Blocks of central -group extensions

Let and be finite groups that have a common central -subgroup for a prime number , and let and respectively be -blocks of and induced by -blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special -bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .

     

Exceptional curves on smooth rational surfaces with not nef and of self-intersection zero

A -curve is a smooth rational curve of self-intersection , where is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has -curves. In this paper we prove that for such a surface , the set of -curves on is finite but non-empty, and that may have no -curves. Related facts are also considered.

     

Filling analytic sets by the derivatives of -smooth bumps

If is an infinite-dimensional Banach space, with separable dual, and is an analytic set such that any point can be reached from  by a continuous path contained (except for the point ) in the interior of , then is the range of the derivative of a -smooth function on  with bounded nonempty support.

     

Semi-continuity of metric projections in -direct sums

Let be a proximinal subspace of finite codimension of . We show that is proximinal in and the metric projection from onto is Hausdorff metric continuous. In particular, this implies that the metric projection from onto is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.

     

Tietze extension theorem for Hilbert -modules

We prove the following generalization of the noncommutative Tietze extension theorem: if is a countably generated Hilbert -module over a -unital -algebra, then the canonical extension of a surjective morphism of Hilbert -modules to extended (multiplier) modules, , is also surjective.

     

On restricted weak type : The continuous case

Let denote the unit circle. An example of a sublinear translation-invariant operator acting on is given such that is of restricted weak type but not of weak type .

     

Measures of concordance determined by -invariant measures on

A measure, , on is said to be -invariant if its value for any Borel set is invariant with respect to the symmetries of the unit square. A function, , generated in a certain way by a measure, , on is shown to be a measure of concordance if and only if the generating measure is positive, regular, -invariant, and satisfies certain inequalities. The construction examined here includes Blomqvist's beta as a special case.

     

Real

The famous problem involves applying two maps: and to positive integers. If is even, one applies , if it is odd, one applies . The conjecture states that each trajectory of the system arrives to the periodic orbit . In this paper, instead of choosing each time which map to apply, we allow ourselves more freedom and apply both and independently of . That is, we consider the action of the free semigroup with generators and on the space of positive real numbers. We prove that this action is minimal (each trajectory is dense) and that the periodic points are dense. Moreover, we give a full characterization of the group of transformations of the real line generated by and .

     

New properties of Mrowka's space

We extend the technique of Mrowka to show that his space has the property that dim while ind , assuming his extra set-theoretic hypothesis. We also show that is compact, so assuming the extra axiom, there is an compact metric space with no compact completion.

     

More on convexity numbers of closed sets in

The convexity number of a set is the least size of a family of convex sets with . is countably convex if its convexity number is countable. Otherwise is uncountably convex.

Uncountably convex closed sets in have been studied recently by Geschke, Kubis, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all , it is consistent that there is an uncountably convex closed set whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of .

Moreover, we construct a closed set whose convexity number is and that has no uncountable -clique for any 1$">. Here is a -clique if the convex hull of no -element subset of is included in . Our example shows that the main result of the above-named authors, a closed set either has a perfect -clique or the convexity number of is in some forcing extension of the universe, cannot be extended to higher dimensions.

     

Properly -realizable groups

A finitely presented group is said to be properly -realizable if there exists a compact -polyhedron with and whose universal cover has the proper homotopy type of a (p.l.) -manifold with boundary. In this paper we show that, after taking wedge with a -sphere, this property does not depend on the choice of the compact -polyhedron with . We also show that (i) all -ended and -ended groups are properly -realizable, and (ii) the class of properly -realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that -ended groups are properly -realizable, assuming -ended groups are).

     

Spectrally bounded -derivations on Banach algebras

Applying the density theorem on algebras with -derivations, we show that if a -derivation of a unital Banach algebra is spectrally bounded, then . Also, if and only if , where denotes the spectral radius of .

     

Irreducibility of the -classes on smooth rational surfaces

We give a characterization for a -divisor on a smooth rational surface to be irreducible under the assumption that an anticanonical divisor of is nef. Here is nef means for every effective divisor on , and a -divisor is a divisor such that the two numerical conditions hold.

As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of -curves. A -curve is a smooth rational curve of self-intersection .

     

Transitive families of projections in factors of type

We introduce a notion of transitive family of subspaces relative to a type factor, and hence a notion of transitive family of projections in such a factor. We show that whenever is a factor of type and is generated by two self-adjoint elements, then contains a transitive family of projections. Finally, we exhibit a free transitive family of projections that generate a factor of type .

     

-additive families and the invariance of Borel classes

We prove that any -additive family of sets in an absolutely Souslin metric space has a -discrete refinement provided every partial selector set for is -discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps -sets to -sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves -sets, is shown as an application of the previous result.

     

Self-adjointness of the perturbed wave operator on

We give classes of unbounded real-valued for which is self-adjoint on , , where is the wave operator defined on .

     

On the algebra of functions -extendable for each finite

For each positive integer we construct a -function of one real variable, the graph of which has the following property: there exists a real function on which is -extendable to , for each finite, but it is not -extendable.

     

Discrete spectra of -algebras and orthogonally closed submodules in Hilbert -modules

Let and be -algebras and let be an --imprimitivity bimodule. Then it is shown that if the spectrum of (resp. of ) is discrete, then every closed --submodule of is orthogonally closed in , and conversely that if (resp. ) is a -space and if every closed --submodule of is orthogonally closed in , then (resp. ) is discrete.

     

A new representation of the Dedekind completion of -spaces

A new representation of the Dedekind completion of is given. We present a necessary and sufficient condition on a compact Hausdorff space for which the Dedekind completion of is , the space of real valued bounded functions on some set .

     

Real -flats tangent to quadrics in

Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.