Analytic quotients homepage

This is a homepage of my book Analytic quotients: theory of liftings for quotients over analytic ideals on the integers (Memoirs of the American Mathematical Society, 148, No. 702, 177 pp.), which is an abridged, revised, and then extended, version of my PhD thesis.
Order it here.

The book mainly consists of four chapters (Chapters 1, 3, 4 and 5), each of which corresponds to a different line of research. (Chapter 2 is very short and it contains only some background material.)

The basic question considered in this monograph is:
How does a change of the ideal I on the natural numbers effect the change of its quotient Boolean algebra, P(N)/I?
The ideal I is always assumed to be analytic (as a matter of fact, all of the considered ideals are Borel).

The main recent development on the subject shows that this was not the most general (or most interesting) question along these lines that one could ask. My result that Todorcevic's Axiom implies all automorphisms of the Calkin algebra are inner shows that the rigidity theory developed in Analytic quotients is also applicable to the quantum analog of P(N)/Fin.

Remarks below are intended for a reader familiar with Analytic quotients or one of survey papers Completely additive liftings or Rigidity conjectures.

Status of open problems

The following describes the current status (to my best knowledge - please send any new information to ifarah (at) mathstat (dot) yorku (dot) ca) of open problems from Analytic quotients.

Chapter 1

An extension of results from this chapter was obtained in Luzin gaps:
Both ideals NWD(Q) and NULL(Q) have the Radon-Nikodym property. NWD(Q) is the first known ideal that has the RNP that is not an F-ideal.

Chapter 3

The main result of Chapter 3, `OCA lifting theorem' was extended in Luzin gaps, using PFA instead of OCA and MA. It was proved that the conclusion of OCA lifting theorem holds for all a class of ideals that are `strongly countably generated by closed approximations.' This class includes all analytic P-ideals, all F-sigma ideals and all F-sigma-delta ideals known to me.
This provides some information on Problem 3.14.1: all nonpathological F-sigma ideals, as well as the ideals NWD(Q) and NULL(Q) belong to the class of analytic ideals described in this problem. Unfortunately, Problem 3.14.1 unneccessarily involves a more general form of (wide open) Question 1.14.1. This problem was separated into two parts in Rigidity conjectures. Some answers to Problem 3.14.1 are given there: Theorem 8.6 and Corollary 8.7 imply that all definable ideals have a somewhat weaker property (a local version, as in section 3.12 of Analytic quotients).

Question 3.14.3 has been solved:
In 2001 Juris Steprans has proved that there is a family of continuum many coanalytic ideals that have pairwise non-isomorphic quotient algebras. Moreover, the regular open algebras of these quotient algebras are nonisomorphic.
Mike Oliver (a student of Greg Hjorth at UCLA) has proved (December 2002) that there are continuum many pairwise nonisomorphic quotients over analytic P-ideals. This solves Question 3.14.3.

All quotients over ideals I-alpha and W-alpha (see Analytic quotients, section 1.14) are countably saturated, hence isomorphic under the Continuum Hypothesis. Thus, with a possible exception of some density ideals (see Section 1.12 of Analytic quotients), quotients over all analytic ideals mentioned in Analytic quotients fall into one of categories (i)-(vi) on page 115. It also turns out that all the quotients over Louveau-Velickovic ideals are isomorphic under CH.

Consider the class of dense ideals of the form Exh(phi), where phi is a supremum of a family of pairwise orthogonal lower semicontinuous measures. This class includes all dense density ideals and all summable ideals. Under CH there are exactly six nonisomorphic quotients over ideals in this class.

These results appear in How many Boolean algebras P(N)/I are there?.

Question 3.7.6 turned out to have a simple answer (yes). E.g., the ideal NWD(Q) of all nowhere dense subsets of the rationals is homogeneous (thanks to Micha(e)l Hrusak for turning my attention to this interesting ideal).
This question was motivated by a question of Just and Krawczyk: if I, J are F-sigma-delta ideals that are not F-sigma and such that their quotients are homogeneous, are their quotients isomorphic? The answer turns out to be negative (see Two F-sigma-delta ideals, a joint paper with S. Solecki).

Chapter 4

The results of Chapter 4 have been considerably improved in Dimension phenomena associated with beta-N spaces and Powers of N*. An n-dimensional generalization of Theorem 4.2.1 was proved in Functions essentially depending on at most one variable.

Chapter 5

Every quotient over an F-sigma P-ideal (and many other analytic P-ideals) other than Fin contains an analytic Hausdorff gap (Analytic Hausdorff gaps). This implies that Fin is the only F-sigma P-ideal such that OCA implies that its gap-spectrum does not contain (omega_2,omega_2)-gaps. The latter fact has immediate implications to Problem 5.13.1 and Question 5.13.7, and it excludes F-sigma P-ideals from both classes of ideals considered in Question 5.13.2, Problem 5.13.5 and Problem 5.13.6.

In 2004 I have announced there are no analytic Hausdorff gaps over Z_0 or any other EU-ideal. I am retracting this claim, for a partial result see Analytic Hausdorff gaps II: the density zero ideal. The proof uses ultraproducts and has an amusing characterization of the Boolean algebra P(N)/Z_0 as a forcing iteration of P(N)/Fin and the measure algebra of Maharam character continuum as a byproduct.

Regarding Question 5.13.9: It turns out that the answer is positive, as there is a natural separation property (weaker than Todorcevic's, yet still rather useful) that all analytic P-ideals satisfy. These results appear in Luzin gaps.

Last but not least - a compact, self-contained, and error-free exposition of the `OCA lifting theorem' was written by David Fremlin and it is available from his web site (among `notes').
(Last revised: November 2007.)