Let F be a finite group. We consider the lamplighter group L=F≀Z over F. We prove that L has a classifying space for proper actions normalE̲L which is a complex of dimension 2. We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that the assembly map μiL:KiLfalse(E̲0.16emLfalse)→Kifalse(C∗Lfalse)false(i=0,1false) is an isomorphism. Actually, K0false(C∗Lfalse) is free abelian of countable rank, with an explicit basis consisting of projections in C∗L, while K1false(C∗Lfalse) is infinite cyclic, generated by the unitary of C∗L implementing the shift. Finally we show that, for F abelian, the C∗‐algebra C∗L is completely characterized by |F| up to isomorphism.

We provide a new computation of the K-theory of the group C * -algebra of the solvable Baumslag-Solitar group BS(1, n) (n = 1); our computation is based on the Pimsner-Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product Z[1/n] Z. We deduce from it a new proof of the Baum-Connes conjecture with trivial coefficients for BS(1, n).

Given p ∈ (1, ∞), let G be a countable Powers group, and let (G, A, α) be a separable nondegenerately representable isometric G-L p operator algebra. We show that if A is G-simple and unital then the reduced L p operator crossed product of A by G, F p r (G, A, α), is simple. This generalizes special cases of some results due to de la Harpe and Skanadalis in the C * -algebra context. We will also show that the result is not true for p = 1. Moreover, we prove that traces on F p r (G, A, α) are in natural bijection with G-invariant traces on A via the standard conditional expectation. As a consequence, for a countable powers group G the reduced L p operator group algebra, F p r (G), is simple and has a unique normalized trace.2010 Mathematics Subject Classification. Primary 46H05; Secondary 46H35, 47L10.

In the literature one may encounter certain infinite tridiagonal matrices, the principal minors of which, constitute the Fibonacci or Lucas sequence. The major purpose of this article is to find new infinite matrices with this property. It is interesting to mention that the matrices found are not tridiagonal which have been investigated before. Furthermore, we introduce the sequences composed of Fibonacci and Lucas k-numbers for the positive integer k and we construct the infinite matrices the principal minors of which generate these sequences.

Abstract. The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see R. Bacher. Determinants of matrices related to the Pascal triangle. J. Théor. Nombres Bordeaux, 14:19-41, 2002). This article presents a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Toeplitz matrix, and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant of a Toeplitz matrix. This equality allows for the evaluation of a few determinants of generalized Pascal matrices associated with certain sequences. In particular, families of quasi-Pascal matrices are obtained whose leading principal minors generate any arbitrary linear subsequences (F nr+s ) n≥1 or (L nr+s ) n≥1 of the Fibonacci or Lucas sequence. New matrices are constructed whose entries are given by certain linear non-homogeneous recurrence relations, and the leading principal minors of which form the Fibonacci sequence.