All automorphisms of the Calkin algebra are inner

Farah, Ilijas

doi:10.4007/annals.2011.173.2.1

Cited by 69 publications

(74 citation statements)

References 35 publications

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“…The name was again suggested by the Editor of this Bulletin (see, also, [35]). Originally in [108] we have used the name Open Coloring Axiom for TA but the name has already been used in [1] for a rather different axiom which, in our terminology here, applies to graphs G = (X, E) on separable metric spaces X of size ℵ 1 with edge relations E clopen rather than open and asserts that the vertex set X can be covered by countably many sets that are either G-complete or G-discrete.…”

Section: Theorem 112 ([108]) the Proper Forcing Axiom Implies Thatmentioning

confidence: 99%

Combinatorial Dichotomies in Set Theory

Todorčević

2011

Bull. symb. log

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In this article we give an overview of a line of research in set theory that has reached a level of maturity and which, in our opinion, merits its being exposed to a more general audience. This line of research is concentrated on finding to which extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered at the level of some other not so large cardinal, the most interesting being of course the level of the second uncountable cardinal. While this is of great interest to set theorists, one of the main forces behind this line of research stems from its applicability to other areas of mathematics where one encounters structures that are not necessarily countable, and therefore one is likely to encounter problems of this kind. We have split this overview into three parts each presenting a set theoretic combinatorial principle imposing a degree of compactness at some small cardinal. The three principles are natural dichotomies about chromatic numbers for graphs and about ideals of countable subsets of some index set. They are all relatively easy to state and apply and are therefore accessible to mathematicians working in areas outside of set theory. Each of these three dichotomies has its axiomatic as well as its mathematical side and we went into some effort to show the close relationship between these. For example, we have tried to show that an abstract analysis of one of these three set theoretic principles can sometimes lead us to results that do not require additional axioms at all but which could have been otherwise difficult to discover directly. We have also tried to select examples with as broad range as possible but their choices could still reflect a personal taste. We have therefore included an extensive reference list where the reader can find a more complete view on this area of current research in set theory. Finally we mention that while this article is meant for a larger audience which is typically interested in a general overview, we have tried to make the article also interesting to experts working in this area by including some of the technicalities especially if they appear to us as important tools in this area. For the same reason we will be pointing out a number of possible directions for further research.Our terminology and notation follows that of standard texts of set theory (see, for example, [53] and [67]). Recall that the Singular Cardinals Hypothesis, SCH, is the statement (∀θ)It is a strong structural assumption about the universe of sets as it answers all questions about the arithmetic of infinite cardinals. It is for this reason one of the most studied such hypotheses of set theory especially in the context of covering properties of inner models of set theory. Another set of principles which is even more relevant from this point of view are the square principles such as 2 κ and 2(κ). Before we introduce these principles, recall that C α (α < θ) is a C-sequence if C α is a closed and unbounded subset of α for all α < θ. These combinator...

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Section: Theorem 112 ([108]) the Proper Forcing Axiom Implies Thatmentioning

confidence: 99%

Combinatorial Dichotomies in Set Theory

Todorčević

2011

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“…Chapter 28: See [49] for more on the intuition of C(l 2 ) as a noncommutative analog of βN \ N. Theorem 28.4 is from [35]. The argument we give here is based on a proof due to Farah [12]. Chapter 29: Farah's theorem is proven in [12].…”

Section: Notesmentioning

confidence: 99%

“…The argument we give here is based on a proof due to Farah [12]. Chapter 29: Farah's theorem is proven in [12]. The fact used in Lemma 29.1 that π(D 0 ) is maximal abelian in C(l 2 ) is proven in [23].…”

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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“…The most notorious results concerning the triviality of automorphisms of corona of non-commutative C*-algebras concern the Calkin algebra. It has been shown ( [23] and [8]) that the statement that all the automorphisms of the Calkin algebra are inner, is independent from ZFC. Refer to [18, §1] for a short overview of the some of the important results about triviality of the isomorphisms between corona algebras.…”

Section: Theorem 12mentioning

confidence: 99%

Reduced Products of Metric Structures: A Metric Feferman–vaught Theorem

Ghasemi¹

2016

J. symb. log.

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Abstract. We extend the classical Feferman-Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum Hypothesis. We also prove the existence of two separable C*-algebras of the form i M k(i) (C) such that the assertion that their coronas are isomorphic is independent from ZFC, which gives the first example of genuinely non-commutative coronas of separable C*-algebras with this property.

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All automorphisms of the Calkin algebra are inner

Cited by 69 publications

References 35 publications

Combinatorial Dichotomies in Set Theory

Combinatorial Dichotomies in Set Theory

Back Matter

Reduced Products of Metric Structures: A Metric Feferman–vaught Theorem

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