Almost maximally almost-periodic group topologies determined by T-sequences

A sequence $\{a_n\}$ in a group $G$ is a {\em $T$-sequence} if there is a Hausdorff group topology $\tau$ on $G$ such that $a_n\stackrel\tau\longrightarrow 0$. In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a $T$-sequence, and investigate special sequences in the Pr\"ufer groups $\mathbb{Z}(p^\infty)$. We show that for $p\neq 2$, there is a Hausdorff group topology $\tau$ on $\mathbb{Z}(p^\infty)$ that is determined by a $T$-sequence, which is close to being maximally almost-periodic--in other words, the von Neumann radical $\mathbf{n}(\mathbb{Z}(p^\infty),\tau)$ is a non-trivial finite subgroup. In particular, $\mathbf{n}(\mathbf{n}(\mathbb{Z}(p^\infty),\tau)) \subsetneq \mathbf{n}(\mathbb{Z}(p^\infty),\tau)$. We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a $T$-sequence with non-trivial finite von Neumann radical.Comment: v2 - accepted (discussion on non-abelian case is removed, replaced by new results on direct sums of finite abelian groups

The Canada Day Theorem

The Canada Day Theorem is an identity involving sums of $k \times k$ minors of an arbitrary $n \times n$ symmetric matrix. It was discovered as a by-product of the work on so-called peakon solutions of an integrable nonlinear partial differential equation proposed by V. Novikov. Here we present another proof of this theorem, which explains the underlying mechanism in terms of the orbits of a certain abelian group action on the set of all $k$-edge matchings of the complete bipartite graph $K_{n,n}$.Comment: 16 pages. pdfLaTeX + AMS packages + TikZ. Fixed a hyperlink problem and a few typo

CLT for random walks of commuting endomorphisms on compact abelian groups

International audienceLet S be an abelian group of automorphisms of a probability space (X, A, µ) with a finite system of generators (A 1 , ..., A d). Let A ℓ denote A ℓ1 1 ...A ℓ d d , for ℓ = (ℓ 1 , ..., ℓ d). If (Z k) is a random walk on Z d , one can study the asymptotic distribu-tion of the sums n−1 k=0 f • A Z k (ω) and ℓ∈Z d P(Z n = ℓ) A ℓ f , for a function f on X. In particular, given a random walk on commuting matrices in SL(ρ, Z) or in M * (ρ, Z) acting on the torus T ρ , ρ ≥ 1, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on T ρ after normalization? In this paper, we prove a central limit theorem when X is a compact abelian con-nected group G endowed with its Haar measure (e.g. a torus or a connected extension of a torus), S a totally ergodic d-dimensional group of commuting algebraic automor-phisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on the spectral properties of the action of S, on random walks and on the variance of the associated ergodic sums

Semilattices of groups and nonstable K-theory of extended Cuntz limits

We give an elementary characterization of those abelian monoidsM that are direct limits of countable sequences of finite direct sums of monoids of the form either (Z/nZ) ⊔ {0} or Z ⊔ {0}. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of all subgroups of M (viewed as a semigroup), and it makes it pos- sible to express M as a certain submonoid of a direct product ×G, where is a distributive semilattice with zero and G is an abelian group. When applied to the monoids V (A) appearing in the nonstable K-theory of C*-algebras, our results yield a full description of V (A) for C*-inductive limits A of finite sums of full matrix algebras over either Cuntz algebras On, where 2 ≤ n < ∞, or corners of O1 by projections, thus extending to the case including O1 earlier work by the authors together with K.R. Goodearl