All Questions
Tagged with operator-theory von-neumann-algebras
309
questions
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81
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Convergence of a Sequence in the GNS Space of a von Neumann Algebra with Semi-finite Trace under the $\sigma$-Weak Topology
Let $ M $ be a von Neumann algebra. A Semi-finite trace on $ M^+ $ is a function $\phi$ on $ M^+ $, taking non-negative, possibly infinite, real values, possessing the following properties:
Linearity:...
- 2,007
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1
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57
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Question about hyperstandard von Neumann algebras and selfpolar cones
Consider the following fragment from the book "Lectures on von Neumann algebras" by Stratila and Zsido, second edition:
I have two questions:
(1) Why is $\mathscr{M}_q= q\mathscr{M}q \...
- 840
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1
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38
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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....
- 840
2
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1
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54
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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 ...
- 840
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1
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57
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Making sense of certain vector-valued integrals in von Neumann algebra theory.
Let $M\subseteq B(H)$ be a von Neumann algebra and $\varphi$ a weight on $M$ with modular automorphism group $\{\sigma_t\}_{t\in \mathbb{R}}$.
We define
$M_\infty$ to be the set of all elements $m\in ...
- 840
1
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2
answers
58
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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. ...
- 629
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53
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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. ...
- 629
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51
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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 ...
- 840
0
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34
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Bounded homomorphism
Let $H,K$ be Hilbert spaces and $X,Y$ be subsets of $\mathcal{B}(H,K)$ and $\mathcal{B}(K,H),$ respectively. By $[Y X]$ we denote the w*-closure of the linear span of the set consisting of operators $...
- 382
3
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1
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45
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Basic lemma on normality of map between von Neumann algebras
Let $f: M\to N$ be a bounded linear map between von Neumann algebras.
Assume that for every net $0 \le x_i\nearrow x$, we have that $f(x_i)\to f(x)$ $\sigma$-strongly. Is it true that $f$ is normal, i....
- 840
2
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1
answer
57
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$\phi$ uwo continuous faithful tracial state on a VNA with GNS.
Let $M \subset B(\mathcal{H})$ be a von neumann algebra and $\phi$ be a uwo continuous faithful tracial state. I understand that we obtain the GNS $(\mathcal{H}_\phi, \pi_\phi, \xi_\phi)$ with $\xi_\...
2
votes
1
answer
76
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$\sigma$-weakly closed subalgebra of direct product of matrix algebras is again a direct product of matrix algebras
Let $A$ be a $\sigma$-weakly closed $*$-subalgebra of the $W^*$-algebra $\prod_{i\in I}^{\ell^\infty} M_{n_i}(\mathbb{C})$. I believe that we must have $A\cong \prod_{j\in J}^{\ell^\infty} M_{m_j}(\...
- 840
1
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1
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39
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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 ...
1
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0
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54
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Free Product of interpolated, free group factors
Let $L(\mathbb F_2)$ be the group von Neumann algebra of the free group on two generators. The interpolated free group factors of Dykema and Radulescu are defined as $L(\mathbb F_r)=L(\mathbb F_2)^{1/\...
0
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1
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117
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Is centre of a Von Neumann algebra trivial? [closed]
Is the centre of a dense sub algebra $A$ of the Von Neumann algebra $M$ is trivial ($Z_A \subset A$) then shall we conclude $Z_M(M)$ is trivial hence $M$ is factor?
Remember that we see $Z_A(A) \...