This paper shows that for the Moore spectrum MG associated with any abelian group G, and for any positive integer n, the order of the Postnikov k-invariant kn+1(MG) is equal to the exponent of the homotopy group πnMG. In the case of the sphere spectrum S, this implies that the exponents of the homotopy groups of S provide a universal estimate for the exponent of the kernel of the stable Hurewicz homomorphism hn: πnX → En(X) for the homology theory E*(—) corresponding to any connective ring spectrum E such that π0E is torsion-free and for any bounded below spectrum X. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism hn: En(X) → Hn(X; π0E), induced by the 0-th Postnikov section of E, is obtained for any connective spectrum E. An application of these results enables us to approximate in a universal way both kernel and cokernel of the unstable Hurewicz homomorphism between the algebraic K-theory of any ring and the ordinary integral homology of its linear group.

Introduction. Let K be an algebraically closed field, G = SL(3, K) the group of 3 × 3 matrices over K of determinant 1. Let denote the monoid of all 3 × 3 matrices over K. If e is an idempotent in , then

are opposite parabolic subgroups of G in the usual sense [1], [28]. However the map

does not exhaust all pairs of opposite parabolic subgroups of G. Now consider the representation ϕ:G → SL(6, K) given by

By a compactum, we mean a compact metric space. A continuum is a connected compactum. A curve is a 1-dimensional continuum. Let X be a continuum and let C(X) be the hyperspace of (nonempty) subcontinua of X, C(X) is metrized with the Hausdorff metric (e.g., see [12] or [18]). One of the most convenient tools in order to study the structure of C(X) is a monotone map ω:C(X) → [0, ω(X)] defined by H. Whitney [25]. A map ω:C(X) → [0, ω(X)] is said to be a Whitney map for C(X) provided that

The continua {ω−1} (0 < t < ω(X)) are called the Whitney continua of X. We may think of the map ω as measuring the size of a continuum. Note that ω−1(0) is homeomorphic to X and ω−1(ω(X)) = {X}. Naturally, we are interested in the structures of ω−1(t)(0 < t < ω(X)). In [14], J. Krasinkiewicz proved that if X is a circle-like continuum and ω is any Whitney map for C(X), then for any 0 < t < ω(X)ω−1(t) is shape equivalent to X, i.e., Sh ω−1(t) = Sh X (e.g., see [1] or [17]). In [8], we proved the following: If one of the conditions (i) and (ii) is satisfied, then the shape morphism

which is defined in [7] and [8], is a shape equivalence.

In [15; 16; 17], Horst Leptin introduced what he called generalized group algebras. These Banach *-algebras are formed by letting a locally compact group G act on a Banach *-algebra A both by *-automorphisms and by a cocycle with values in the multiplier algebra, M (A ), of A. We will review the precise construction later, but for now we remark that examples include the group algebra of a group extension, the covariance algebras of quantum field theory, the “projective group algebras” of a group G (that is, for each complex-valued cocycle λ, called a multiplier in the literature, the Banach *-algebra whose nondegenerate *-representations are in bijective correspondence with the λ-projective representations of G), and the twisted group algebras of Edwards and Lewis [8; 9].

For linear nonuniform cellular automata (NUCA) which are local perturbations of linear CA over a group universe G and a finite-dimensional vector space alphabet V over an arbitrary field k, we investigate their Dedekind finiteness property, also known as the direct finiteness property, i.e., left or right invertibility implies invertibility. We say that the group G is $L^1$-surjunctive, resp. finitely $L^1$-surjunctive, if all such linear NUCA are automatically surjective whenever they are stably injective, resp. when in addition k is finite. In parallel, we introduce the ring $D^1(k[G])$ which is the Cartesian product $k[G] \times (k[G])[G]$ as an additive group but the multiplication is twisted in the second component. The ring $D^1(k[G])$ contains naturally the group ring $k[G]$ and we obtain a dynamical characterization of its stable finiteness for every field k in terms of the finite $L^1$-surjunctivity of the group G, which holds, for example, when G is residually finite or initially subamenable. Our results extend known results in the case of CA.

Let m be any real or complex number, and let Gm be the group generated by the 2 X 2 matrices A = (1, m\ 0, 1) and B = (1, 0; m, 1), where we use the notation (C11, C12; c21, C22) to denote (by rows) the elements of a 2 X 2 matrix C. Thus, Gm is the set of all finite products (or words) of the form

… Ah(3)Bh(2)Ah(1)

where the h(i) are nonzero integers with h﹛\) possibly zero. If a non-trivial word of this form equals the identity / = (1, 0; 0, 1), then Gm is non-free; otherwise, Gm is free.

In the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4

In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem

A decomposition of a topological space X is a partitioning of X into non-empty, disjoint sets called elements of the decomposition. An element of a decomposition is non-degenerate if it contains more than one point. Associated with each decomposition D of X is a topological space D*, called the hyperspace of the decomposition. A classical problem on decompositions of topological spaces is to find conditions under which D* is homeomorphic to X. Often decompositions arise from mappings: if g is a mapping of a space X onto a space Y, then D = {g-1(y) |y ∊ Y} is a decomposition of X.

