Metabelian $p$-groups of maximal class
R. J.
Miech
331-373
Abstract: This paper deals with the classification of the metabelian $p$-groups of maximal class and order $ {p^n}$ where $ p$ is odd and, roughly, $n \geqq 2p$.
On the cohomology of stable two stage Postnikov systems
John R.
Harper
375-388
Abstract: We study the cohomology of certain fibre spaces. The spaces are the total spaces of stable two stage Postnikov systems. We study their cohomology as Hopf algebras over the Steenrod algebra. The first theorem determines the cohomology as a Hopf algebra over the ground field, the algebra structure being known previously. The second theorem relates the action of the Steenrod algebra to the Hopf algebra structure and other available structures. The work is in the direction of explicit computations of these structures but is not quite complete with regard to the action of the Steenrod algebra. The ideas of Massey and Peterson [7], Mem. Amer. Math. Soc. No. 74, are used extensively, and $\bmod 2$ cohomology is used throughout.
A surface in $E\sp{3}$ is tame if it has round tangent balls
L. D.
Loveland
389-397
Abstract: R. H. Bing has asked if a $2$-sphere $S$ in ${E^3}$ is tame when it is known that for each point $ p$ in $S$ there exist two round balls which are tangent to each other at $p$ and which lie, except for $p$, in opposite complementary domains of $S$. The main result in this paper is that Bing's question has an affirmative answer.
The dual topology for the principal and discrete series on semisimple groups
Ronald L.
Lipsman
399-417
Abstract: For a locally compact group $G$, the dual space $\hat G$ is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull-kernel topology. We prove three results about $ \hat G$ in the case that $ G$ is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of $\hat G$; (2) the ``discrete series'' of square-integrable representations does in fact inherit the discrete topology from $\hat G$; (3) the topology of the reduced dual ${\hat G_r}$, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups.
Spectral representation of certain one-parametric families of symmetric operators in Hilbert space
A. E.
Nussbaum
419-429
Abstract: It is proved that if a one-parameter family of symmetric operators acting in a Hilbert space has the semigroup property on a dense linear manifold and is weakly continuous, then the operators are essentially selfadjoint and permute in the sense of permuting spectral projections of the selfadjoint extensions. It follows from this that the operators have a joint spectral integral representation.
Functional analytic properties of topological semigroups and $n$-extreme amenability
Anthony To-ming
Lau
431-439
Abstract: Let $S$ be a topological semigroup, $ \operatorname{LUC} (S)$ be the space of left uniformly continuous functions on $ S$, and $\Delta (S)$ be the set of multiplicative means on $ \operatorname{LUC} (S)$. If $( \ast )\operatorname{LUC} (S)$ has a left invariant mean in the convex hull of $ \Delta (S)$, we associate with $S$ a unique finite group $G$ such that for any maximal proper closed left translation invariant ideal $I$ in $\operatorname{LUC} (S)$, there exists a linear isometry mapping $\operatorname{LUC} (G)/I$ one-one onto the set of bounded real functions on $G$. We also generalise some recent results of T. Mitchell and E. Granirer. In particular, we show that $ S$ satisfies $ ( \ast )$ iff whenever $ S$ is a jointly continuous action on a compact hausdorff space $ X$, there exists a nonempty finite subset $F$ of $X$ such that $sF = F$ for all $s \in S$. Furthermore, a discrete semigroup $ S$ satisfies $ ( \ast )$ iff whenever $\{ {T_s};s \in S\}$ is an antirepresentation of $S$ as linear maps from a norm linear space $ X$ into $X$ with $ \vert\vert{T_s}\vert\vert \leqq 1$ for all $s \in S$, there exists a finite subset $\sigma \subseteq S$ such that the distance (induced by the norm) of $x$ from $ {K_X} =$ linear span of $\{ x - {T_s}x;x \in X,s \in S\}$ in $ X$ coincides with distance of $O(\sigma ,x) = \{ (1/\vert\sigma \vert)\sum\nolimits_{a \in \sigma } {{T_{at}}(x);t \in S\} }$ from 0 for all $x \in X$.
Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions
Donald P.
Ballou
441-460
Abstract: This paper is concerned with the existence of weak solutions to certain nonlinear hyperbolic Cauchy problems. A condition on the curves of discontinuity is used which guarantees uniqueness in the class of piecewise smooth weak solutions. The method of proof is geometric in nature and is constructive in the manner of A. Douglis and Wu Cho-Chün; that is, for certain types of initial data the method of characteristics is employed to construct piecewise smooth weak solutions. A limiting process is then used to obtain existence for bounded, measurable initial data. The solutions in some cases exhibit interesting, new phenomena. For example, a certain class of initial data having one jump gives rise to a solution having a curving contact discontinuity which does not enter the region of intersecting characteristics.
A generalization of parallelism in Riemannian geometry, the $C\sp{\omega }$ case
Alan B.
