Horocycle flows on certain surfaces without conjugate points
Patrick
Eberlein
1-36
Abstract: We study the topological but not ergodic properties of the horocycle flow $\{ {h_t}\}$ in the unit tangent bundle SM of a complete two dimensional Riemannian manifold M without conjugate points that satisfies the ``uniform Visibility'' axiom. This axiom is implied by the curvature condition $K \leqslant c < 0$ but is weaker so that regions of positive curvature may occur. Compactness is not assumed. The method is to relate the horocycle flow to the geodesic flow for which there exist useful techniques of study. The nonwandering set ${\Omega _h} \subseteq SM$ for $\{ {h_t}\}$ is classified into four types depending upon the fundamental group of M. The extremes that $ {\Omega _h}$ be a minimal set for $\{ {h_t}\}$ and that ${\Omega _h}$ admit periodic orbits are related to the existence or nonexistence of compact ``totally convex'' sets in M. Periodic points are dense in $ {\Omega _h}$ if they exist at all. The only compact minimal sets in ${\Omega _h}$ are periodic orbits if M is noncompact The flow $\{ {h_t}\}$ is minimal in SM if and only if M is compact. In general $\{ {h_t}\}$ is topologically transitive in ${\Omega _h}$ and the vectors in ${\Omega _h}$ with dense orbits are classified. If the fundamental group of M is finitely generated and ${\Omega _h} = SM$ then $\{ {h_t}\}$ is topologically mixing in SM.
On analytically invariant subspaces and spectra
Domingo A.
Herrero
37-44
Abstract: Let T be a bounded linear operator from a complex Banach space $\mathfrak{X}$ into itself. Let ${\mathcal{A}_T}$ and $ \mathcal{A}_T^a$ denote the weak closure of the polynomials and the rational functions (with poles outside the spectrum $\sigma (T)$ of T) in T, respectively. The lattice $ {\operatorname{Lat}}\;\mathcal{A}_T^a$ of (closed) invariant subspaces of $\mathcal{A}_T^a$ is a very particular subset of the invariant subspace lattice $ {\operatorname{Lat}}\;{\mathcal{A}_T} = {\operatorname{Lat}}\;T$ of T. It is shown that: (1) If the resolvent set of T has finitely many components, then $ {\operatorname{Lat}}\;\mathcal{A}_T^a$ is a clopen (i.e., closed and open) sublattice of $ {\operatorname{Lat}}\;T$, with respect to the ``gap topology'' between subspaces. (2) If $ {\mathfrak{M}_1},{\mathfrak{M}_2} \in {\operatorname{Lat}}\;T,{\mathfrak{M}_1} \cap {\mathfrak{M}_2} \in {\operatorname{Lat}}\;\mathcal{A}_T^a$ and ${\mathfrak{M}_1} + {\mathfrak{M}_2}$ is closed in $ \mathfrak{X}$ and belongs to $ {\operatorname{Lat}}\;\mathcal{A}_T^a$, then $ {\mathfrak{M}_1}$ and ${\mathfrak{M}_2}$ also belong to $ {\operatorname{Lat}}\;\mathcal{A}_T^a$. (3) If $\mathfrak{M} \in {\operatorname{Lat}}\;T,R$ is the restriction of T to $\mathfrak{M}$ and $\bar T$ is the operator induced by T on the quotient space $ \mathfrak{X}/\mathfrak{M}$, then $\sigma (T) \subset \sigma (R) \cup \sigma (\bar T)$. Moreover, $\sigma (T) = \sigma (R) \cup \sigma (\bar T)$ if and only if $\mathfrak{M} \in {\operatorname{Lat}}\;\mathcal{A}_T^a$. The results also include an analysis of the semi-Fredholm index of R and $ \bar T$ at a point $\lambda \in \sigma (R) \cup \sigma (\bar T)\backslash \sigma (T)$ and extensions of the results to algebras between $ {\mathcal{A}_T}$ and $\mathcal{A}_T^a$.
On the spectra of the restrictions of an operator
Domingo A.
