Intersection properties of balls and subspaces in Banach spaces
Ȧsvald
Lima
1-62
Abstract: We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with ${L_1}(\mu )$ dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with $ {L_1}(\mu )$ dual spaces whose unit ball contains extreme points.
$K$-theory and Steenrod homology: applications to the Brown-Douglas-Fillmore theory of operator algebras
Jerome
Kaminker;
Claude
Schochet
63-107
Abstract: The remarkable work of L. G. Brown, R. Douglas and P. Fillmore on operators with compact self-commutators once again ties together algebraic topology and operator theory. This paper gives a comprehensive treatment of certain aspects of that connection and some adjacent topics. In anticipation that both operator theorists and topologists may be interested in this work, additional background material is included to facilitate access.
The isomorphism problem for two-generator one-relator groups with torsion is solvable
Stephen J.
Pride
109-139
Abstract: The theorem stated in the title is obtained by determining (in a sense to be made precise) all the generating pairs of an arbitrary two-generator one-relator group with torsion. As a consequence of this determination it is also deduced that every two-generator one-relator group G with torsion is Hopfian, and that the automorphism group of G is finitely generated.
Chern-Simons invariants of reductive homogeneous spaces
Harold
Donnelly
141-164
Abstract: The geometric characteristic classes of Chern-Simons are computed for certain connections on the canonical bundle and tangent bundle over a reductive homogeneous space. This includes globally symmetric spaces with the Levi-Civita connection of any bi-invariant metric.
The fundamental form of an inseparable extension
Murray
Gerstenhaber
165-184
Abstract: If K is a finite purely inseparable extension of a field k, then the symmetric multiderivations of K (symmetric maps $f:K \times \cdots \times K\;(n\;{\text{times}}) \to K$ which are derivations as functions of each single variable) form a ring under the symmetrized cup product. This ring contains an element $\Gamma (K/k)$ called the fundamental form of K over k, which is defined up to multiplication by a nonzero element of K and has the property that if B is any intermediate field between K and k, then $ \Gamma (K/B)$ divides $\Gamma (K/k)$.
Functions satisfying elementary relations
Michael F.
Singer
185-206
Abstract: In this paper we deal with the following problems: When do the solutions of a collection of differential equations satisfy an elementary relation, that is, when is there an equation of the form $R = 0$ where R is some algebraic combination of logarithmic, exponential and algebraic functions involving solutions of our differential equations? If such relations exist, what can they look like? These problems are given an algebraic setting and general forms for such relations are exhibited. With these, we are able to show that certain classes of functions satisfy no elementary relations.
Homological algebra and set theory
Paul C.
Eklof
207-225
Abstract: Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of $ {\operatorname{Ext}}_\Lambda ^1$ for rings $\Lambda$ of global dimension 1. Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC. Applications are given to abelian group theory, including a partial solution (assuming ${\text{V}} = {\text{L}}$) to a problem of Baer on the splitting of abelian groups. Some independence results in abelian group theory are also proved.
Derivatives of entire functions and a question of P\'olya
Simon
Hellerstein;
Jack
Williamson
227-249
Abstract: An old question of Pólya asks whether an entire function f which has, along with each of its derivatives, only real zeros must be of the form $\displaystyle f(z) = {z^m}{e^{ - a{z^2} + bz + c}}\prod\limits_n {\left( {1 - \frac{z}{{{z_n}}}} \right)} {e^{z/{z_n}}}$ where $a \geqslant 0,b$ and the ${z_n}$ are real, and ${\Sigma _n}z_n^{ - 2} < \infty$. This note answers this question (essentially in the affirmative) if f is of finite order; indeed, it is established that if $f,f'$, and $f''$ have only real zeros (f of finite order), then either f has the above form or f has one of the forms $\displaystyle f(z) = a{e^{bz}},\quad f(z) = a({e^{icz}} - {e^{id}})$ where a, b, c, and d are constants, b complex, c and d real.
A relation between two biharmonic Green's functions on Riemannian manifolds
Dennis
Hada
251-261
Abstract: The biharmonic Green's function $\gamma$ whose values and Laplacian are identically zero on the boundary of a region and the biharmonic Green's function $\Gamma$ whose values and normal derivative vanish on the boundary originated in the investigation of thin plates whose edges are simply supported or clamped, respectively. A relation between these two biharmonic Green's functions known for planar regions is extended to Riemannian manifolds thereby establishing that any Riemannian manifold for which $\gamma$ exists must also carry $\Gamma$.
