All Questions
Tagged with von-neumann-algebras lattice-theory
9 questions
2
votes
0
answers
69
views
Link between Carathéodory's criterion and commutation in an orthomodular lattice?
In the theory of outer measures, Carathéodory's criterion constructs from an outer measure $\mu^*$ on $X$ a $\sigma$-algebra $\Sigma$ of subsets of $X$, on which the restriction of $\mu^*$ is a ...
- 193
2
votes
1
answer
170
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Defining states on von Neumann algebras from filters on the projection lattices
Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for ...
- 2,830
2
votes
1
answer
141
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A congruence relation on the projection lattice
This question is a continuation of what I asked here. Tristan Bice showed the following nice result there:
Let $A$ be a von Neumann algebra and $P$ its projection lattice, ordered by $p\leq q\...
- 1,731
8
votes
2
answers
496
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Which complete orthomodular lattices arise from von Neumann algebras?
Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice.
Question 1: Is the construction $A \mapsto \Pi(A)$ a ...
- 63.9k
7
votes
1
answer
732
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To what extent can a von Neumann algebra be determined by its projection lattice structure?
Let $ M, N $ be von Neumann algebras, $ P $ (resp. $Q$) the projection lattice of $M$ (resp. $N$). Any isomorphism $ \varphi : M \to N $ on the level of involutive algebras induces an isomorphism $ \...
- 1,586
5
votes
1
answer
158
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Does every non-type-I factor's projection lattice admit a dense embedding of the standard continuum-collapsing poset?
Let $R$ be a non-type-I factor acting on a separable Hilbert space.
Let $P(R)$ be the set of $R$'s projections with the usual ordering ($x \leq y \iff$ range$(x) \subseteq$ range$(y)$) under which it ...
- 271
3
votes
1
answer
238
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How much of a factor's structure is determined by the order-type of its projection lattice?
H. A. Dye showed that a type II or III factor $R$ is determined, up to *-algebraic isomorphism or anti-isomorphism, by the ortholattice-isomorphism type of its projection lattice ("ortholattice-...
- 271
6
votes
1
answer
486
views
Dye's Theorem for real von Neumann algebras
Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically ...
- 388
6
votes
2
answers
919
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Type III factor representation
Does there exist any theorem which permits, under suitable hypotheses, to represent a particular complete orthomodular lattice as the projection lattice of a Type III von Neumann factor?
- 275