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 frames for countably generated Hilbert -modules

Let be a countably generated Hilbert -module over a -algebra We prove that a sequence is a standard frame for if and only if the sum converges in norm for every and if there are constants such that for every We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert -modules over the -algebra of all compact operators on some Hilbert space is discussed.

     

The number of Hall -subgroups of a -separable group

We observe a simple formula to compute the number of Hall -subgroups of a -separable finite group in terms of only the action of a fixed Hall -subgroup of on a set of normal -sections of . As a consequence, we obtain that divides whenever is a subgroup of a finite -separable group . This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro's result.

     

Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

On quasi-affine transforms of Read's operator

We show that C. J. Read's example of an operator on which does not have any non-trivial invariant subspaces is not the adjoint of an operator on a predual of . Furthermore, we present a bounded diagonal operator such that even though is unbounded, the operator is a bounded operator on with invariant subspaces, and is adjoint to an operator on .

     

Small volume on big -spheres

We consider Riemannian metrics on the -sphere for such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the -dimensional case where Berger has shown that .

     

Perfect cliques and colorings of Polish spaces

A coloring of a set is any subset of , where 1$"> is a natural number. We give some sufficient conditions for the existence of a perfect -homogeneous set, in the case where is and is a Polish space. In particular, we show that it is sufficient that there exist -homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect -clique if it contains any uncountable -clique, where is a natural number or (a set is a -clique in if the convex hull of any of its -element subsets is not contained in ).

     

Weak properties of weighted convolution algebras

Suppose that is a weighted convolution algebra on with the weight normalized so that the corresponding space of measures is the dual space of the space of continuous functions. Suppose that is a continuous nonzero homomorphism, where is also a convolution algebra. If is norm dense in , we show that is (relatively) weak dense in , and we identify the norm closure of with the convergence set for a particular semigroup. When is weak continuous it is enough for to be weak dense in . We also give sufficient conditions and characterizations of weak continuity of . In addition, we show that, for all nonzero in , the sequence converges weak to 0. When is regulated, converges to 0 in norm.

     

Multiplicative bijections of

Let be a compact Hausdorff space which satisfies the first axiom of countability, let and let , be the set of all continuous functions from to If , ,is a bijective multiplicative map, then there exist a homeomorphism and a continuous map such that for all and for all

     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

On -supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let be a saturated formation containing . Suppose that is a group with a normal subgroup such that .

(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .

     

Discrete spectra of -algebras and complemented submodules in Hilbert -modules

Let be a -algebra and let be a full (right) Hilbert -module. It is shown that if the spectrum of is discrete, then every closed --submodule of is complemented in , and conversely that if is a -space and if every closed --submodule of is complemented in , then is discrete.

     

Modular group algebras of -separable -groups

Under the assumptions of MA and CH, it is proved that if is a field of prime characteristic and is an -separable abelian -group of cardinality , then an isomorphism of the group algebras and implies an isomorphism of and .

     

Uniqueness of structures for connective covers

We refine our earlier work on the existence and uniqueness of structures on -theoretic spectra to show that the connective versions of real and complex -theory as well as the connective Adams summand at each prime have unique structures as commutative -algebras. For the -completion we show that the McClure-Staffeldt model for is equivalent as an ring spectrum to the connective cover of the periodic Adams summand . We establish a Bousfield equivalence between the connective cover of the Lubin-Tate spectrum and .

     

-injective rings and -stable primes

The notion of stability of the highest local cohomology module with respect to the Frobenius functor originates in the work of R. Hartshorne and R. Speiser. R. Fedder and K.-i. Watanabe examined this concept for isolated singularities by relating it to -rationality. The purpose of this note is to study what happens in the case of non-isolated singularities and to show how this stability concept encapsulates a few of the subtleties of tight closure theory. Our study can be seen as a generalization of the work by Fedder and Watanabe. We introduce two new ring invariants, the -stability number and the set of -stable primes. We associate to every ideal generated by a system of parameters and an ideal of multipliers denoted and obtain a family of ideals . The set is independent of and consists of finitely many prime ideals. It also equals prime ideal such that is -stable. The maximal height of such primes defines the -stability number.

     

The -exponent of the -local spectrum

Let be a fixed odd prime. In this paper we prove an exponent conjecture of Bousfield, namely that the -exponent of the spectrum is for . It follows from this result that the -exponent of is at least for and , where denotes the -connected cover of .

     

Pseudocompact spaces and -spaces

Let be a completely regular Hausdorff space, and let be the space of continuous real-valued functions on endowed with the compact-open topology. We find various equivalent conditions for to be a -space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for , Grothendieck's -spaces do not coincide with Jarchow's -spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for , our original motivation.

     

-function of an operator: A white noise approach

Let be the canonical framework of white noise analysis over the Gel'fand triple and be the space of continuous linear operators from to . Let be a self-adjoint operator in with spectral representation . In this paper, it is proved that under appropriate conditions upon , there exists a unique linear mapping such that for each . The mapping is then naturally used to define as , where is the Dirac -function. Finally, properties of the mapping are investigated and several results are obtained.

     

Une propriété de continuité du temps local

Let denote the local time (at 0) associated with a martingale . The aim of this note is to prove that the mapping is continuous from into weak-.