The topology on the primitive ideal space of transformation group $C\sp{\ast} $-algebras and C.C.R. transformation group $C\sp{\ast} $-algebras
Dana P.
Williams
335-359
Abstract: If $(G,\Omega )$ is a second countable transformation group and the stability groups are amenable then ${C^ \ast }(G,\Omega )$ is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In addition, partial results relating closed orbits to C.C.R. algebras are obtained in the nonseparable case. In several cases, the topology of the primitive ideal space is calculated explicitly. In particular, if the stability groups are all contained in a fixed abelian subgroup $H$, then the topology is computed in terms of $ H$ and the orbit structure, provided $ {C^ \ast }(G,\Omega )$ and $ {C^ \ast }(H,\Omega )$ are $ EH$-regular. These conditions are automatically met if $G$ is abelian and $ (G,\Omega )$ is second countable.
The discontinuous initial value problem of a reacting gas flow
Lung An
Ying;
Ching Hua
Wang
361-387
Abstract: We show that the local solvability of the solution of a reacting gas flow system $(1.1) - (1.4)$ with initial values $ (1.5)$, which has a large jump at the point $x' = 0$ and the structure of the solution near the origin $(0,0)$ are identical to those of the Riemann problem of the homogeneous system corresponding to the Cauchy problem $(1.1) - (1.5)$.
The cyclic connectivity of homogeneous arcwise connected continua
David P.
Bellamy;
Lewis
Lum
389-396
Abstract: A continuum is cyclicly connected provided each pair of its points lie together on some simple closed curve. In 1927, G. T. Whyburn proved that a locally connected plane continuum is cyclicly connected if and only if it contains no separating points. This theorem was fundamental in his original treatment of cyclic element theory. Since then numerous authors have obtained extensions of Whyburn's theorem. In this paper we characterize cyclic connectedness in the class of all Hausdorff continua. Theorem. The Hausdorff continuum $ X$ is cyclicly connected if and only if for each point $x \in X$, $x$ lies in the relative interior of some arc in $X$ and $X - \{ x\}$ is arcwise connected. We then prove that arcwise connected homogeneous metric continua are cyclicly connected.
Nonweakly compact operators from order-Cauchy complete $C(S)$ lattices, with application to Baire classes
Frederick K.
Dashiell
397-413
Abstract: This paper is concerned with the connection between weak compactness properties in the duals of certain Banach spaces of type $C(S)$ and order properties in the vector lattice $C(S)$. The weak compactness property of principal interest here is the condition that every nonweakly compact operator from $C(S)$ into a Banach space must restrict to an isomorphism on some copy of $ {l^\infty }$ in $ C(S)$. (This implies Grothendieck's property that every ${w^ \ast }$-convergent sequence in $C{(S)^ \ast }$ is weakly convergent.) The related vector lattice property studied here is order-Cauchy completeness, a weak type of completeness property weaker than $\sigma$-completeness and weaker than the interposition property of Seever. An apphcation of our results is a proof that all Baire classes (of fixed order) of bounded functions generated by a vector lattice of functions are Banach spaces satisfying Grothendieck's property. Another application extends previous results on weak convergence of sequences of finitely additive measures defined on certain fields of sets.
A counterexample to the bounded orbit conjecture
Stephanie M.
Boyles
415-422
Abstract: A long outstanding problem in the topology of Euclidean spaces is the Bounded Orbit Conjecture, which states that every homeomorphism of ${E^2}$ onto itself, with the property that the orbit of every point is bounded, must have a fixed point. It is well known that the conjecture is true for orientation preserving homeomorphisms. We provide a counterexample to the conjecture by constructing a fixed point free orientation reversing homeomorphism which satisfies the hypothesis of the conjecture.
Isomorphism theorems for octonion planes over local rings
Robert
Bix
423-439
Abstract: It is proved that there is a collineation between two octonion planes over local rings if and only if the underlying octonion algebras are isomorphic as rings. It is shown that every isomorphism between the little or middle projective groups of two octonion planes over local rings is induced by conjugation with a collineation or a correlation of the planes when the local rings contain $\frac{1} {2}$.