Herrero
45-58
Abstract: Let T be a bounded linear operator from a complex Banach space $\mathfrak{X}$ into itself and let $\mathfrak{M}$ be a closed invariant subspace of T. Let $ T\vert\mathfrak{M}$ denote the restriction of T to $\mathfrak{M}$ and let $\sigma$ denote the spectrum of an operator. The main results say that: (1) If $\mathfrak{X}$ is the closed linear span of a family $ \{ {\mathfrak{M}_v}\}$ of invariant subspaces, then every component of $\sigma (T)$ intersects the closure of the set $ { \cup _v}\sigma (T\vert{\mathfrak{M}_v})$ and every point of $\sigma (T)\backslash { \cup _v}\sigma (T\vert{\mathfrak{M}_v})$ is an approximate eigenvalue of T. (2) If $ \mathfrak{X}$ is the closed linear span of a finite family $ \{ {\mathfrak{M}_1}, \ldots ,{\mathfrak{M}_n}\}$ of invariant subspaces, and the spectra $\sigma (T\vert{\mathfrak{M}_j}),j = 1,2, \ldots ,n$, are pairwise disjoint, then $\mathfrak{X}$ is actually equal to the algebraic direct sum of the $ {\mathfrak{M}_j}$'s, the $ {\mathfrak{M}_j}$'s are hyperinvariant subspaces of T and $\sigma (T) = \cup _{j = 1}^n\sigma (T\vert{\mathfrak{M}_j})$. This last result is sharp in a certain specified sense. The results of (1) have a ``dual version'' $ (1')$; (1) and $ (1')$ are applied to analyze the spectrum of an operator having a chain of invariant subspaces which is ``piecewise well-ordered by inclusion", extending in several ways recent results of J. D. Stafney on the spectra of lower triangular matrices.
The Mackey Borel structure on the spectrum of an approximately finite-dimensional separable $C\sp*$-algebra
George A.
Elliott
59-68
Abstract: It is shown that the Mackey Borel structures on the spectra of any two approximately finite-dimensional separable $ {C^\ast}$-algebras not of type I are isomorphic.
Topological irreducibility of nonunitary representations of group extensions
Floyd L.
Williams
69-84
Abstract: A functional-analytic approach to the study of the topological irreducibility of certain nonunitary induced representations is set forth. The methods contrast, and in some sense, encompass those first initiated by E. Thieleker in [4], and are amenable to complete irreducibility questions as well. Several sufficient conditions for topological irreducibility are established. A sufficient condition for reducibility is also presented--the latter serving to explain an interesting counterexample due to J. M. G. Fell.
Embedding of closed categories into monoidal closed categories
Miguel L.
Laplaza
85-91
Abstract: S. Eilenberg and G. M. Kelly have defined a closed category as a category with internal homomorphism functor, left Yoneda natural arrows, unity object and suitable coherence axioms. A monoidal closed category is a closed category with an associative tensor product which is adjoint to the int-hom. This paper proves that a closed category can be embedded in a monoidal closed category: the embedding preserves any associative tensor product which may exist. Besides the usual tools of the theory of closed categories the proof uses the results of B. Day on promonoidal structures.
Plugging flows
Peter B.
Percell;
F. Wesley
Wilson
93-103
Abstract: A plug construction is a local modification of a nonsingular flow which severs certain kinds of recurrence properties. In this paper we investigate the effect of plug constructions on minimal sets, the nonwandering set, and the chain recurrent set and the explosions of these sets when a plug construction is perturbed.
Quasi-multipliers
Kelly
McKennon
105-123
Abstract: A quasi-multiplier m on an algebra A is a bilinear mapping from $A \times A$ into itself such that $m(ax,yb) = am(x,y)b$ for all $a,x,y,b \in A$. An introduction to the theory of quasi-multipliers on Banach algebras with minimal approximate identities is given and applications to $ {C^\ast}$-algebras and group algebras are developed.
Almost split sequences for group algebras of finite representation type
Idun
Reiten
125-136
Abstract: Let k be an algebraically closed field of characteristic p and G a finite group such that p divides the order of G. We compute all almost split sequences over kG when kG is of finite representation type, or more generally, for a finite dimensional k-algebra $\Lambda$ given by a Brauer tree. We apply this to show that if $\Lambda$ and $\Lambda '$ are stably equivalent k-algebras given by Brauer trees, then they have the same number of simple modules.
Knotting a $k$-connected closed ${\rm PL}$ $m$-manifold in $E\sp{2m-k}$
Jože
Vrabec
137-165
Abstract: Embeddings of a k-connected closed PL m-manifold $(0 \leqslant k \leqslant m - 3)$ in $ (2m - k)$-dimensional euclidean space are classified up to isotopy. Thus this paper completes the results stated, and partly proved, in J. F. P. Hudson's Piecewis linear topology.