Let $G$ be a locally compact group. Let ${{A}_{M}}\left( G \right)\,\left( {{A}_{0}}\left( G \right) \right)$ denote the closure of $A\left( G \right)$, the Fourier algebra of $G$ in the space of bounded (completely bounded) multipliers of $A\left( G \right)$. We call a locally compact group $\text{M}$-weakly amenable if ${{A}_{M}}\left( G \right) $ has a bounded approximate identity. We will show that when $G$ is $\text{M}$-weakly amenable, the algebras ${{A}_{M}}\left( G \right) $ and ${{A}_{0}}\left( G \right)$ have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of topologically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra.

For G a group, S a subset of G which generates G, the length problem in G with respect to S is to find, for g ∈ G, the least integer r such that g can be written as the product of r elements of S. For G an orthogonal group Of(F) (here F is a field, and the elements of Of(F) preserve the quadratic form f) and S the set of reflections in Of(F) the length problem has been studied by E. Cartan [2], J. Dieudonné [4, 5], E. Ellers [7], P. Scherk [8], and others. In all of these investigations, however, the problem posed by requiring that S be a single conjugacy class of reflections in Of(F) has been ignored. And it is generally the case that the reflections in Of(F) fall into several conjugacy classes.

In [14] we began a study of C*-algebras corresponding to projective representations of the discrete Heisenberg group, and classified these C*-algebras up to *-isomorphism. In this sequel to [14] we continue the study of these so-called Heisenberg C*-algebras, first concentrating our study on the strong Morita equivalence classes of these C*-algebras. We recall from [14] that a Heisenberg C*-algebra is said to be of class i, i ∊ {1, 2, 3}, if the range of any normalized trace on its K0 group has rank i as a subgroup of R; results of Curto, Muhly, and Williams [7] on strong Morita equivalence for crossed products along with the methods of [21] and [14] enable us to construct certain strong Morita equivalence bimodules for Heisenberg C*-algebras.

In the theory of the asymptotic solution or stability of ordinary differential equations most attention has been given to linear or nearly-linear cases. Investigations in this field, starting primarily with those of Kneser (7) on the equation y″ + f(x)y = 0, have by now mostly been summed up in results on the vector-matrix system dy/dx = Ay + f (y, x), where y and f denote n-vectors of functions, and A an n-by-n matrix, frequently assumed constant.

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.

This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.

1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

The topic of homotopy pull-backs and push-outs has recently been discussed by a number of authors; Boardman and Vogt [5], Bousfield and Kan [6], Fantham [7], Mather [11], and Vogt [16]. Mather develops the theory with an eye to applications and of particular interest is his cube theorem which appears in this paper as Theorem (1.10); the significance of this theorem to applications is shown in [11]. As often occurs in homotopy theory the dual is not true. The purpose of this paper is to examine approximations to the dual in order to obtain new information concerning classical problems of duality.

Soient A un anneau unitaire, G un groupe. L'anneau de groupe AG est le A-module libre ayant pour base les éléments de G, la multiplication étant définie par :

avec

Le premier résultat de cet article concerne les annulateurs des idéaux à gauche engendrés par les sous-groupes finis de G, il permet d'obtenir une démonstration facile de la caractérisation des anneaux de groupes parfaits à gauche.

Let $\Gamma $ be a rank-one arithmetic subgroup of a semisimple Lie group $G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of $G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma $-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Gamma $ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma $ follows from results of DeGeorge-Wallach and Delorme.

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if

(1.1)

T is called sub-Markovian if

(1.2)

All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.

For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let

(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if

(1.3)

and

(1.4)

Let (E, T) be a Hausdorff topological vector space whose topological dual separates points of E, X be a non-empty weakly compact convex subset of E and W be the relative weak topology on X. If F: (X, W) → 2(E,T) is continuous (respectively, upper semi-continuous if £ is locally convex), approximation and fixed point theorems are obtained which generalize the corresponding results of Fan, Park, Reich and Sehgal-Singh-Smithson (respectively, Ha, Reich, Park, Browder and Fan) in several aspects.

This paper represents a general study of the (Yoneda) Ext-algebra A* of a finite dimensional K-algebra A. Our motivation lies in the problem of establishing conditions under which (i) the species of A* coincides with the dual species of A and (ii) the quasi-heredity of A (or A*) yields the quasi-heredity of A* (or A, respectively). These questions are closely related to the Kazhdan—Lusztig Theory as presented by [CPS2]. The main results include introducing the concept of a solid algebra and the relevant Theorem 4.5 as well as a rather complete description of the situation in the case of monomial algebras in Section 5.

In (3) we proved that if A is a complex n-square normal matrix with characteristic roots α1 … , αn,then

1

If A is positive semi-definite hermitian, the inequality (1) becomes

2

This inequality partially answers the problem of determining the maximum permanent of a positive semi-definite hermitian matrix with prescribed characteristic roots (6). In (1), Brualdi and Newman proved that (2) also holds when A is an n-square circulant with non-negative entries. In a recent conversation Dr. Newman raised the question of determining the cases of equality in (1). In the present note we answer this question.