Borel classes of sets of extreme and exposed points in

It is known that the sets of extreme and exposed points of a convex Borel subset of are Borel. We show that for there exist convex subsets of such that the sets of their extreme and exposed points coincide and are of arbitrarily high Borel class. On the other hand, we show that the sets of extreme and of exposed points of a convex set of additive Borel class are of ambiguous Borel class . For proving the latter-mentioned results we show that the union of the open and the union of the closed segments of are of the additive Borel class if is a convex set of additive Borel class .

     

Relations approximated by continuous functions

Let be a Tychonoff space, let be the space of all continuous real-valued functions defined on and let be the hyperspace of all nonempty closed subsets of . We prove the following result. Let be a locally connected, countably paracompact, normal -space without isolated points, and let . Then is in the closure of in with the locally finite topology if and only if is the graph of a cusco map. Some results concerning an approximation in the Vietoris topology are also given.

     

Remarks concerning linear characters of reflection groups

Let be a finite group generated by unitary reflections in a Hermitian space , and let be a root of unity. Let be a subspace of , maximal with respect to the property of being a -eigenspace of an element of , and let be the parabolic subgroup of elements fixing pointwise. If is any linear character of , we give a condition for the restriction of to to be trivial in terms of the invariant theory of , and give a formula for the polynomial , where is the dimension of the -eigenspace of . Applications include criteria for regularity, and new connections between the invariant theory and the structure of .

     

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 variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

Spectra of upper triangular operator matrices

Let be given Banach spaces. For and , let be the operator defined on by . We give sufficient conditions on to get where runs over a large class of spectra. We also discuss the case of some spectra for which the latter equality fails.

     

The linear space of generalized Brownian motions with applications

In this paper, we define, motivated by recent works of Chang and Skoug, stochastic integrals for a generalized Brownian motion ( ) and then use it to study the representation problem on the linear space spanned by . We next establish a translation theorem for -functionals of , , and then use this translation to establish an integration by parts formula for -functionals of .

     

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 .

     

The classical monotone convergence theorem of Beppo Levi fails in noncommutative -spaces

Let be a complex Hilbert space and let be a von Neumann algebra over equipped with a faithful, normal state . Then is a prehilbert space with respect to the inner product , whose completion is given by the Gelfand-Naimark-Segal representation theorem, according to which there exist a one-to-one -homomorphism of into the algebra of all bounded linear operators acting on and a cyclic, separating vector such that for all . Given any separable Hilbert space , we construct a faithful, normal state on and an increasing sequence of positive operators acting on such that is bounded, but fails to converge both bundlewise and in -norm. We also present an example of an increasing sequence of positive operators which has a subsequence converging both bundlewise and in -norm, but the whole sequence fails to converge in either sense. Finally, we observe that our results are linked to a previous one by R. V. Kadison.

     

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 with -power character degrees

Let be a -block of a finite group . If is a -power for all , then is nilpotent.

     

Perturbing a product of stable flows

Suppose that and are axiom A flows with attractors and . Then the attractor for the product flow on the product manifold is no longer hyperbolic (although there is a hyperbolic action of ).

It is easy to see that the attractor cannot explode but we show here that it cannot implode: for any flow sufficiently close to any attractor whose basin is not too thin is -dense in .

     

On decompositions of Banach spaces of continuous functions on Mrówka's spaces

It is well known that if is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space of continuous functions on has complemented copies of , i.e., . We address the question if this could be the only type of decompositions of into infinite-dimensional summands for infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrówka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.

     

Hopf algebras of dimension

Let be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If is not semisimple and for some odd integer , then or is not unimodular. Using this result, we prove that if for some odd prime , then is semisimple. This completes the classification of Hopf algebras of dimension .

     

The content of a Gaussian polynomial is invertible

Let be an integral domain and let be a nonzero polynomial in . The content of is the ideal generated by the coefficients of . The polynomial is called Gaussian if for all . It is well known that if is an invertible ideal, then is Gaussian. In this note we prove the converse.

     

Strong pseudo-contractions perturbed by compact operators in Banach spaces

Let be a (real) Banach space, let be an open subset of , and let denote the collection of all nonempty bounded and closed subsets of . Suppose is continuous from into with respect to the Hausdorff metric and strongly pseudo-contractive, while is compact from into . Then has a fixed point if it satisfies the classical Leray-Schauder condition on the boundary of .

     

Countable dense homogeneity of definable spaces

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel the following are equivalent: (1) is in , (2) is CDH and (3) is homeomorphic to or to . Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr ns and Zhou, by showing that is not CDH.

     

The Artin-Stafford gap theorem

Let be an algebraically closed field, and let be a finitely graded -algebra which is a domain. We show that cannot have Gelfand-Kirillov dimension strictly between and .

     

Bounds for the index of the centre in capable groups

A group is called capable if it is isomorphic to for some group . Let be a capable group. I. M. Isaacs (2001) showed that if is finite, then the index of the centre is bounded above by some function of . We show that if , then with some constant and this bound is essentially best possible. We complete a result of Isaacs, showing that if is a cyclic group, then .

     

-actions with

Suppose that is a smooth -action on a closed smooth -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each connected component of the fixed point set vanish in positive dimension. This paper shows that if 2^k\dim F$"> and each -dimensional part possesses the linear independence property, then bounds equivariantly, and in particular, is the best possible upper bound of if is nonbounding.