In 1967, Powers [7] proved that there exists a one-parameter family of pairwise non-isomorphic hyperfinite factors of type III. Powers′ result on hyperfinite factors has been extended by Araki and Woods [1]. Connes [4], and Williams [11], with different proofs, showed that there exists a continuous family of mutually non-isomorphic non-hyperfinite factors of type III.

Let (X, d) be a metric space and Y and Z subsets of X. We say that Z is a bisector in Y and write Y⊳Z iff Y⊃Z and there are two distinct points y1, y2 ∈ Y such that Z = ={z:d(z, y1) = d(z, y2) and z∈Y}. By a reduced bisector chain in (X, d) of length n we understand a chain X = such that dim Xn≤0 and dimXn-1>0). By r(X, d) we denote the maximum length of reduced bisector chains in (X, d). For a metrizable topological space X we introduce the topological invariant r(X) as the minimum of r(X, d) taken over the set of all metrizations d of X. We prove that the function r(X) coincides with the dimension of X on the class of compact metric spaces.

Le théorème de Weierstrass affirme que toute fonction réelle continue sur l'intervalle [0, 1] est limite uniforme d'une suite de polynômes (voir [5]). Dans [4], Ch. Müntz généralise ce résultat de la façon suivante.

Soit α1, α2, α3,… une suite strictement croissante et non bornée de nombres réels.

Every ⊥-symmetric relatively semi-orthocomplemented lattice is M-symmetric. This answers the Problem 1 in [2] in the affirmative and provides a new proof to a result on ⊥-symmetric lattices proved in [2] (Corollary below). The notation and terminology are as in [2].

Let 〈L; ⋀, V〉 be a lattice. Two elements a and b of L are said to form a modular pair, in symbols aMb, if

The relation aM✶b is defined dually.

We show that a locally compact semigroup S is topological left amenable iff a certain space of left uniformly continuous functions on the convolution semigroup of probability measures M0(S) on S is left amenable or equivalently iff the convolution semigroup M0(S) has the fixed point property for uniformly continuous affine actions on compact convex sets.

A random map is a discrete time process in which one of a number of maps, 𝓜, is chosen at random at each stage and applied. In this note we study a random map, where 𝓜 is a set of piecewise linear Markov maps on [0, 1]. Sufficient conditions are presented which allow the determination of the unique absolutely continuous invariant measure of the process.

A lower bound for a family of Ramsey numbers is derived using a geometrical argument.

As is well known, any module M over a ring possesses injective hulls, i. e., injective essential extensions [3], unique up to isomorphisms which map M identically, but the various proofs for this all require some transfinite arguments and hence provide little indication as to how these hulls may actually be constructed for a given module.

Let λ be a map from the additive Euclidean n-group Rn into the space R of real numbers and define a multiplication * on Rn by v * w = (λ(w))v. Then (Rn, + , *) is a topological nearring if and only if λ is continuous and λ(av) = aλ(v) for every v € Rn and every a in the range of λ. For any such map λ and any topological space X we denote by Nλ(X, Rn) the nearring of all continuous functions from X into (Rn, +, *) where the operations are pointwise. The ideals of Nλ(X, Rn) are investigated in some detail for certain λ and the results obtained are used to prove that two compact Hausdorff spaces X and Y are homeomorphic if and only if the nearrings Nλ(X, Rn) and Nλ(Y, Rn) are isomorphic.

We construct the quantum $s$-tuple subfactors for an AFD $\text{I}{{\text{I}}_{1}}$ subfactor with finite index and depth, for an arbitrary natural number $s$. This is a generalization of the quantum multiple subfactors by Erlijman and Wenzl, which in turn generalized the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.

A bounded basis {xn} of a Banach space X is called perfectly homogeneous if every bounded block basic sequence {yn} of {xn} is equivalent to {xn}. By a result of M. Zippin [4], a basis in a Banach space is perfectly homogeneous if and only if it is equivalent to the unit vector basis of c0 or lp, 1 ≤ p < + ∞. A basis {xn} of a Banach space X is called symmetric, if every permutation {xσ(n)} of {xn} is a basis of X, equivalent to the basis {xn}. It is clear that every perfectly homogeneous basis is symmetric.

In this paper, we find a lower bound on the number of cyclic function fields of prime degree $l$ whose class numbers are divisible by a given integer $n$. This generalizes a previous result of D. Cardon and R. Murty which gives a lower bound on the number of quadratic function fields with class numbers divisible by $n$.

Let $R$ be a ring with 1 and $P(R)$ the periodic radical of $R$. We obtain necessary and sufficient conditions for $P(\text{RG})=0$ when $\text{FG}$ is the group ring of an $\text{FC}$ group $G$ and $R$ is commutative. We also obtain a complete description of $P\left( I(X,R) \right)$ when $I(X,R)$ is the incidence algebra of a locally finite partially ordered set $X$ and $R$ is commutative.

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti–Tate group. We extend this result to 1-motives.

Let (𝓜,g) be a Riemannian manifold and let 1, 2, 3 be mutually orthogonal distributions on 𝓜 of dimensions p1, p2,P3 such that p1 + p2 + p3 = dim 𝓜. We assume that , and 1, ⊕ 2 are integrable and that all the geodesies of 𝓜 with initial tangent vector in 2 remain tangent to 2. Then, we prove that Pontk(2, ⊕ 3) = 0 for k > p2 + 2p3, where Pontk(2, ⊕ 3) is the subspace of the Pontrjagin algebra of 2 ⊕ 3 generated by forms of degree k.

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {Um}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m0 = m0(ɛ, K) such that

A theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

It is shown that in a dense-in-itself Hausdorff space if every set having a dense interior is open, then every compact set is finite.

This paper deals with the relative trace formula in the case of base change. Two truncations of the kernel are introduced, both based on the ideas of Arthur, and their integrals are shown to be asymptotic to each other. We also consider products of the kernel with automorphic forms, as these appear when comparing trace formulae (see [5]).