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.

     

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.

     

Local existence of -sets, projective tensor products, and Arens regularity for

Theorem. If are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets such that (i) is either a Kronecker set or (ii) for some , is a translate of a -set all of whose elements have order , and (iii) is isomorphic to the projective tensor product .

This extends what was previously known for groups such as or for the case to the general locally compact abelian group. Old results concerning the local existence of Kronecker and -sets are improved.

     

-bounding and -induction

Working in the base theory of , we show that for all , the bounding principle for -formulas ( ) is equivalent to the induction principle for -formulas ( ). This partially answers a question of J. Paris.

     

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 .

     

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, .

     

-adic -basis and regular local ring

We introduce the concept of -adic -basis as an extension of the concept of -basis. Let be a regular local ring of prime characteristic and a ring such that . Then we prove that is a regular local ring if and only if there exists an -adic -basis of and is Noetherian.

     

Interpolation between and

We show that if is a rearrangement invariant space on that is an interpolation space between and and for which we have only a one-sided estimate of the Boyd index 1/p, 1 < p < \infty$">, then is an interpolation space between and . This gives a positive answer for a question posed by Semenov. We also present the one-sided interpolation theorem about operators of strong type and weak type .

     

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

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

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 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.

     

A prediction problem in

For a nonnegative integrable weight function on the unit circle , we provide an expression for , in terms of the series coefficients of the outer function of , for the weighted distance , where is the normalized Lebesgue measure and ranges over trigonometric polynomials with frequencies in , , . The problem is open for .

     

A concrete description of -spaces as -spaces and its applications

We prove that for a compact Hausdorff space without isolated points, and are isometrically Riesz isomorphic spaces under a certain topology on . Moreover, is a closed subspace of . This provides concrete examples of compact Hausdorff spaces such that the Dedekind completion of is (= the set of all bounded real-valued functions on ) since the Dedekind completion of is ( and spaces as Banach lattices).

     

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 .

     

Lightface -indescribable cardinals

-absoluteness for forcing means that for any forcing , . `` inaccessible to reals' means that for any real , . To measure the exact consistency strength of `` -absoluteness for forcing and is inaccessible to reals', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) -indescribable cardinal; has this property exactly if it is inaccessible and .

     

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.

     

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.

     

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.

     

Codes over and and Hermitian lattices over imaginary quadratic fields

We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over and , and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly modular lattice . We identify some of the hermitian lattices constructed and observe an interesting pair of nonisomorphic 3/2 dimensional codes over that give rise to isomorphic hermitian lattices when constructed at the lowest level 7 but nonisomorphic lattices at higher levels. The results show that the two alphabets and are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.