Markov neighborhoods for zero-dimensional basic sets
Dennis
Pixton
431-462
Abstract: We extend the local stable and unstable laminations for a zero-dimensional basic set to semi-invariant laminations of a neighborhood, and use these extensions to construct the appropriate analog of a Markov partition, which we call a Markov neighborhood. The main applications we give are in the perturbation theory for stable and unstable manifolds; in particular, we prove a transversality theorem. For these applications we require not only that the basic sets be zero dimensional but that they satisfy certain tameness assumptions. This leads to global results on improving stability properties via small isotopies.
On first countable, countably compact spaces. I. $(\omega \sb{1},\,\omega \sp{\ast} \sb{1})$-gaps
Peter J.
Nyikos;
Jerry E.
Vaughan
463-469
Abstract: This paper is concerned with the $ ({\omega_1},\omega_1^{\ast})$-gaps of F. Hausdorff and the topological spaces defined from them by Eric van Douwen. We construct special gaps in order that the associated gap spaces will have interesting topological properties. For example, the gap spaces we construct show that in certain models of set theory, there exist countably compact, first countable, separable, nonnormal ${T_2}$-spaces.
On the location of zeros of oscillatory solution
H.
Gingold
471-496
Abstract: The location of zeros of solutions of second order singular differential equations is provided by a new asymptotic decomposition formula. The approximate location of zeros is provided with high accuracy error estimates in the neighbourhood of the point at infinity. The same asymptotic formula suggested is applicable to the neighbourhood of most types of singularities as well as to the neighbourhoods of regular points.
The structure of $\omega \sb{1}$-separable groups
Paul C.
Eklof
497-523
Abstract: A classification theorem is proved for $ {\omega_1}$-separable ${\omega_1}$-free abelian groups of cardinality ${\omega_1}$ assuming Martin's Axiom $($MA$)$ and ${2^{\aleph_0}} > {\aleph_1}$. As a consequence, several structural results about direct sum decompositions of $ {\omega_1}$-separable groups are proved. These results are proved independent of ZFC, and, in addition, another structural property is proved undecidable in ${\text{ZFC}} + {\text{MA}} + {2^{\aleph_0}} > {\aleph_1}$. The problem of classifying these groups in a model of ${2^{\aleph_0}} = {\aleph_1}$ is also investigated.
An inequality with applications in potential theory
Boris
Korenblum;
Edward
Thomas
525-536
Abstract: An analytic inequality (announced previously) is proved and a certain monotonicity condition is shown to be essential for its validity, contrary to an earlier conjecture. Then, a generalization of the inequality, which takes into account the extent of nonmonotonicity, is established.
Global solvability on two-step compact nilmanifolds
Jacek M.
Cygan;
Leonard F.
Richardson
537-554
Abstract: We apply the methods of representation theory of nilpotent Lie groups to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations $Df = g$ in $ {C^\infty }$ of a two-step compact nilmanifold. We show that, under algebraically well-defined conditions on $D$ in the complexified Lie algebra, smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution. Such strong estimates are not possible on multidimensional tori.
Mean values of subsolutions of elliptic and parabolic equations
William P.
Ziemer
555-568
Abstract: Integral averages of weak subsolutions (and supersolutions) in $ {R^n}$ of quasilinear elliptic and parabolic equations are investigated. The important feature is that these integral averages are defined in terms of measures that reflect interesting geometric phenomena. Harnack type inequalities are established in terms of these integral averages.
Doubly slice knots and the Casson-Gordon invariants
Daniel
Ruberman
569-588
Abstract: We find knots in all dimensions which are algebraically but not geometrically doubly slice. Our new obstructions involve the Casson-Gordon invariants of the finite cyclic covers in odd dimensions and of the infinite cyclic cover in even dimensions. These same invariants provide new criteria for amphicheirality and invertibility of even-dimensional knots.
Gradings of ${\bf B}\sb{n}$ and ${\bf C}\sb{n}$ of finite representation type
Ibrahim
Assem;
Oscar
Roldán
589-609
Abstract: It was shown by Bongartz and Gabriel that the classification of simplyconnected algebras (i.e. finite-dimensional, basic, of finite representation type and with a simply-connected Auslander-Reiten graph) can be reduced to the study of certain numerical functions, called gradings, operating on a tree. Here, we classify in terms of their bounden species the simply-connected algebras arising from gradings of the Dynkin trees $ {{\mathbf{B}}_n}$ and ${{\mathbf{C}}_n}$, and show that these are exactly the tilted algebras of types ${{\mathbf{B}}_n}$ and ${{\mathbf{C}}_n}$, respectively.
Espaces $l\sp{p}$ dans les sous-espaces de $L\sp{1}$
S.
Guerre;
M.
Levy
611-616
Abstract: It is shown that every subspace $E$ of ${L^1}$ contains a subspace isomorphic to ${l^{p(E)}}$, where $p(E)$ is the upper bound of the set of real $ p$'s such that $ E$ is of type $ p$-Rademacher. As $ p(E)$ is also the upper bound of the set of real $p$'s such that $E$ embeds into ${L^p}$, this result answers a question of H. P. Rosenthal. The proof uses the theory of stable Banach spaces developed by J. L. Krivine and B. Maurey.
