n)} (\mathbb{C})$. ...
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construct an element in $\prod M_{k(n)} (\mathbb{C})$

Suppose $A$ is a $C^*$ algebra,$\oplus_{c_0} M_{k(n)} (\mathbb{C} )$ is a essential ideal of $A$ and there is an element $(x_n) \in A$ such that $(x_n) \in \prod M_{k(n)} (\mathbb{C})$ and $tr(x_n) \...
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Minimal or Maximal Von Neumann algebra contained in a given $C^*$ algebra

Let $A,B \subset B(H)$ be two concrete von Neumann algebra. Is $A\cap B$ a von Neumann algebra, too? What about the intrinsic analogy of this question, as follows: Let $C$ be a $C^*$ ...
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Classification of $C^*$ algebras whose subalgebra generated by projections is a von neumann algebra

Inspired by this question we ask the following question: Is there a complete classification of all unital $C^*$ algebra $A$ for which the following subalgebra $B$ is a von Neumann algebra? Is ...
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$C_0$ direct sum of complex matrices

How many elements in the $C_0$ direct sum of complex matrices,say$\oplus_n M_n(\mathbb{C})$? $\oplus_n M_n(\mathbb{C})=\{(x_n)\in \prod_n M_n(\mathbb{C}):\|x_n\| \to 0\}$ I think there are ...
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3answers
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orthogonal projections in $C^*$ algebra

Suppose $A$ is an arbitrary $C^*$ algebra,can $A$ be linear spanned by all orthogonal projections of in it ? If not,is there a relationship between a $C^*$ algebra and all orthogonal projections in ...
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a question on gns construction

If $\pi$ is a zero representation of $C^*$ algebra $A$,there is no state$\tau$ on $A$ such that $\tau(a)=(\pi(a)\xi,\xi)$. When we talk about GNS constuction,Should the zero representation be ...
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nondegenerate representation of a $C^*$ algebra

Every representation $(\pi,H)$ of a $C^*$ algebra $A$ can be reduced to the case of a non-degenerate representation.Usually,we take $K=[\pi(A) H]$,then we get a non-degenerate representation $(\pi_K,...
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1answer
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proper ideals of $\prod_{i\in I}A_i$

$\oplus_{i\in I}A_i$ denotes the $c_0$ direct sum of $C^*$ algebras $A_i$, $\prod_{i\in I}A_i$ is the $l^\infty$ direct sum of $A_i$.We know that $\oplus_{i\in I}A_i$ is the essential ideal of $\prod_{...
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essential ideal of a $C^*$ algebra

There is a well known fact:$A=\oplus_{i\in I}A_i$($c_0$ direct sum) is an essential ideal $\prod_{i\in I}A_i$($l^\infty$ direct sum),where each $A_i$ is a $C^*$ algebra. I have two questions: 1.If $\...
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1answer
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Kadison's Theorem for operator subsystems of commutative $C^*$-algebras

By a result of Kadison, every operator subsystem of a commutative $C^*$-algebra is isomorphic to the space of continuous affine functions on its state space. In other words, if $X$ is a compact ...
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About the C*-algebra of the Schrodinger representation of the Weyl C*-algebra

We start with the Weyl C*-algebra $\mathcal{W}$ for a finite dimensional symplectic space and we consider the irreducible Schrodinger representation $\pi:\mathcal{W}\rightarrow \mathcal{B}(\mathcal{H})...
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tracial state on a unital infinite dimensional simple $C^*$ algebra

If $A$ is a finite dimensional simple $C^*$ algebra,then it has the form of $M_n(\mathbb{C})$,which has unique tracial state. My question is:If $A$ is an unital infinite dimensional simple $C^*$ ...
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2answers
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annihilator of an ideal in $C^*$ algebra

If $H$ is a hilbert space,K is a closed subspace of $H$,then $H=K\oplus K^\perp$. If $A$ is a $C^*$ algebra,$I$ is a closed ideal of $A$.Does there exist a similar decomposition $A=I\oplus I^\perp$?
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tracial state on a non-unital simple $C^*$ algebra

I think that there is no tracial state on non-unital simple $C^*$ algebras.Is my thought correct? I 'll appreciate it if anyone can supply me a counterexample.
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infinite dimensional quotient of a $C^*$ algebra [closed]

If $A$ is a noncommutative non unital $C^*$ algebra,does there exist a nontrivial ideal $J$ of $A$ such that the dimension of $A/J$ is infinite?
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State spaces of a von Neumann algebra and of a dense C* subalgebra.

Assume that A is a von Neumann algebra on a separable Hilbert space H and D a dense separable sub-algebra of A. What type of a state on A is an extension of a state on D. Is it always a normal state ...
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non-existence of nonzero special linear map on $B(H)$

I saw an exercise :There is no nonzero linear map $\phi:B(H) \rightarrow \mathbb{C}$ satisfying $\phi(ab)=\phi(ba)$ when $dim(H)=\infty$. I have no idea.Can anyone give me some hints,thanks!
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examples about Stinespring's theorem for $C^*$ algebras

The Stinespring's theroem is a little abstract.I wonder whether there exist concrete examples which clarify the theorem.
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center of quotients of $C^*$ algebra

Does there exist a non-unital $C^*$ algebra $A$ such that $\mathcal{Z}(A/I)\neq 0$,where $ \mathcal{Z}(A/I)$ denotes the center of the $A/I$.
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two commuting $*$ homomorphisms

Suppose there is a nonzero non-trivial $*$ homomorphism $\pi_1:A\rightarrow B(H),\pi_1(A)\neq kId $,Can we construct another nonzero non-trivial $*$ homomorphism $\pi_2:A\rightarrow B(H)$ such that $\...
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tracial state on a non-unital $C^*$ algebra

