Maharam algebras and Cohen reals

We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real.

     

Combined use of liquid chromatography-nuclear magnetic resonance spectroscopy and liquid chromatography-mass spectrometry for the characterization of an acarbose degradation product

Directly coupled LC-MS and LC-NMR were applied to identify and structurally characterize an acarbose degradation product A in acidic media. A comparative analysis of the stop-flow LC-NMR (1H and TOCSY) and LC-MS data provided evidence that A is structurally related to acarbose, differing from the parent compound in a number of subunits present in the molecule. Spectral analysis revealed that A was the alpha-glucosidase inhibitor amylostatin XG. Complementary information obtained from the two methods led to the structural elucidation of A which was later corroborated by high-resolution NMR spectroscopy of the isolated molecule.     

Some Calkin algebras have outer automorphisms

We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, assuming some restriction of the Generalized Continuum Hypothesis, there are many outer automorphisms.     

Basis Problem for Turbulent Actions II: c0-Equalities

Let (Xⁿ,dⁿ) be a sequence of finite metric spaces of uniformlybounded diameter. An equivalence relation D on the product defined by if and only if is a c₀-equality.A systematic study is made of c₀-equalities and Borel reductionsbetween them. Necessary and sufficient conditions, expressedin terms of combinatorial properties of metrics d_n, are obtainedfor a c₀-equality to be effectively reducible to the isomorphismrelation of countable structures. It is proved that every Borelequivalence relation E reducible to a c₀-equality D either reducesa c₀-equality D' additively reducible to D, or is Borel-reducibleto the equality relation on countable sets of reals. An appropriatelydefined sequence of metrics provides a c₀-equality which isa turbulent orbit equivalence relation with no minimum turbulentequivalence relation reducible to it. This answers a questionof Hjorth. It is also shown that, whenever E is an F-equivalencerelation and D is a c₀-equality, every Borel equivalence relationreducible to both D and to E has to be essentially countable.2000 Mathematics Subject Classification: 03E15.     

The commutant of L(H) in its ultrapower may or may not be trivial

Kirchberg asked in 2004 whether the commutant of L(H) in its (norm) ultrapower is trivial. Assuming the Continuum Hypothesis, we prove that the answer depends on the choice of the ultrafilter.     

Trivial automorphisms

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, σ, is equal to ?₁ and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\) . Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.     

Borel subgroups of Polish groups

We study three classes of subgroups of Polish groups: Borel subgroups, Polishable subgroups, and maximal divisible subgroups. The membership of a subgroup in each of these classes allows one to assign to it a rank, that is, a countable ordinal, measuring in a natural way complexity of the subgroup. We prove theorems comparing these three ranks and construct subgroups with prescribed ranks. In particular, answering a question of Mauldin, we establish the existence of Borel subgroups which are -complete, α?3, and -complete, α?2, in each uncountable Polish group. Also, for every α<ω₁ we construct an Abelian, locally compact, second countable group which is densely divisible and of Ulm length α+1. All previously known such groups had Ulm length 0 or 1.     

Analytic Hausdorff gaps II: The density zero ideal

We prove two results about the quotient over the asymptotic density zero ideal. First, it is forcing equivalent to % MathType!End!2!1!, where % MathType!End!2!1! is the homogeneous probability measure algebra of characterc. Second, if it has analytic Hausdorff gaps, then they look considerably different from proviously known gaps of this form. Partially supported by NSERC.     

A coherent family of partial functions on

We prove that there is a family of partial functions is a tower in such that every surjection is associated to a cohomologically different Hausdorff gap (see Talayco). This improves a result of Talayco.

     

Omitting types and AF algebras

We prove that the classes of UHF algebras and AF algebras, while not axiomatizable, can be characterized as those C*-algebras that omit certain types in the logic of metric structures.