Embedding Stieltjes-Volterra integral equations in Stieltjes integral equations
William L.
Gibson
263-277
Abstract: J. A. Reneke has shown that the linear Stieltjes-Volterra integral equations studied by D. B. Hinton can be transformed into Stieltjes integral equations of the type studied by J. S. Mac Nerney. By taking advantage of the nonlinear nature of Mac Nerney's results, Reneke was able to extend Hinton's existence theorem to a nonlinear setting. In this paper, we use Reneke's embedding technique to generalize several other of Hinton's results, and we characterize completely, in the linear case, the range of Reneke's embedding transformation.
The topological structure of $4$-manifolds with effective torus actions. I
Peter Sie
Pao
279-317
Abstract: Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of ${S^4},{S^2} \times {S^2},C{P^2}, - C{P^{ - 2}},{S^1} \times {S^3}$, and three families of elementary 4-manifolds, $ R(m,n),T(m,n;m',n'),L(n;p,q;m)$ has been obtained (where m, n, $ m',n'$, p, and q are integers). In addition, a stable homeomorphic relation for the manifolds $R(m,n)$ and $T(m,n;m',n')$ can also be found in [9]. But the topological classification of R's, T's, and especially L's were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R's and T's, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds $ L(n;p,q;m)$ form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of $L(n;p,q;m)$ is finite cyclic of order n. And it is proved that ${\pi _1}(L(n;p,q;m))$ and $({S^2} \times {S^2})\char93 \cdots \char93 \;({S^2} \times {S^2})$ ($n - 1$ copies), even though $L(n;p,q;m)$ and
Structure of subalgebras between $L\sp{\infty }$ and $H\sp{\infty }$
Sun Yung A.
Chang
319-332
Abstract: Let B be a closed subalgebra of $ {L^\infty }$ of the unit circle which contains $ {H^\infty }$ properly. Let $ {C_B}$ be the $ {C^\ast}$-algebra generated by the inner functions that are invertible in B. It is shown that the linear span ${H^\infty } + {C_B}$ is equal to B. Also, a closed subspace (called $VM{O_B}$) of BMO (space of functions of bounded mean oscillation) is identified to which B bears the same relation as $ {L^\infty }$ does to BMO.
Ulm's theorem for partially ordered structures related to simply presented abelian $p$-groups
Laurel A.
Rogers
333-343
Abstract: If we have an abelian p-group G, a multiplication by p for each element of G is defined by setting $ px = x + x + \cdots + x$, where p is the number of terms in the sum. If we forget about the addition on G, and just keep the multiplication by p, we have the algebraic structure called a p-basic tree. A natural partial order can be defined, the graph of which is a tree with 0 as root. A p-basic tree generates a simply presented abelian p-group, and provides a natural direct sum decomposition for it. Ulm invariants may be defined directly for a p-basic tree so that they are equal to the Ulm invariants of the corresponding group. A central notion is that of a stripping function between two p-basic trees. Given a stripping function from X onto Y we can construct an isomorphism between the groups corresponding to X and Y; in particular, X and Y have the same Ulm invariants. Conversely, if X and Y have the same Ulm invariants, then there is a map from X onto Y that is the composition of two stripping functions and two inverses of stripping functions. These results constitute Ulm's theorem for p-basic trees, and provide a new proof of Ulm's theorem for simply presented groups.
The classifying space of a permutation representation
James V.
Blowers
345-355
Abstract: In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. An example of this classifying space is given by a generalization of the infinite join construction that defines the standard example of a classifying space of a group. In a previous paper of the author, the join of two permutation representations was defined, and it was shown that the cohomology ring of the join was trivial. In this paper the classifying space of the join of two permutation representations is shown to be the topological join of the two classifying spaces and from this the triviality of the cup-product is derived topologically.
One-parameter groups of isometries on Hardy spaces of the torus: spectral theory
Earl
Berkson;
Horacio
Porta
357-370
Abstract: The spectral theory of the infinitesimal generator of an arbitrary one-parameter group of isometries on ${H^p}$ of the torus, $1 \leqslant p < \infty ,p \ne 2$, is considered. In particular, the spectrum of the generator is determined.