Does there exist a non-unital $C^*$ algebra which have uncountable tracial states?
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commutant of a non unital $C^*$ algebra

If $A$ is a non-unital $C^*$ algebra,is the commutant of $A$ empty? Does there exist a theorem which states that every $C^*$ algebra has commutant(centralizer)?
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Non normal state

Does anybody know an example of a state on a von Neumann algebra that is not normal? If it has relevance to physics it would be nice.
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Predual of $W^*$-subalgebra

I've seen many references claiming that if $\mathcal{N}$ is a $\sigma$-weakly closed *-subalgebra of a von Neumann algebra $\mathcal{M}$, then by taking $\mathcal{N}_\bot:=\{\phi\in\mathcal{M}_*|a(\...
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state on a non-unital $C^*$ algebra

Suppose $\tau$ is a state on a non-unital $C^*$ algebra $A$.There is a well-known inequality: $$\tag{$*$}|\tau(a)|^2\leq\tau(a^*a),\ \text{ for all } a\in A.$$ Does there exist some nonzero element $...
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On existence of Singular value decomposition in von Neumann algebras

For which class of von Neumann algebras we will have singular value decomposition of each element of the algebra?
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Hilbert subspaces whose vectors are each cyclic for a von Neumann factor

Let $\mathcal{R}$ be a type III factor acting on separable Hilbert space $H$, and $S \subseteq H$ a closed linear subspace such that every nonzero $v \in S$ is cyclic for $R$; can there exist a ...
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Approximating states of the enveloping von Neumann Algebra

Let $A$ be a $C^*$-Algebra and $A''$ its enveloping von Neumann Algebra. Is the state space $S(A)$ of $A$ weak*-dense in the state space of $S(A'')$? I Know that every state on $A$ extends as a vector-...
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On classifying von Neumann algebras with respect to some property

For which class of von Neumann algebras have the property$:T \in B(\mathcal{H})\text{ such that } \text{Co}_{M}(T)^{-}\cap M'\text{ nonempty }$. Where $\text{Co}_{M}(T)^{-}$ is weak operator closure ...
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Conditional Expectation on von Neumann algebras

A linear map $\phi$ from a von Neumann algebra M to the sub algebra N is called a conditional expectation when $\phi$ has the following properties. 1)$\phi(I)=I$, 2) $\phi(x_{1}yx_{2})=x_{1}\phi(y)x_{...
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Introduction to von Neumann algebras

I'm learning the basics of von Neumann algebras. Every reference on the subject I can find turns to the study of projections, introduces factors and the type classification immediately after having ...
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Dixmier averaging Theorem

We know the standard result of Dixmier averaging Theorem for von Neumann algebras. Is Dixmier averaging Theorem still holds for $C^{*}$-algebras??
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Confusion on construction of type III factors

Well $G$ is group of translations: $\mathbb{R} \ni s:\mapsto as+b $ where $a, b \in \mathbb{Q}$, modulus of $a\neq1$. Under this action of $G$ on $\mathbb{R}$ why Lebesgue measure is not invariant? ...
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1answer
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Enough existence of faithful normal states on von Neumann algebra acting on separable Hilbert space

Under which condition , given a von Neumann algebra $M$ acting on separable Hilbert space $\mathcal{H}$ have uncountable number of faithful normal states?
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1answer
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On clarification on representation of operators on direct sum of Hilbert spaces as matrices

Suppose $G$ is a countable group, consider $\mathcal{H}=\oplus\{\mathcal{H}_{g}:g\in G\}$, Suppose $T$ is an operator in $\mathcal{H}$, then how to give a isomorphism from $B(\mathcal{H})$ into ...
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1answer
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Two ordered relations on projections.

Let $A$ be a vn-algebra. Let $p$ and $q$ be two projections. In the literature, we say $p$ is majorised by $q$ if $pq=p$. Q. Suppose that $q-p$ is a positive element in $A$ (meaning $q-p=x^*x$ for ...
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A relation between projections

Let $A$ be a vn-algebra. Let $e,p$ and $q$ be projections in $A$. Suppose that $p\leq q$. True or false? $$q\wedge e-p\wedge e\leq q-p$$
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Can a projection in factor $R$'s commutant fail to meet a maximal projection in a factor that includes $R$?

If $\mathcal{R} \subseteq \mathcal{S}$ are factors acting on Hilbert space $H$, with $\mathcal{S}$ type I and $\mathcal{R}$ not type I, and $P$ is a maximal projection in $\mathcal{S}$ (meaning the ...
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Conditional Expectation for von Neumann algebra.

Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...
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1answer
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Pure states on subalgebras of $\mathcal{B}(\mathcal{H})$ in finite dimensions.

I consider only finite-dimensional Hilbert spaces. We know that pure states on $\mathcal{B}(\mathcal{H})$ are exactly the vector states or in terms on density matrices, the rank one projections. My ...
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On direct integral of states of von Neumann algebras

Suppose we consider a direct integral of GNS states of a measure space in von Neumann algebra, get the new state by direct integral. Does the GNS represenation of the state breaks down into direct ...
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definition of complete vector lattice

Suppose $M$ is a von-Neumann algebra, $L=M\cap M'$ is the centre of $M$. The last line on page 29, C*-algebras and their automorphism groups, states that the self-adjoint part $L_{sa}$ of $L$ is a ...
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About unitary group of a von Neumann algebra

Is unitary group of a von Neumann algebra is locally compact in some topology? Can we make sense of integration with respect to Haar measure?