Set theory and operator algebras

Farah, Ilijas; Wofsey, Eric

doi:10.1017/cbo9781139208574.004

Cited by 31 publications

(43 citation statements)

References 48 publications

(31 reference statements)

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“…This problem has received a lot of attention (see e.g. [FHS11,GS14] and the survey [Fa14]). The connection between the two problems stems from the continuous version of the Keisler-Shelah theorem [Ke61,Sh71].…”

Section: Introductionmentioning

confidence: 99%

II1 factors with nonisomorphic ultrapowers

Boutonnet¹,

Chifan²,

Ioana³

2017

Duke Math. J.

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Abstract. We prove that there exist uncountably many separable II1 factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II1 factors originally introduced by McDuff [MD69a,MD69b] are such examples. This entails the existence of a continuum of non-elementarily equivalent II1 factors, thus settling a well-known open problem in the continuous model theory of operator algebras.

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Section: Introductionmentioning

confidence: 99%

II1 factors with nonisomorphic ultrapowers

Boutonnet¹,

Chifan²,

Ioana³

2017

Duke Math. J.

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“…Together with the Brown-DouglasFillmore characterization of unitary equivalence modulo compact perturbation of essentially normal operators, this implies that a positive answer to Problem 8.1 is equivalent to the consistency of the existence of normal operators a and b in C(H) and an automorphism Φ of C(H) such that Φ(a) = b but for every inner automorphism Ψ of C(H) we have Ψ(a) = b. An argument using [2] shows that if an automorphism Φ sends the standard atomic masa to itself then Φ cannot sendṠ toṠ * (see [18,Prop. 7.7]).…”

Section: Discussionmentioning

confidence: 99%

All automorphisms of the Calkin algebra are inner

Farah¹

2011

Ann. Math.

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“…Chapter 24: The standard source for consequences of Martin's axiom is [15]. The version of Theorem 16.4 in which CH is weakened to MA was proven by Farah and the author ( [13], Theorem 6.46). Chapter 25: This material can also be found in [37] and [9].…”

Section: Notesmentioning

confidence: 99%

Back Matter

2014

Forcing for Mathematicians

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In this book we have defined forcing notions concretely as families of sets ordered by inclusion. The standard definition treats them abstractly as preordered sets. Our version simplifies the exposition a little because the order relation is built in and does not have to be added separately. It may appear to be less general than the preorder definition, but the two are actually equivalent. The purpose of this appendix is to explain this equivalence.Recall that a preordered set is a set equipped with a relation that is reflexive and transitive, but not necessarily antisymmetric -we can have p ≤ q and q ≤ p for distinct p and q. (Beware: in the forcing literature the term "partial order" sometimes means "preorder".) In the abstract approach, it is convenient to use preorders when we get to iterated forcing. The notions of extension, compatible elements, dense set, ideal, and generic ideal all generalize straightforwardly to preordered sets, just as in Chapter 23.We should also mention that according to the usual convention an extension of p lies below p, not above it, and one is correspondingly interested in the dual notion of filters rather than ideals. We have reversed this convention in order to avoid habitually ordering sets by reverse inclusion. Having forcing notions grow downward carries no particular benefit, although it does make sense in the context of Boolean-valued forcing.Preorders do not create any genuinely greater generality over partial orders. We can always factor out the equivalence relation which sets p ∼ q if p ≤ q and q ≤ p, and it is more or less obvious that forcing with the resulting poset is equivalent in every significant sense to forcing with the original preordered set. It should also be clear that replacing one poset with another one to which it is order-isomorphic does not meaningfully affect the

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Set theory and operator algebras

Cited by 31 publications

References 48 publications

II1 factors with nonisomorphic ultrapowers

II1 factors with nonisomorphic ultrapowers

All automorphisms of the Calkin algebra are inner

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