Poritz
461-494
Abstract: The concept of parallelism along a curve in a Riemannian manifold is generalized to parallelism along higher dimensional immersed submanifolds in such a way that the minimal immersions are self parallel and hence correspond to geodesics. Let $g:N \to M$ be a (not necessarily isometric) immersion of Riemannian manifolds. Let $G:T(N) \to T(M)$ be a tangent bundle isometry along $g$, that is, $G$ covers $g$ and maps fibers isometrically. By mimicing the construction used for isometric immersions, it is possible to define the mean curvature vector field of $ G.G$ is said to be parallel along $g$ if this vector field vanishes identically. In particular, minimal immersions have parallel tangent maps. For curves, it is shown that the present definition reduces to the definition of Levi-Civita. The major effort is directed toward generalizations, in the real analytic case, of the two basic theorems for parallelism. On the one hand, the existence and uniqueness theorem for a geodesic in terms of data at a point extends to the well-known existence and uniqueness of a minimal immersion in terms of data along a codimension one submanifold. On the other hand, the existence and uniqueness theorem for a parallel unit vector field along a curve in terms of data at a point extends to a local existence and uniqueness theorem for a parallel tangent bundle isometry in terms of mixed initial and partial data. Since both extensions depend on the Cartan-Kahler Theorem, a procedure is developed to handle both proofs in a uniform manner using fiber bundle techniques.
Decomposing manifolds into homologically equivalent submanifolds
J. Scott
Downing
495-501
On the structure of primary abelian groups of countable Ulm type
Doyle O.
Cutler
503-518
Abstract: In this paper we will give structure theorems for abelian $p$-groups of countable Ulm type utilizing the notion of high subgroup introduced by John M. Irwin and its generalization, $N$-high subgroup, introduced by Irwin and E. A. Walker. The general technique employed is to give conditions under which automorphisms of these subgroups extend to automorphisms of the group.
On the solutions of a class of linear selfadjoint differential equations
Larry R.
Anderson;
A. C.
Lazer
519-530
Abstract: Let $L$ be a linear selfadjoint ordinary differential operator with coefficients which are real and sufficiently regular on $( - \infty ,\infty )$. Let ${A^ + }({A^ - })$ denote the subspace of the solution space of $Ly = 0$ such that $y \in {A^ + }(y \in {A^ - })$ iff ${D^k}y \in {L^2}[0,\infty )({D^k}y \in {L^2}( - \infty ,0])$ for $k = 0,1, \ldots ,m$ where $ 2m$ is the order of $ L$. A sufficient condition is given for the solution space of $Ly = 0$ to be the direct sum of $ {A^ + }$ and $ {A^ - }$. This condition which concerns the coefficients of $ L$ reduces to a necessary and sufficient condition when these coefficients are constant. In the case of periodic coefficients this condition implies the existence of an exponential dichotomy of the solution space of $Ly = 0$.
Actions of the torus on $4$-manifolds. I
Peter
Orlik;
Frank
Raymond
531-559
Abstract: Smooth actions of the $2$-dimensional torus group $SO(2) \times SO(2)$ on smooth, closed, orientable $ 4$-manifolds are studied. A cross-sectioning theorem for actions without finite nontrivial isotropy groups and with either fixed points or orbits with isotropy group isomorphic to $ SO(2)$ yields an equivariant classification for these cases. This classification is made numerically specific in terms of orbit invariants. A topological classification is obtained for actions on simply connected $4$-manifolds. It is shown that such a manifold is an equivariant connected sum of copies of complex projective space $ C{P^2}, - C{P^2}$ (reversed orientation), $ {S^2} \times {S^2}$ and the other oriented ${S^2}$ bundle over ${S^2}$. The latter is diffeomorphic (but not always equivariantly diffeomorphic) to $C{P^2}\char93 - C{P^2}$. The connected sum decomposition is not unique. Topological actions on topological manifolds are shown to reduce to the smooth case. In an appendix certain results are extended to torus actions on orientable $4$-dimensional cohomology manifolds.
Group algebra modules. III
S. L.
Gulick;
T.-S.
Liu;
