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.

     

-spaces and finite unions

This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) ``nice" subspaces? Our approach is based on the notion of a -space introduced by E. van Douwen and on a generalization of this notion, the notion of -space. It is proved that if a space is the union of a finite family of subparacompact subspaces, then is an -space. Under , it follows that if a separable normal -space is the union of a finite number of subparacompact subspaces, then is Lindelöf. It is also established that if a regular space is the union of a finite family of subspaces with a point-countable base, then is a -space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.

     

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 .

     

Covering by graphs of -ary functions and long linear orderings of Turing degrees

A point is covered by a function iff there is a permutation of such that .

By a theorem of Kuratowski, for every infinite cardinal exactly -ary functions are needed to cover all of . We show that for arbitrarily large uncountable it is consistent that the size of the continuum is and is covered by -ary continuous functions.

We study other cardinal invariants of the -ideal on generated by continuous -ary functions and finally relate the question of how many continuous functions are necessary to cover to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.

     

Simple corona -algebras

Let be a non-unital and -unital simple -algebra. We show that if is simple, then is purely infinite. We also show that is simple if and only if has a continuous scale provided that is not isomorphic to the compact operators.

     

Characterizations of elements with compact support in the dual spaces of -modules of

For a locally compact group and , let be the Figà-Talamanca-Herz algebra and let be its dual Banach space. For a Banach -module of , we denote the norm closure of the subspace of the elements in with compact support by . We prove that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with and . In particular, we have that an element of is in if and only if for any 0$">, there exists a compact subset of such that for all with . If has an orthogonal complement in , we characterize by the following condition: is in if and only if for any 0$"> and any compact subset of , there exists some with and such that \Vert u\Vert - \epsilon $">. Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking and as some -subalgebras of .

     

On embeddings of full amalgamated free product C-algebras

We examine the question of when the -homomorphism of full amalgamated free product C-algebras, arising from compatible inclusions of C-algebras , and , is an embedding. Results giving sufficient conditions for to be injective, as well as classes of examples where fails to be injective, are obtained. As an application, we give necessary and sufficient conditions for the full amalgamated free product of finite-dimensional C-algebras to be residually finite dimensional.

     

Stability of -algebras associated to graphs

We characterize stability of graph -algebras by giving five conditions equivalent to their stability. We also show that if is a graph with no sources, then is stable if and only if each vertex in can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph -algebra. Specifically, if is a graph and is the graph formed by adding a head to each vertex of , then is the stabilization of ; that is, .

     

Distinguished representations and poles of twisted tensor -functions

Let be a quadratic extension of -adic fields. If is an admissible representation of that is parabolically induced from discrete series representations, then we prove that the space of -invariant linear functionals on has dimension one, where is the mirabolic subgroup. As a corollary, it is deduced that if is distinguished by , then the twisted tensor -function associated to has a pole at . It follows that if is a discrete series representation, then at most one of the representations and is distinguished, where is an extension of the local class field theory character associated to . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

     

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.