Packing and covering constants for certain families of trees. II
A.
Meir;
J. W.
Moon
167-178
Abstract: In an earlier paper we considered the problem of determining the packing and covering constants for families of trees whose generating function y satisfied a relation $y = x\phi (y)$ for some power series $\phi$ in y. In the present paper we consider the problem for some families of trees whose generating functions satisfy a more complicated relation.
Weak $L\sb{1}$ characterizations of Poisson integrals, Green potentials and $H\sp{p}$ spaces
Peter
Sjögren
179-196
Abstract: Our main result can be described as follows. A subharmonic function u in a suitable domain $\Omega$ in $ {{\mathbf{R}}^n}$ is the difference of a Poisson integral and a Green potential if and only if u divided by the distance to $\partial \Omega $ is in weak $ {L_1}$ in $\Omega$. Similar conditions are given for a harmonic function to be the Poisson integral of an ${L_p}$ function on $ \partial \Omega$. Iterated Poisson integrals in a polydisc are also considered. As corollaries, we get weak ${L_1}$ characterizations of $ {H^p}$ spaces of different kinds.
A superposition theorem for unbounded continuous functions
Raouf
Doss
197-203
Abstract: Let ${R^n}$ be the n-dimensional Euclidean space. We prove that there are 4n real functions ${\varphi _q}$ continuous on ${R^n}$ with the following property: Every real function f, not necessarily bounded, continuous on ${R^n}$, can be written $ f(x) = \Sigma _{q = 1}^{2n + 1}g({\varphi _q}(x)) + \Sigma _{q = 2n + 2}^{4n}h({\varphi _q}(x)),x \in {R^n}$, where g, h are 2 real continuous functions of one variable, depending on f.
The closed leaf index of foliated manifolds
Lawrence
Conlon;
Sue
Goodman
205-221
Abstract: For M a closed, connected, oriented 3-manifold, a topological invariant is computed from the cohomology ring ${H^\ast}(M;{\mathbf{Z}})$ that provides an upper bound to the number of topologically distinct types of closed leaves any smooth transversely oriented foliation of M can contain. In general, this upper bound is best possible.
The diameter of orbits of compact groups of isometries; Newman's theorem for noncompact manifolds
David
Hoffman
223-233
Abstract: The diameter of orbits of a compact isometry group G of a Riemannian manifold M cannot be uniformly small. If the sectional curvature of M is bounded above by $ {b^2}$ (b real or pure imaginary), then explicit bounds are found for $D(M)$, where $D(M)$ is defined to be the largest number such that: If every orbit G has diameter less than $ D(M)$, then G acts trivially on M. These bounds depend only on b and the injectivity radius of M. The proofs involve an investigation of various types of convex sets and an estimate for distance contraction of the exponential map on a manifold with bounded curvature.
Some properties of families of convex cones
Meir
Katchalski
235-240
Abstract: The purpose of this paper is to study properties of finite families of convex cones in n-dimensional Euclidean space ${R^n}$, whose members all have the origin as a common apex. Of special interest are such families of convex cones in ${R^n}$ which have the following property: Each member of the family is of dimension n, the intersection of any two members is at least $(n - 1)$-dimensional, ..., the intersection of any n members is at least 1-dimensional and the intersection of all the members is the origin.
On the asymptotic distribution of closed geodesics on compact Riemann surfaces
Burton
Randol
241-247
Abstract: The set of lengths of closed geodesics on a compact Riemann surface is related to the Selberg zeta function in a manner which is evocative of the relationship between the rational primes and the Riemann zeta function. In this paper, this connection is developed to derive results about the asymptotic distribution of these lengths.
Disintegration of measures on compact transformation groups
Russell A.
Johnson
249-264
Abstract: Let G be a compact metrizable group which acts freely on a locally compact Hausdorff space X. Let X, $ \mu$ be a measure on $ X,\pi :X \to X/G \equiv Y$ the projection, $ \nu = \pi (\mu )$. We show that there is a $\nu$-Lusin-measurable disintegration of $ \mu$ with respect to it. We use this result to prove a structure theorem concerning T-ergodic measures on bitransformation groups (G, X, T) with G metric and X compact. We finish with some remarks concerning the case when G is not metric.
