Approximate Homomorphisms II: Group Homomorphisms

Received October 13, 1998     

Reductions between meager ideals

We construct a nonmeager ideal that is not a P-ideal yet Fin × ⊘ is not reducible to it. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 213–219, 2005.     

The relative commutant of separable C-algebras of real rank zero

We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C^∗-algebra in its ultrapower depend on the choice of the ultrafilter?     

Approximate Homomorphisms

which ``almost everywhere' looks like an ultrafilter has to be close to some fixed ultrafilter. Received: August 20, 1997     

Von Neumann's Problem and Large Cardinals

It is a well-known problem of Von Neumann to discover whetherthe countable chain condition and weak distributivity of a completeBoolean algebra imply that it carries a strictly positive probabilitymeasure. It was shown recently by Balcar, Jech and Pazák,and by Velikovi, that it is consistent with ZFC, modulo theconsistency of a supercompact cardinal, that every ccc weaklydistributive complete Boolean algebra carries a contiuous strictlypositive submeasure – that is, it is a Maharam algebra.We use some ideas of Gitik and Shelah and implications fromthe inner model theory to show that some large cardinal assumptionsare necessary for this result. 2000 Mathematics Subject Classification03E55, 28A60 (primary), 03E75 (secondary).     

Four and more

We isolate several large classes of definable proper forcings and show how they include many partial orderings used in practice.     

Nonseparable UHF algebras I: Dixmier's problem

There are three natural ways to define UHF (uniformly hyperfinite) C^∗-algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.     

Identification of impurities in acarbose by using an integrated liquid chromatography-nuclear magnetic resonance and liquid chromatography-mass spectrometry approach

The usefulness of applying an integrated LC-NMR and LC-MS approach to acarbose bulk drug impurity profiling is demonstrated. LC-MS and LC-NMR methodologies were employed for the online separation and structural elucidation of a final drug product. Combining data provided by the stop-flow LC-NMR and LC-MS experiments made it possible to identify the main components present in the acarbose sample. Spectral analysis revealed that A and B were known impurities while C was an unknown compound. LC-MS and LC-NMR analyses revealed that C was a pentasaccharide differing from the acarbose in number and nature of sugar subunits in the molecule. It was subsequently isolated and its structure was confirmed by the offline 1- and 2-D NMR experiments, and atom assignment was made.     

Powers of

We prove that the Cech-Stone remainder of the integers, , maps onto its square if and only if there is a nontrivial map between two of its different powers, finite or infinite. We also prove that every compact space that maps onto its own square maps onto its own countable infinite product.

     

Luzin gaps

We isolate a class of ideals on that includes all analytic P-ideals and all ideals, and introduce `Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients . We also prove that the ideals and have the Radon-Nikodým property, and (using OCA) a uniformization result for -coherent families of continuous partial functions.