Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered

We establish the following result.

Theorem. Let be a integrable bounded group representation whose Arveson spectrum is scattered. Then the subspace generated by all eigenvectors of the dual representation is dense in Moreover, the closed subalgebra generated by the operators () is semisimple.

If, in addition, does not contain any copy of then the subspace spanned by all eigenvectors of is dense in Hence, the representation is almost periodic whenever it is strongly continuous.

     

On covering multiplicity

Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

     

Catenarity in module-finite algebras

The main theorem says that any module-finite (but not necessarily commutative) algebra over a commutative Noetherian universally catenary ring is catenary. Hence the ring is catenary if is Cohen-Macaulay. When is local and is a Cohen-Macaulay -module, we have that is a catenary ring, for any , and the equality holds true for any pair of prime ideals in and for any saturated chain of prime ideals between and .

     

Quadratic base change for as a theta correspondence I: Occurrence

The local theta correspondence is considered for reductive dual pairs where is a -adic field of characteristic zero and is the orthogonal group attached to a quaternary quadratic form with coefficients in and of Witt rank one over . It is shown that certain representations of occur in the correspondence.

     

Positive differentials, theta functions and Hardy kernels

Let be a planar regular region whose Schottky double has genus and set . For fixed we determine the range of the function where is the Riemann theta function on . Also we introduce two weighted Hardy spaces to study the problem when the matrix is positive definite. The proof relies on new theta identities using Fay's trisecants formula.

     

The Weierstrass approximation theorem and a characterization of the unit circle

We study real algebraic morphisms from nonsingular real algebraic varieties with into nonsingular real algebraic curves . We show, among other things, that the set of real algebraic morphisms from into is never dense in the space of all maps from into , unless is biregularly isomorphic to a Zariski open subset of the unit circle.

     

Relative Brauer groups of discrete valued fields

Let be a non-trivial finite Galois extension of a field . In this paper we investigate the role that valuation-theoretic properties of play in determining the non-triviality of the relative Brauer group, , of over . In particular, we show that when is finitely generated of transcendence degree 1 over a -adic field and is a prime dividing , then the following conditions are equivalent: (i) the -primary component, , is non-trivial, (ii) is infinite, and (iii) there exists a valuation of trivial on such that divides the order of the decomposition group of at .

     

Degenerations for modules over representation-finite algebras

Let be a representation-finite algebra. We show that a finite dimensional -module degenerates to another -module if and only if the inequalities hold for all -modules . We prove also that if for any indecomposable -module , then any degeneration of -modules is given by a chain of short exact sequences.

     

Open covers and partition relations

An open cover of a topological space is said to be an -cover if there is for each finite subset of the space a member of the cover which contains the finite set, but the space itself is not a member of the cover. We prove theorems which imply that a set of real numbers has Rothberger's property if, and only if, for each positive integer , for each -cover of , and for each function from the two-element subsets of , there is a subset of such that is constant on , and each element of belongs to infinitely many elements of (Theorem 1). A similar characterization is given of Menger's property for sets of real numbers (Theorem 6).

     

Rules and reals

A ``-rule" is a sequence of pairwise disjoint sets , each of cardinality and subsets . A subset (a ``real') follows a rule if for infinitely many , .

Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all -rules, , and the least number of -rules with no real that follows all of them, .

Call a bounded rule if is a -rule for some . Let be the least cardinality of a set of bounded rules with no real following all rules in the set.

We prove the following: and for all . However, in the Laver model, .

An application of is in Section 3: we show that below one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over . The consistency of such a family is still open.