A. C. M.
van Rooij
561-579
Abstract: Let $\Gamma$ be a locally compact group and $ K$ a Banach space. The left ${L^1}(\Gamma )$ module $K$ is by definition absolutely continuous under the composition $\ast$ if for $k \in K$ there exist ${L^1}(\Gamma )$ module--the main object we study. If $ Y \subseteq X$ is measurable, let ${L_Y}$ consist of all functions in ${L^1}(X)$ vanishing outside $ Y$. For $\Omega \subseteq \Gamma $ not locally null and $ B$ a closed linear subspace of $K$, we observe the connection between the closed linear span (denoted ${L_\Omega } \ast B$) of the elements $f \ast k$, with $f \in {L_\Omega }$ and $k \in B$, and the collection of functions of $ B$ shifted by elements in $ \Omega$. As a result, a closed linear subspace of ${L^1}(X)$ is an ${L_Z}$ for some measurable $Z \subseteq X$ if and only if it is closed under pointwise multiplication by elements of ${L^\infty }(X)$. This allows the theorem stating that if $ \Omega \subseteq \Gamma$ and $Y \subseteq X$ are both measurable, then there is a measurable subset $Z$ of $X$ such that ${L_\Omega } \ast {L_Y} = {L_Z}$. Under certain restrictions on $\Gamma$, we show that this $Z$ is essentially open in the (usually stronger) orbit topology on $X$. Finally we prove that if $ \Omega$ and $ Y$ are both relatively sigma-compact, and if also ${L_\Omega } \ast {L_Y} \subseteq {L_Y}$, then there exist ${\Omega _1}$ and ${Y_1}$ locally almost everywhere equal to $\Omega$ and $Y$ respectively, such that ${\Omega _1}{Y_1} \subseteq {Y_1}$; in addition we characterize those $\Omega$ and $Y$ for which ${L_\Omega } \ast {L_\Omega } = {L_\Omega }$ and ${L_\Omega } \ast {L_Y} = {L_Y}$.
Group algebra modules. IV
S. L.
Gulick;
T.-S.
Liu;
A. C. M.
van Rooij
581-596
Abstract: Let $\Gamma$ be a locally compact group, $ \Omega$ a measurable subset of $\Gamma$, and let $ {L_\Omega }$ denote the subspace of $ {L^1}(\Gamma )$ consisting of all functions vanishing off $\Omega$. Assume that ${L_\Omega }$ is a subalgebra of ${L^1}(\Gamma )$. We discuss the collection ${\Re _\Omega }(K)$ of all module homomorphisms from ${L_\Omega }$ into an arbitrary Banach space $ K$ which is simultaneously a left $ {L^1}(\Gamma )$ module. We prove that $ {\Re _\Omega }(K) = {\Re _\Omega }({K_0}) \oplus {\Re _\Omega }({K_{\text{abs} }})$, where ${K_0}$ is the collection of all $k \in K$ such that $fk = 0$, for all $f \in {L^1}(\Gamma )$, and where ${K_{\text{abs} }}$ consists of all elements of $ K$ which can be factored with respect to the module composition. We prove that ${\Re _\Omega }({K_0})$ is the collection of linear continuous maps from ${L_\Omega }$ to ${K_0}$ which are zero on a certain measurable subset of $X$. We reduce the determination of $ {\Re _\Omega }({K_{\text{abs} }})$ to the determination of $ {\Re _\Gamma }({K_{\text{abs} }})$. Denoting the topological conjugate space of $K$ by ${K^ \ast }$, we prove that $ {({K_{\text{abs} }})^ \ast }$ is isometrically isomorphic to ${\Re _\Omega }({K^ \ast })$. Finally, we discuss module homomorphisms $R$ from $ {L_\Omega }$ into $ {L^1}(X)$ such that for each $ f \in {L_\Omega },Rf$ vanishes off $Y$.
Semigroups through semilattices
J. H.
Carruth;
Jimmie D.
Lawson
597-608
Abstract: Presented in this paper is a method of constructing a compact semigroup $S$ from a compact semilattice $X$ and a compact semigroup $ T$ having idempotents contained in $X$. The notions of semigroups (straight) through chains and (straight) through semilattices are introduced. It is shown that the notion of a semigroup through a chain is equivalent to that of a generalized hormos. Universal objects are obtained in several categories including the category of clans straight through a chain and the category of clans straight through a semilattice relative to a chain. An example is given of a nonabelian clan $S$ with abelian set of idempotents $E$ such that $S$ is minimal (as a clan) about $E$.
Cohomology of $F$-groups
Peter
Curran
609-621
Abstract: Let $G$ be a group of Möbius transformations and $V$ the space of complex polynomials of degree $ \leqq$ some fixed even integer. Using the action of $G$ on $V$ defined by Eichler, we compute the dimension of the cohomology space $ {H^1}(G,V)$, first for $ G$ an arbitrary $ F$-group (a generalization of Fuchsian group) and then for the free product of finitely many $F$-groups. These results extend those which Eichler obtained in a 1957 paper, where a correspondence was established between elements of ${H^1}(G,V)$ and cusp forms on $G$.
On automorphism groups of $C\sp*$-algebras
Mi-soo Bae
Smith
623-648
Closedness of coboundary modules of analytic sheaves
Yum-tong
Siu;
Günther
Trautmann
649-658
Abstract: Suppose $ A$ is a subvariety of a complex space $X$ and $ \mathcal{J}$ is a coherent analytic sheaf on $X$. It is shown that, if the analytic sheaf $ \mathcal{H}_{A}^{\nu}(\mathcal{J})$ of local cohomology is coherent for $0 \leqq \nu \leqq q$, then for $0 \leqq \nu \leqq q$ the local cohomology group $\mathcal{H}_{A}^{\nu}(X, \mathcal{J})$ with its natural topology is Hausdorff and hence is a Fréchet space.