Symmetry properties of the zero sets of nil-theta functions
Sharon
Goodman
441-460
Abstract: Let $N$ denote the three dimensional Heisenberg group, and let $\Gamma$ be a discrete two-generator subgroup of $ N$ such that $ N/\Gamma$ is compact. Then we may decompose $ {L^2}(N/\Gamma )$ into primary summands with respect to the right regular representation $R$ of $N$ on $ {L^2}(N/\Gamma )$ as follows: ${L^2}(N/\Gamma ) = \oplus \sum\nolimits_{m \in {\mathbf{Z}}} {{H_m}(\Gamma )}$. It can be shown that for $m \ne 0,{H_m}(\Gamma )$ is a multiplicity space for the representation $ R$ of multiplicity $\left\vert m \right\vert$. The distinguished subspace theory of $ {\text{L}}$. Auslander and ${\text{J}}$. Brezin singles out a finite number of the decompositions of ${H_m}(\Gamma ),m \ne 0$, which are in some ways nicer than the others. They define algebraically an integer valued function, called the index, on the set ${\Omega _m}$ of irreducible closed $ R$-invariant subspaces of ${H_m}(\Gamma )$ such that the distinguished subspaces have index one. In this paper, we give an analytic-geometric interpretation of the index. Every space in $ {\Omega _m}$ contains a unique (up to constant multiple) special function, called a nil-theta function, that arises as a solution of a certain differential operator on $N/\Gamma$. These nil-theta functions have been shown to be closely related to the classical theta functions. Since the classical theta functions are determined (up to constant multiple) by their zero sets, it is natural to attempt to classify the spaces in ${\Omega _m}$ using various properties of the zero sets of the nil-theta functions lying in these spaces. We define the index of a nil-theta function in ${H_m}(\Gamma )$ using the symmetry properties of its zero set. Our main theorem asserts that the algebraic index of a space in $ {\Omega _m}$ equals the index of the unique nil-theta function lying in that space. We have thus an analytic-geometric characterization of the index. We then use these results to give a complete description of the zero sets of those nil-theta functions of a fixed index. We also investigate the behavior of the index under the multiplication of nil-theta functions; i.e. we discuss how the index of the nil-theta function $FG$ relates to the indices of the nil-theta functions $F$ and $G$.
Generalizations of Ces\`aro continuous functions and integrals of Perron type
Cheng Ming
Lee
461-481
Abstract: The linear space of all the Cesàro continuous functions of any order is extended by introducing pointwisely Cesàro continuous functions and exact generalized Peano derivatives. Then six generalized integrals of Perron type are defined and studied. They are based on three recent monotonicity theorems and each depends on an abstract upper semilinear space of certain functions. Some of the integrals are more general than all the integrals in the Cesàro-Perron scale provided that the abstract semilinear space is taken to be the linear space of all the pointwisely Cesàro continuous functions or all the exact generalized Peano derivatives. That such a concrete general integral is possible follows from the fact proved here that each exact generalized Peano derivative is in Baire class one and has the Darboux property. Relations between the pointwisely Cesàro continuous functions or the exact generalized Peano derivatives and functions defined by means of the values of certain Schwartz's distributions at "points" are also established.
On purely inseparable algebras and P.H.D. rings
Shizuka
Satô
483-498
Abstract: M. E. Sweedler has considered purely inseparable algebras over rings. We define a stronger notion for purely inseparable algebras over rings and we study the fundamental properties of purely inseparable algebras. Moreover, we consider the relations between purely inseparable algebras and P.H.D. rings.
Liapounoff's theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures
Thomas E.
Armstrong;
Karel
Prikry
499-514
Abstract: Liapounoff's theorem states that if $ (X,\Sigma )$ is a measurable space and $\mu :\Sigma \to {{\mathbf{R}}^d}$ is nonatomic, bounded, and countably additive, then $ \mathcal{R}(\mu ) = \{ \mu (A):A \in \Sigma \}$ is compact and convex. When $ \Sigma$ is replaced by a $ \sigma$-complete Boolean algebra or an $F$-algebra (to be defined) and $\mu$ is allowed to be only finitely additive, $ \mathcal{R}(\mu )$ is still convex. If $\Sigma$ is any Boolean algebra supporting nontrivial, nonatomic, finitely-additive measures and $Z$ is a zonoid, there exists a nonatomic measure on $\Sigma$ with range dense in $Z$. A wide variety of pathology is examined which indicates that ranges of finitely-additive, nonatomic, finite-dimensional, vector-valued measures are fairly arbitrary.
