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.

     

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.

     

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.

     

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.

     

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.

     

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 .

     

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).

     

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 .

     

Finite -arc transitive Cayley graphs and flag-transitive projective planes

In this paper, a characterisation is given of finite -arc transitive Cayley graphs with . In particular, it is shown that, for any given integer with and , there exists a finite set (maybe empty) of -transitive Cayley graphs with such that all -transitive Cayley graphs of valency are their normal covers. This indicates that -arc transitive Cayley graphs with are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

     

The number of planar central configurations for the -body problem is finite when mass positions are fixed

In the -body problem a central configuration is formed when the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. Lindstrom showed for and for 4$"> that if masses are located at fixed points in the plane, then there are only a finite number of ways to position the remaining th mass in such a way that they define a central configuration. Lindstrom leaves open the case . In this paper we prove the case using as variables the mutual distances between the particles.

     

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 .

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

The vanishing of implies that is regular

Let be an excellent local ring of positive prime characteristic. We show that if , then is regular. This improves a result of Schoutens, in which the additional hypothesis that was an isolated singularity was required for the proof.

     

Examples of non-formal closed -connected manifolds of dimensions

We construct closed -connected manifolds of dimensions that possess non-trivial rational Massey triple products. We also construct examples of manifolds such that all the cup-products of elements of vanish, while the group is generated by Massey products: such examples are useful for the theory of systols.

     

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 .

     

-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.

     

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.