Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
10 views

Normal u.c.p extension of Schur-multiplier

I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all. Let $\Gamma$ be a discrete group and ...
0
votes
0answers
10 views

How to identify infinite tensor product of c-star algebras

We know the definition of the infinite tensor product of $C^{*}$-algebras as an inductive limit. What is the infinite tensor product of $\mathbb{C}$ is isomorphic to? Similar for abelian $C^{*}$-...
2
votes
1answer
22 views

Does the von Neumann algebra generated by a normal operator contain all commuting projections?

Let $H$ be a Hilbert space and $T\in B(H)$ a bounded normal operator. Let $\mathscr{A}$ be the von Neumann algebra generated by $T$. Is it true that $\mathscr{A}$ contains every orthogonal projection ...
1
vote
1answer
25 views

Importance of pure states in $C^{*}$ algebras

In a tensor product of $C^{*}$-algebras I have seen some proofs are used by the argument of norms by pure states. Why pure states are essential to study related to vN algebras and $C^{*}$-algebras?
0
votes
0answers
32 views

On c-star tensor norms and von Neumann tensor norm

Does on finite tensor products of matrix algebras, all the $C^*$-norms and von Neumann tensor product norm coincide, or rather they are algebraically isomorphic? What about abelian von Neumann ...
1
vote
1answer
55 views

Weakly dense C*-algebra in a commutative von Neumann algebra and order convergence

Let $H$ be a Hilbert space and $\mathscr{A}$ a commutative norm-closed unital $*$-subalgebra of $\mathcal{B}(H)$. Let $\mathscr{M}$ be the weak operator closure of $\mathscr{A}$. Question: For given ...
0
votes
1answer
35 views

What is the definition of identity bimodule in the context of von Neumann or $\ast$-algebras?

I will just repeat the question in the title: What is the definition of identity bimodule in the context of von Neumann or $\ast$-algebras? I know what a bimodule is but I never heard up to ...
2
votes
1answer
28 views

About “names” of von Neuman algebra morphisms

I have actually a basic quastion about maps between von Neumann algebras. If I have a map $f:N \to M$ being $N$ and $M$ von Neumann algebras. when this map is considered: completely positive, normal ...
3
votes
0answers
59 views

Is strong topology on a W*-algebra a dual topology?

I've come up with this question when reading Shoichiro Sakai's book $C^{*}$-Algebras and $W^{*}$-algebras Chapter 1.8. Let $A$ be a $W^{*}$-algebra (a $C^{*}$-algebra with a Banach space predual) and ...
-1
votes
1answer
49 views

Proving difficulty in MASA [closed]

Prove that $ L^\infty{(\mathbb{R},\mu)}$ is a masa in $B(L^2(\mathbb{R},\mu))$, $\mu$ is sigma finite measure in particular Lebesgue, and what is the cyclic vector of it?
3
votes
0answers
29 views

What does the enveloping von Neumann algebra functor do to locally compact Hausdorff spaces?

Given a $C^*$-algebra $A$, we have an enveloping von Neumann algebra $A^{**}$ which is adjoint to the forgetful functor from the category of $W^*$-algebras to $C^*$-algebras. Because commutativity is ...
3
votes
1answer
32 views

Bell type inequalities - von Neumann Algebras

I have a problem with proving part (4) of the following Lemma 2.1. It concerns Bell-type Inequalities. I attach the relevant definitions the lemma and the suggested proof (It begins with "Since..."). ...
0
votes
0answers
18 views

Existence of infinite family of orthogonal projections in finite vN algebra

Can we always have an infinite family of orthogonal projections in type $II_{1}$ vN algebra such each of which has full central support?
5
votes
1answer
59 views

Type III Von Neumann algebras and spectra of the modular operators.

I´m studying the paper of Fredenhagen. There he said that he would prove that the algebra of local observables under certain conditions is of type III, by showing that all modular operators satisfy $...
0
votes
1answer
22 views

dimension of a von Neumann algebra

Is there any dimension on a von Neumann algebra? Is there any relationship between finite von Neumann algebras and finite dimensional von Neumann algebras?
0
votes
2answers
66 views

Example of norm separable c-star algebras [closed]

I want to know enough examples of norm separable $C^{*}$-algebras which are neither finite dimensional nor commutative.
0
votes
1answer
33 views

examples of non-unital commutative $C^*$-algebras

I know that all the non-unital commutative $C^*$ algebras are isomorphic to $C_0(\Omega)$,where $\Omega$ is a locally compact space. Can anyone show me some common non-unital commutative examples.I ...
-1
votes
1answer
35 views

non isomorphic finite dimensional $C*$ algebras

How many non isomorphic finite dimensional $C^*$ algebras if the dimensions without a bound? Is it countable or uncountable?
0
votes
1answer
26 views

spectrum of $C^*$ algebras

When $A=\bigoplus B(\Bbb C^n)$ ($c_0$ direct sum),how to compute the spectrum of $A$ ?What about the conclusion If we replace the $\ell ^\infty $ direct sum with $c_0$ direct sum?
1
vote
1answer
34 views

