Rigidity Conjectures

Vignati, Alessandro

doi:10.48550/arxiv.1812.01306

Cited by 3 publications

(9 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 6.10 inspired the second part of Conjecture 3.6, asserting that under Forcing Axioms every isomorphism between coronas of separable C *algebras is topologically trivial. Building on [90] and [92], and a noncommutative version of the OCA Lifting Theorem from the latter reference in particular, this conjecture was confirmed in [143]. The outline of the proof of Theorem 6.15 follows the proof of Theorem 6.10, but is much more difficult.…”

Section: čEch-stone Remainders Of Zero-dimensional Spacesmentioning

confidence: 81%

“…For example, for abelian C * -algebras, an algebraically trivial isomorphism between C(βX \ X) and C(βY \ Y ) is dual to an algebraically trivial homeomorphism as in Definition 3.8 ([143, Proposition 2.7]). Also, a very natural description of algebraically trivial isomorphisms can be made for reduced products of unital separable C * -algebras which do not have central projections (see [143,Proposition 5.3]). In case of the Calkin algebra, algebraic triviality coincides with innerness.…”

Section: Multipliers and Coronasmentioning

confidence: 99%

“…The converse is now known (at least, unconditionally on A and B), and it is tightly related to Ulam-stability phenomena (see §4). We record this as a question for future reference (see [143,Conjecture 5.1]). Question 3.11.…”

Section: Multipliers and Coronasmentioning

confidence: 99%

“…). A simple computation (regrettably absent from [143]) shows that when ϕ : A 1 → B 1 is an ε- * -homomorphism as in Definition 4.14, then f : A → B defined by f (x) = x ϕ(x/ x ) for x = 0 and f (0) = 0 is a 4ε-homomorphism in the sense of (4.3) and that f (x) − ϕ(x) ≤ ε for all x ∈ A 1 . The thesis follows by connecting the dots.…”

Section: Question 413 Is Every Topologically Trivial Boolean Algebra ...mentioning

confidence: 99%

“…However, in the case of more complicated objects such as the Calkin algebra Q(H) it is not difficult to define a topologically trivial automorphism that cannot be lifted by a * -homomorphism ( §3; the obstruction is similar to that in the case of Boolean algebras-the automorphism is conjugation by a unitary with a nontrivial Fredholm index). We therefore import the following Gordian definition from [143]. Definition 1.3.…”

Section: Introduction: the General Rigidity Questionmentioning

confidence: 99%

See 4 more Smart Citations

Corona Rigidity

Farah¹,

Ghasemi²,

Vaccaro³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

In 1956 W. Rudin proved that the Continuum Hypothesis (CH) implies that the Čech-Stone remainder of N (with the discrete topology), βN \ N, has 2 c homeomorphisms. In 1979, Shelah described a forcing extension of the universe in which every autohomeomorphism of βN\N is the restriction of a continuous map of βN into itself. Since there are only c such maps, the conclusion contradicts Rudin's. Rudin's result is, by today's standards, trivial: By the Stone Duality, autohomeomorphisms of βN \ N correspond to automorphisms of the Boolean algebra P(N)/ Fin. This algebra is countably saturated hence CH implies that it is fully saturated. A standard back-and-forth argument produces a complete binary tree of height ℵ1 = c whose branches are distinct automorphisms. The fact that the theory of atomless Boolean algebras admits elimination of quantifiers is not even used in this argument.On the other hand, Shelah's result is, unlike most of the 1970s memorabilia, still as formidable as when it appeared. Extensions of Shelah's argument (nowadays facilitated by Forcing Axioms) show that this rigidity of P(N)/ Fin is shared by other similar quotient structures, and that is what this survey is about. Contents 1. Introduction: The general rigidity question.I. FARAH, S. GHASEMI, A. VACCARO, AND A. VIGNATI 4. Ulam-stability 21 4.1. From Borel to asymptotically additive 22 4.2. Asymptotic additivity 24 4.3. Boolean algebras 24 4.4. C * -algebras 26 4.5. Reduced products of C * -algebras 27 4.6. The Kadison-Kastler conjecture 28 5. Independence results, I. Nontrivial isomorphisms 29 5.1. Countable saturation 30 5.2. The failure of countable saturation 36 5.3. Stratifying Q(H) 39 5.4. Cohomology and automorphisms 40 5.5. Higher-dimensional Čech-Stone remainders 43 5.6. Other models for non-rigidity 44 6. Independence results, II. Rigidity 46 6.1. Shelah's proof 46 6.2. Quotients of P(N) and zero-dimensional remainders 49 6.3. Coronas and Forcing Axioms 52 6.4. Other models for rigidity 57 7. Endomorphisms 59 8. Larger Calkin algebras 62 9. Uniform Roe coronas 64 10. This paper was too short for. . .

show abstract

Section: čEch-stone Remainders Of Zero-dimensional Spacesmentioning

confidence: 81%

Section: Multipliers and Coronasmentioning

confidence: 99%

Section: Multipliers and Coronasmentioning

confidence: 99%

Section: Question 413 Is Every Topologically Trivial Boolean Algebra ...mentioning

confidence: 99%

Section: Introduction: the General Rigidity Questionmentioning

confidence: 99%

See 3 more Smart Citations

Corona Rigidity

Farah¹,

Ghasemi²,

Vaccaro³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Trivial Endomorphisms of the Calkin Algebra

Vaccaro¹

2019

Preprint

View full text Add to dashboard Cite

We prove that it is consistent with ZFC that every (unital) endomorphism of the Calkin algebra Q(H) is unitarily equivalent to an endomorphism of Q(H) which is liftable to a (unital) endomorphism of B(H). We use this result to classify all unital endomorphisms of Q(H) up to unitary equivalence by the Fredholm index of the image of the unilateral shift. Finally, we show that it is consistent with ZFC that the class of C * -algebras that embed into Q(H) is not closed under countable inductive limit nor tensor product.

show abstract

Between reduced powers and ultrapowers, II

Farah¹,

Shelah²

2020

Preprint

View full text Add to dashboard Cite

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.The trivializing effect of the Continuum Hypothesis (CH) to the structure of the continuum has been known at least since the times of Sierpiński and Gödel ([19]). The particular instance of this phenomenon that we are concerned with in the present paper is the existence of highly non-canonical isomorphisms between massive quotient structures of cardinality c = 2 ℵ 0 . The operation of taking a reduced product F A n of a sequence (A n ) of first-order structures often results in a countably saturated structure. 1 This is the case with the two most commonly used reduced products: ultraproducts associated with nonprincipal ultrafilters on N and reduced products associated with the Fréchet filter. If each A n has the cardinality of at most c, then so does F A n , and the CH implies that the latter structure is saturated. By a classical theorem of Keisler, elementarily equivalent and saturated first-order structures of the same cardinality are isomorphic (see [5, Theorem 5.1.13]). Therefore the isomorphism of such reduced products reduces (no pun intended) to elementary equivalence.In [11], this observation was combined with computation of the theory of the structure (K denotes the Cantor space)

show abstract

Rigidity Conjectures

Cited by 3 publications

References 21 publications

Corona Rigidity

Corona Rigidity

Trivial Endomorphisms of the Calkin Algebra

Between reduced powers and ultrapowers, II

Contact Info

Product

Resources

About