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
The basic question considered in this monograph is:
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
An extension of results from this chapter was obtained in
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.
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
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
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
This question was motivated by a question of
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
F-sigma-delta ideals, a joint paper
with S. Solecki).
The results of Chapter 4 have been considerably improved
Dimension phenomena associated with
and Powers of N*.
An n-dimensional generalization of Theorem 4.2.1 was proved in
Functions essentially depending
on at most one variable.
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.)