On characterizing the standard quantum logics
W. John
Wilbur
265-282
Abstract: Let $\mathcal{L}$ be a complete projective logic. Then $\mathcal{L}$ has a natural representation as the lattice of $\langle { \cdot , \cdot } \rangle$-closed subspaces of a left vector space V over a division ring D, where $\langle {\cdot,\cdot} \rangle$ is a definite $ \theta$-bilinear symmetric form on V, $\theta$ being some involutive antiautomorphism of D. Now a well-known theorem of Piron states that if D is isomorphic to the real field, the complex field or the sfield of quaternions, if $ \theta$ is continuous, and if the dimension of $ \mathcal{L}$ is properly restricted, then $ \mathcal{L}$ is just one of the standard Hilbert space logics. Here we also assume $\mathcal{L}$ is a complete projective logic. Then if every $\theta$-fixed element of D is in the center of D and can be written as $\pm \,d\theta (d)$, some $d \in D$, and if the dimension of $\mathcal{L}$ is properly restricted, we show that $\mathcal{L}$ is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron's theorem to discontinuous $\theta$. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.
Restrictions of convex subsets of $C(X)$
Per
Hag
283-294
Abstract: The main result of this paper is a theorem giving a measure-theoretic condition which is necessary and sufficient for a closed convex subset S of $C(X)$ to have the so-called bounded extension property with respect to a closed subset F of X. This theorem generalizes well-known results on closed subspaces by Bishop, Gamelin and Semadeni.
Ideals of coadjoint orbits of nilpotent Lie algebras
Colin
Godfrey
295-307
Abstract: For f a linear functional on a nilpotent Lie algebra g over a field of characteristic 0, let $J(f)$ be the ideal of all polynomials in $ S(g)$ vanishing on the coadjoint orbit through f in ${g^\ast}$, and let $I(f)$ be the primitive ideal of Dixmier in the universal enveloping algebra $U(g)$, corresponding to the orbit. An inductive method is given for computing generators ${P_1}, \ldots ,{P_r}$ of $J(f)$ such that $\varphi {P_1}, \ldots ,\varphi {P_r}$ generate $ I(f),\varphi$ being the symmetrization map from $S(g)$ to $U(g)$. Upper bounds are given for the number of variables in the polynomials ${P_i}$ and a counterexample is produced for upper bounds proposed by Kirillov.
On curves on formal groups
Robert A.
Morris;
Bodo
Pareigis
309-319
Abstract: The structure of the group of curves on a formal group is determined when the formal group is on a truncated power series algebra over a commutative ring. The resulting curve functor is faithful but not full. Applications to the Lie algebra of the formal group are given.
The $p$-adic log gamma function and $p$-adic Euler constants
Jack
Diamond
321-337
Abstract: We define $ {G_p}$, a p-adic analog of the classical log gamma function and show it satisfies relations similar to the standard formulas for log gamma. We also define p-adic Euler constants and use them to obtain results on $G{'_p}$ and on the logarithmic derivative of Morita's $ {\Gamma _p}$.
Matrix representation of simple halfrings
H. E.
Stone
339-353
Abstract: The structure of halfrings which are strong direct sums of minimal subtractive right ideals is studied. A class of right simple hemirings which contains both division hemirings and differential subsemirings of division rings is introduced and studied extensively as a tool in this investigation. A matrix representation is obtained for a class of halfrings which properly includes differential subsemirings of simple Artinian rings.
Deformations of Lie subgroups
Don
Coppersmith
355-366
Abstract: We give rigidity and universality theorems for embedded deformations of Lie subgroups. If $K \subset H \subset G$ are Lie groups, with $ {H^1}(K,g/h) = 0$, then for every ${C^\infty }$ deformation of H, a conjugate of K lies in each nearby fiber $ {H_s}$. If $H \subset G$ with $ {H^2}(H,g/h) = 0$, then there is a universal ``weak'' analytic deformation of H, whose base space is a manifold with tangent plane canonically identified with $\operatorname{Ker} {\delta ^1}$.
Addendum to: ``Knots with infinitely many spanning surfaces'' (Trans. Amer. Math. Soc. {\bf 229} (1977), 329--349)
Julian R.
Eisner
367-369