k-Sums in abelian groups

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.Comment: 15 page

Some new results about a conjecture by Brian Alspach

In this paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\mathbb{Z}_n\setminus \{0\}$ of size $k$ such that $\sum_{z\in A} z\not= 0$, it is possible to find an ordering $(a_1,\ldots,a_k)$ of the elements of $A$ such that the partial sums $s_i=\sum_{j=1}^i a_j$, $i=1,\ldots,k$, are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $k\leq 11$ in cyclic groups of prime order. Here, we extend such result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $\mathbb{Z}_n$. We also consider a related conjecture, originally proposed by Ronald Graham: given a subset $A$ of $\mathbb{Z}_p\setminus\{0\}$, where $p$ is a prime, there exists an ordering of the elements of $A$ such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis and Schmitt, based on the Alon's combinatorial Nullstellensatz, we prove the validity of such conjecture for subsets $A$ of size $12$

Semilattices of groups and inductive limits of Cuntz algebras

We characterize, in terms of elementary properties, the abelian monoids which are direct limits of finite direct sums of monoids of the form ðZ=nZÞ t f0g (where 0 is a new zero element), for positive integers n. The key properties are the Riesz refinement property and the requirement that each element x has finite order, that is, ðn þ 1Þx ¼ x for some positive integer n. Such monoids are necessarily semilattices of abelian groups, and part of our approach yields a characterization of the Riesz refinement property among semilattices of abelian groups. Further, we describe the monoids in question as certain submonoids of direct products L G for semilattices L and torsion abelian groups G. When applied to the monoids VðAÞ appearing in the non-stable K-theory of C*-algebras, our results yield characterizations of the monoids VðAÞ for C* inductive limits A of sequences of finite direct products of matrix algebras over Cuntz algebras On. In particular, this completely solves the problem of determining the range of the invariant in the unital case of Rørdam’s classification of inductive limits of the above type

Hecke characters and the KK-theory of totally real and CM number fields

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for $F$ with infinite type equal to a special value of certain $G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l> 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberger-splitting maps in the $l$-primary Quillen localization sequence for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$. We also apply the Stickelberger-splitting maps to construct special elements in the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$, for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general totally real number field, we study the cyclicity of $D(n)_l$ in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if $F$ is CM, special values of our $l$-adic Hecke characters are used to construct Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler system of Gauss sums and of the $K$-theoretic Euler systems constructed in Banaszak-Gajda when $K = \Bbb Q$.Comment: 38 page

On strongly reflexive topological groups

Let Gˆ denote the Pontryagin dual of an abelian topological group G. Then G is reflexive if it is topologically isomorphic to Gˆˆ, strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of Gˆ is reflexive. It is well known that locally compact abelian (LCA) groups are strongly reflexive. W. Banaszczyk [Colloq. Math. 59 (1990), no. 1, 53–57], extending an earlier result of R. Brown, P. J. Higgins and S. A. Morris [Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32], showed that all countable products and sums of LCA groups are strongly reflexive. L. Aussenhofer [Dissertationes Math. (Rozprawy Mat.) 384 (1999), 113 pp.] showed that all Čech-complete nuclear groups are strongly reflexive. It is an open question whether the strongly reflexive groups are exactly the Čech-complete nuclear groups and their duals. A Hausdorff topological group G is almost metrizable if and only if it has a compact subgroup K such that G/K is metrizable [W. Roelcke and S. Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill, New York, 1981]. In this paper it is shown that the annihilator of a closed subgroup of an almost metrizable group G is topologically isomorphic to the dual of the corresponding Hausdorff quotient, and an analogous statement holds for the character group of G. It then follows that an almost metrizable group is strongly reflexive only if its Hausdorff quotients and those of its dual are reflexive