A reciprocity law for certain Frobenius extensions

Let be a finite Galois extension of algebraic number fields with Galois group . Assume that is a Frobenius group and is a Frobenius complement of . Let be the maximal normal nilpotent subgroup of . If is nilpotent, then every Artin L-function attached to an irreducible representation of arises from an automorphic representation over , i.e., the Langlands' reciprocity conjecture is true for such Galois extensions.

     

The range of a ring homomorphism from a commutative -algebra

We prove that if a commutative semi-simple Banach algebra is the range of a ring homomorphism from a commutative -algebra, then is -equivalent, i.e. there are a commutative -algebra and a bicontinuous algebra isomorphism between and . In particular, it is shown that the group algebras , and the disc algebra are not ring homomorphic images of -algebras.

     

On contravariant finiteness of subcategories of modules of projective dimension

Let be an artin algebra. This paper presents a sufficient condition for the subcategory of to be contravariantly finite in , where is the subcategory of consisting of --modules of projective dimension less than or equal to . As an application of this condition it is shown that is contravariantly finite in for each when is stably equivalent to a hereditary algebra.

     

Asymptotic behaviour of certain sets of prime ideals

Let be ideals of the commutative ring , let be a Noetherian -module and let be a submodule of ; also let be an Artinian -module and let be a submodule of . It is shown that, whenever is a sequence of -tuples of non-negative integers which is non-decreasing in the sense that for all and all , then Ass is independent of for all large , and also Att is independent of for all large . These results are proved without any regularity conditions on the ideals , and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.

     

Generalized rational identities of subnormal subgroups of skew fields

Let be a skew field with infinite center such that , and let be a non-central subnormal subgroup of the multiplicative group of . Then there are no non-trivial generalized rational identities of . This generalizes a theorem proved by Makar-Limanov.

     

Three counterexamples for a question concerning Green's functions and circular symmetrization

We construct domains in the plane such that if is the Green's function of with pole at zero, while is the symmetric non-increasing rearrangement of for each fixed and is the Green's function of the circular symmetrization , again with pole at zero, then there are positive numbers and such that

whenever . One of our constructions will have simply connected. We also consider the case where the poles of the Green's functions do not lie at the origin. Our work provides a negative answer to a question of Hayman (1967).

     

Normsets and determination of unique factorization in rings of algebraic integers

The image of the norm map from to (two rings of algebraic integers) is a multiplicative monoid . We present conditions under which is a UFD if and only if has unique factorization into irreducible elements. From this we derive a bound for checking if is a UFD.

     

Almost disjoint permutation groups

A permutation group on a set of (infinite) cardinality is almost disjoint if no element of except the identity has fixed points, i.e., if is an almost disjoint family of subsets of . We show how almost disjoint permutation groups can be constructed from almost disjoint families of sets.

     

Hereditarily aspherical compacta

The notion of (strongly) hereditarily aspherical compacta introduced by Daverman (1991) is modified. The main results are: Theorem. If is a hereditarily aspherical compactum, then ANR. In particular, is strongly hereditarily aspherical.

Theorem. Suppose is a cell-like map of compacta and is shape aspherical for each closed subset of . Then

1.

Y is hereditarily shape aspherical,

2.

is a hereditary shape equivalence,

3.

.

Theorem. Suppose is a group containing integers. Then the following conditions are equivalent:

1.

and ,

2.

.

Theorem. Suppose is a group containing integers. If and , then is hereditarily shape aspherical.

Theorem. Let be a two-dimensional, locally connected and semilocally simply connected compactum. Then, for any compactum

     

On a pattern of reflexive operator spaces

A linear subspace is a separating subspace for an operator space if the only member of annihilating is 0. It is proved in this paper that if has a strictly separating vector and a separating subspace satisfying , then is reflexive. Applying this to finite dimensional leads to more results on reflexivity. For example, if dim , and every nonzero operator in has rank , then is reflexive.

     

Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

On the perturbation theory of -accretive operators in Banach spaces

Let be a real Banach space. Let be -accretive with compact. Let be such that is condensing for some Let and assume that there exists a bounded open set and such that is bounded and

for all Then A basic homotopy result of the degree theory for with condensing and possibly unbounded, is used to improve and/or extend recent results by Hirano and Kalinde.

     

On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any separable unital -subalgebra of ,

In this note, we exhibit a non-separable unital -subalgebra of for which (VDCT) fails.

     

The second iterate of a map with dense orbit

Suppose that is a Hausdorff topological space having no isolated points and that is continuous. We show that if the orbit of a point under is dense in while the orbit of under is not, then the space decomposes into three sets relative to which the dynamics of are easy to describe. This decomposition has the following consequence: suppose that has dense orbit under and that the closure of the set of points of having odd period under has nonempty interior; then has dense orbit under .

     

On purely inseparable extensions and their generators

Let be a field of characteristic and a polynomial algebra in two variables. By a -generator of we mean an element of for which there exist and such that . We also define a -line of to mean any element of whose coordinate ring is that of a -generator. Then we prove that if is such that is a -line of (where is an indeterminate over ), then is a -generator of . This is analogous to the well-known fact that if is such that is a line of , then is a variable of . We also prove that if is a -line of for which there exist and such that , then is in fact a -generator of .

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

Integer-valued polynomials on Krull rings

If is a subring of a Krull ring such that is a valuation ring for every finite index , in Spec, we construct polynomials that map into the maximal possible (for a monic polynomial of fixed degree) power of , for all in Spec simultaneously. This gives a direct sum decomposition of Int, the -module of polynomials with coefficients in the quotient field of that map into , and a criterion when Int has a regular basis (one consisting of 1 polynomial of each non-negative degree).

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .