Strongly 1-Bounded Von Neumann Algebras
By Kenley Jung. 2005
78 citations recorded by the source corpus.
Abstract.Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if $${\mathbb{P}}^\alpha (F) < \infty$$ where $${\mathbb{P}}^\alpha$$ is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an $$x \in F$$ with $$\chi (x) > -\infty$$ . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ0(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ0 is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II1-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of $$SL_n({\mathbb{Z}}), n \geq 3$$ . If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/7aa6c163f8f63baf79eb906c4cba2d0b80d6178a
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In Algebra and Number Theory
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Citations in the corpus
- On Ozawa's Property for Free Group Factors (2006, background): “also [ J2 ]). Recently, Peterson used a completely new approach to prove the primeness of group factors N = L with satisfying β (2) () 6 0 ([Pe]).”
- A random matrix approach to the Peterson-Thom conjecture (2020, background): “It would mean that not only can we find a pair of homomorphisms satisfying the conclusion of Jung’s theorem, but that a randomly chosen pair satisfies Jung’s theorem.”
- L2-rigidity in von Neumann algebras (2009, methodology): “2 again generates a completely positive semigroup and hence 424 J . Peterson”
- Examples of factors which have no Cartan subalgebras (2012, background): “Jung generalized this result to some free product von Neumann algebras [19].”
- Group cocycles and the ring of affiliated operators (2007, background): “Some evidence for this appears in [25, 36], and in [35].”