Weak and pointwise compactness in the space of bounded continuous functions
Robert F.
Wheeler
515-530
Abstract: Let $T$ be a completely regular Hausdorff space, ${C_b}(T)$ the space of bounded continuous real-valued functions on $T$, $M(T)$ the Banach space dual of ${C_b}(T)$. Let $\mathcal{H}$ denote the family of subsets of $ {C_b}(T)$ which are uniformly bounded and relatively compact for the topology $ {\mathfrak{J}_p}$ of pointwise convergence. The basic question considered here is: what is the largest subspace $Z$ of $M(T)$ such that every member of $\mathcal{H}$ is relatively $\sigma ({C_b},Z)$-compact? Classical results of Grothendieck and Ptak show that $ Z = M(T)$ if $ T$ is pseudocompact. In general, ${M_t} \subset Z \subset {M_s};$ assuming Martin's Axiom, a deep result of Talagrand improves the lower bound to ${M_\tau }$. It is frequently, but not always, true that $Z = {M_s};$ counterexamples are given which use Banach spaces in their weak topologies to construct the underlying $T$'s.
On the genus of symmetric groups
Viera Krňanová
Proulx
531-538
Abstract: A new method for determining genus of a group is described. It involves first getting a bound on the sizes of the generating set for which the corresponding Cayley graph could have smaller genus. The allowable generating sets are then examined by methods of computing average face sizes and by voltage graph techniques to find the best embeddings. This method is used to show that genus of the symmetric group ${S_5}$ is equal to four. The voltage graph method is used to exhibit two new embeddings for symmetric groups on even number of elements. These embeddings give us a better upper bound than that previously given by A. T. White.
Integral geometric properties of capacities
Pertti
Mattila
539-554
Abstract: Let $m$ and $n$ be positive integers, $0 < m < n$, and ${C_K}$ and ${C_H}$ the usual potential-theoretic capacities on ${R^n}$ corresponding to lower semicontinuous kernels $ K$ and $H$ on ${R^n} \times {R^n}$ with $ H(x,y) = K(x,y){\left\vert {x - y} \right\vert^{n - m}} \geqslant 1$ for $ \left\vert {x - y} \right\vert \leqslant 1$. We consider relations between the capacities ${C_K}(E)$ and $ {C_H}(E \cap A)$ when $E \subset {R^n}$ and $A$ varies over the $m$-dimensional affine subspaces of ${R^n}$. For example, we prove that if $ E$ is compact, ${C_K}(E) \leqslant c\smallint {C_H}(E \cap A)d{\lambda _{n,m}}A$ where ${\lambda _{n,m}}$ is a rigidly invariant measure and $c$ is a positive constant depending only on $ n$ and $m$.
Global Warfield groups
Roger
Hunter;
Fred
Richman
555-572
Abstract: A global Warfield group is a summand of a simply presented abelian group. The theory of global Warfield groups encompasses both the theory of totally projective $p$-groups, which includes the classical Ulm-Zippin theory of countable $p$-groups, and the theory of completely decomposable torsion-free groups. This paper develops the central results of the theory including existence and uniqueness theorems. In addition it is shown that every decomposition basis of a global Warfield group has a nice subordinate with simply presented torsion cokernel, and that every global Warfield group is a direct sum of a group of countable torsion-free rank and a simply presented group.
Manifolds of nonanalyticity of solutions of certain linear PDEs
E. C.
Zachmanoglou
573-582
Abstract: It is shown that if $ P$ is a linear partial differential operator with analytic coefficients, and if $M$ is an analytic manifold of codimension $ 3$ which is partially characteristic with respect to $P$ and satisfies certain additional conditions, then one can find, in a neighborhood of any point of $ M$, solutions of the equation $Pu = 0$ which are flat or singular precisely on $ M$.
Cartan structures on contact manifolds
G.
Burdet;
M.