arbitrary $n$-dimensional matrix algebras in II$_1$ factor

I'm struggling to show that in a type II$_1$ and any $n$ $\exists$ a subfactor $M$ such that $M \cong M_n$ . I suppose it should follow from the isomorphism between the equivalence classes of ...
0
votes
1answer
27 views

approximate identity element

Let $I\subset \prod_n B(H_{m_n})$ be a separable $C^*$ algebra,where $\prod B(H_{m_n})$ denotes the $\ell ^{\infty}$direct sum of $B(H_n)$ and $dim(H_{m_n})<\infty$. We suppose that there exits a ...
1
vote
1answer
16 views

equivalent projections in finite factors are unitarily equivalent

Why are two Murray von Neuman equivalent projections $p$ and $q$ in a finite factor unitarily equivalent?
1
vote
1answer
25 views

image of some ideal under the quotient map

! I am still confused about the range of $\sigma(I_{\omega})$.Since $\|x_n\|_2\to 0,$we have $\|x_n\|\to 0$,$\sigma(I_{\omega})=0,$then $J=0$,it is trivial. Is my understanding correct?
2
votes
1answer
13 views

Equality of tracial states on dense $C^*$-subalgebra implies equality on generated von Neumann algebra?

Maybe this is a simple question, but I'm not sure about the following: Let $\cal M$, $\cal N$ be von Neumann algebras and $X\subseteq \cal M$ a weakly dense (possibly separable) $C^*$-subalgebra. Let ...
1
vote
1answer
26 views

identify GNS construction as asubalgebra of $R^{\omega}$

I have two questions In lemma6.5.5. 1.why can we identify the GNS representation of $\prod M_{k(n)}(\Bbb C)/\bigoplus M_{k(n)}(\Bbb C)$ with respect to $\tau _{\omega}$ with a subalgebra of $R^{\...
0
votes
1answer
26 views

Some clarifications required

Is the inductive limit of tensor product $L^{\infty}(X,\mu)^{\otimes \infty}$ is isomorphic to $L^{\infty}(X,\mu)$?
0
votes
1answer
38 views

functional calculus

Suppose $A$ is a non-unital $C^*$ algebra,$B$ is another $C^*$ algebra.Suppose $\phi:A\rightarrow B$ is a non-zero $*$ homomorphism and $x_0$ is a normal elememt in $A$,by continuous functional ...
-2
votes
1answer
37 views

strictly positive element

If $A$ is a non-unital separable $C^*$ algebra,does there exist a strictly positive idempotent element in $A$ ?
2
votes
1answer
38 views

Do all *-isomorphisms between von Neumann algebras preserve strong operator topology?

Do all $*$-isomorphisms between von Neumann algebras preserve the strong operator topology? Seems clearly true for $*$-isomorphisms with a unitary implementation, but I don't see the answer for other ...
0
votes
0answers
17 views

On support of a spectral measure

Suppose $A$ is the self-adjoint unbounded operator in Hilbert space $\mathcal{H}$, if spectral measure $E$ of $A$ is supported on $[0,a]$, then prove that the resolution of identity $E_{a}=1_{(-\infty,...
1
vote
1answer
20 views

composition of finite rank projection and bounded operator

If $P\in B(H)$ is a finite rank projection,we assume the rank is $n$,I know the fact $PB(H)P\cong M_n(\mathbb{C})$,but how to construct the isomorphism?
2
votes
2answers
39 views

Extending a $*$-homomorphism between $C^*$-algebras to $*$-homomorphism between generated von Neumann algebras

Let $A \subseteq \cal B(H)$, $B \subseteq \cal B(H')$ be $C^*$-algebras, where $\cal H$, $\cal H'$ are Hilbert spaces and let $\psi: A \rightarrow B$ be a $*$-homomorphism. My question: When does ...
1
vote
1answer
27 views

minimal projection in a $C^*$ algebra

If $(H,\pi)$ is a nonzero representation of a $C^*$ algebra $A$,does there exist a minimal projection $E\in A$ such that $\pi(E)\neq 0$?
1
vote
1answer
21 views

a question on partial isometry

This is a statement from wikipedia.I don't understand why can we deduce that $C$ is a partial isometry if $A^*A=B^*B$.
1
vote
1answer
35 views

On proof of weak operator closed ness of kernel of the representation

Let $\pi:L^{\infty}(X,\mu)\mapsto B(\mathcal{H})$ be a representation of an abelian von Neumann algebra. Where $\mu$ is a probability measure and $\mathcal{H}$ is a seperable Hilbert space. Prove that ...
2
votes
0answers
53 views

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 $...
1
vote
0answers
24 views

$p$ 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 $(pq)^n$ ?
0
votes
1answer
39 views

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?
-1
votes
1answer
27 views

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^*),\;...
1
vote
1answer
27 views

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)...
3
votes
0answers
112 views

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 ...
1
vote
1answer
33 views

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 $C^*$ ...
1
vote
1answer
24 views

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 ...
1
vote
3answers
41 views

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 ...
0
votes
1answer
19 views

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 ...
0
votes
1answer
22 views

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,...
2
votes
1answer
41 views

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_{...
0
votes
1answer
73 views

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 $\...
2
votes
1answer
43 views

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 ...
1
vote
1answer
24 views

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})...