Perrin
583-602
Abstract: Owing to the existence of a dilatation generator of eigenvalues $\pm 2, \pm 1,0$ the symplectic Lie algebra is considered as a $\vert 2\vert$-graded Lie algebra. The corresponding decomposition of the symplectic group ${\text{Sp(2(}}n + 1{\text{),}}{\mathbf{R}}{\text{)}}$ makes the semidirect product (denoted $ {L^0}$) of the $ (2n + 1)$-dimensional Weyl group by the conformal symplectic group $ {\text{CSp(}}2n,{\mathbf{R}}{\text{)}}$ appear as a privileged subgroup and permits one to construct a $2n + 1$-dimensional homogeneous space possessing a natural contact form. Then ${\text{Sp}}(2(n + 1),{\mathbf{R}})$-valued Cartan connections on a ${L^0}$principal fibre bundle over a $2n + 1$-dimensional manifold ${B_{2n + 1}}$ are constructed and called symplectic Cartan connections. The conditions for obtaining a unique symplectic Cartan connection are given. The existence of this unique Cartan connection is used to define the notion of contact structure over ${B_{2n + 1}}$ and it is shown that any $ {L^0}$-structure of degree $ 2$ over ${B_{2n + 1}}$ can be considered as a contact structure on it. Moreover it is shown that a contact structure can be associated in a canonical way to any contact manifold.
On asymptotically almost periodic solutions of a convolution equation
Olof J.
Staffans
603-616
Abstract: We study questions related to asymptotic almost periodicity of solutions of the linear convolution equation $( \ast )\mu \ast x = f$. Here $\mu$ is a complex measure, and $ x$ and $f$ are bounded functions. Basically we are interested in conditions which imply that bounded solutions of $( \ast )$ are asymptotically almost periodic. In particular, we show that a certain necessary condition on $f$ for this to happen is also sufficient, thereby strengthening earlier results. We also include a result on existence of bounded solutions, and indicate a generalization to a distribution equation.
$L\sp{p}$ norms of certain kernels of the $N$-dimensional torus
L.
Colzani;
P. M.
Soardi
617-627
Abstract: In this paper we study a class of kernels ${F_R}$ which generalize the Bochner-Riesz kernels on the $N$-dimensional torus. Our main result consists in upper estimates for the ${L^p}$ norms of ${F_R}$ as $R$ tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces.
Arborescent structures. II. Interpretability in the theory of trees
James H.
Schmerl
629-643
Abstract: The first-order theory of arborescent structures is shown to be completely faithfully interpretable in the first-order theory of trees. It follows from this interpretation that Vaught's conjecture is true for arborescent structures, the theory of arborescent structures is decidable, and every $ {\aleph _0}$-categorical arborescent structure has a decidable theory.
A rigid subspace of $L\sb{0}$
N. J.
Kalton;
James W.
Roberts
645-654
Abstract: We construct a closed infinite-dimensional subspace of ${L_0}(0,1)$ (or ${L_p}$ for $0 < p < 1$) which is rigid, i.e. such that every endomorphism in the space is a multiple of the identity.
Baire category principle and uniqueness theorem
J. S.
Hwang
655-665
Abstract: Applying a theorem of Bagemihl and Seidel (1953), we prove that if $ {S_2}$ is a set of second category in $ (\alpha ,\beta )$, where $0 \leqslant \alpha < \beta \leqslant 2\pi$, and if $f(z)$ is a function meromorphic in the sector $ \Delta (\alpha ,\beta ) = \{ z:0 < \left\vert z \right\vert < \infty ,\alpha < \arg z < \beta \}$ for which ${\underline {{\operatorname{lim}}} _{r \to \infty }}\left\vert {f(r{e^{i\theta }})} \right\vert > 0$, for all $ \theta \in {S_2}$, then there exists a sector $ \Delta (\alpha ',\beta ') \subseteq \Delta (\alpha ,\beta )$ such that $(\alpha ',\beta ') \subseteq {\bar S_2},{S_2}$ is second category in $(\alpha ',\beta ')$, and $f(z)$ has no zero in $\Delta (\alpha ',\beta ')$. Based on this property, we prove several uniqueness theorems.
Erratum to: ``The kinematic formula in complex integral geometry''
Theodore
Shifrin
667