Time-ordered operators. II
Tepper L.
Gill
617-634
Abstract: In this paper, we substantially improve on the work of [G1]. After constructing the general mathematical foundations for linear time-ordered evolution equations, we apply our results to show that both the perturbation expansion and the Feynman diagram method are mathematically sound. We provide a remainder term so that the expansion may be considered exact at all orders. We then show that time-ordered operators naturally induce an operator-valued path integral whenever a transition kernel is given.
Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders
Eugene
Fabes;
Sandro
Salsa
635-650
Abstract: In this paper the authors prove unique solvability of the initial-Dirichlet problem for the heat equation in a cylindrical domain with Lipschitz base, lateral data in ${L^p},p \geqslant 2$, and zero initial values. A Poisson kernel for this problem is shown to exist with the property that its $ {L^2}$-averages over parabolic rectangles are equivalent to $ {L^1}$-averages over the same sets.
A restriction theorem for semisimple Lie groups of rank one
Juan A.
Tirao
651-660
Abstract: Let ${\mathfrak{g}_{\mathbf{R}}} = {\mathfrak{f}_{\mathbf{R}}} + {\mathfrak{p}_{\mathbf{R}}}$ be a Cartan decomposition of a real semisimple Lie algebra $ {\mathfrak{g}_{\mathbf{R}}}$ and let $\mathfrak{g} = \mathfrak{f} + \mathfrak{p}$ be the corresponding complexification. Also let $ {\mathfrak{a}_{\mathbf{R}}}$ be a maximal abelian subspace of $ {\mathfrak{p}_{\mathbf{R}}}$ and let $ \mathfrak{a}$ be the complex subspace of $ \mathfrak{p}$ generated by $ {\mathfrak{a}_{\mathbf{R}}}$. We assume $\dim {\mathfrak{a}_{\mathbf{R}}} = 1$. Now let $G$ be the adjoint group of $\mathfrak{g}$ and let $K$ be the analytic subgroup of $G$ with Lie algebra $ {\text{ad}}_\mathfrak{g}(\mathfrak{f})$. If $ S^\prime(\mathfrak{g})$ denotes the ring of all polynomial functions on $\mathfrak{g}$ then clearly $S^\prime(\mathfrak{g})$ is a $G$-module and a fortiori a $K$-module. In this paper, we determine the image of the subring $ S^\prime{(\mathfrak{g})^K}$ of $K$-invariants in $S^\prime(\mathfrak{g})$ under the restriction map $f \mapsto f{\vert _{\mathfrak{f} + \mathfrak{a}}}(f \in S^\prime{(\mathfrak{g})^K})$.
A generalization of $F$-spaces and some topological characterizations of GCH
Mary Anne
Swardson
661-675
Abstract: Several topological characterizations involving $F$-spaces of the continuum hypothesis are due to R. G Woods and E. K. van Douwen. We extend this work by defining a space $X$ to be an $ {F_\alpha }$-space if the union of $< \alpha$ cozero-sets is ${C^{\ast}}$-embedded in $X$ and by giving, for every infinite cardinal $ \alpha$, topological characterizations involving $ {F_\alpha }$-spaces of the cardinal equality ${2^\alpha } = {\alpha ^ + }$ .
Some applications of direct integral decompositions of $W\sp{\ast} $-algebras
Edward
Sarian
677-689
Abstract: Let $\mathcal{A}$ be a $ {W^{\ast}}$-algebra and let $A \in \mathcal{A}$. $ \mathcal{K}(\mathcal{A})$ and $C(A)$ represent certain convex subsets of $\mathcal{A}$. We prove the following via direct integral theory: (1) If $ \mathcal{A}$ is of type $ {{\text{I}}_\infty }$, $ {\text{II}}_\infty$, or III, then $C(A) = \{ 0\}$ iff ${\text{A}} \in \mathcal{K}(\mathcal{A})$. (2) If $ \mathcal{A}$ is of type I or II, then $ \mathcal{K}(\mathcal{A})$ is strongly dense in $ \mathcal{A}$. (3) If $\mathcal{A}$ is of type ${{\text{I}}_\infty }$, ${\text{II}}_\infty$, or III and $\mathcal{B}$ is a $ {W^{\ast}}$-subalgebra of $\mathcal{A}$, we give sufficient conditions for a Schwartz map $P$ of $ \mathcal{A}$ into $\mathcal{B}$ to annihilate $ \mathcal{K}(\mathcal{A})$. Several preliminary lemmas that are useful for direct integral theory are also proved.
Radial limits of $n$-subharmonic functions in the polydisc
W. C.
Nestlerode;
M.
Stoll
691-703
Abstract: We prove a relation between a certain weighted radial limit of an $ n$-subharmonic function in the polydisc ${U^n}$ and the representing measure of its least $n$-harmonic majorant. We apply this result to functions in $N({U^n})$, the Nevalinna class of ${U^n}$. In particular, we obtain a necessary condition for a function to belong to the component of the origin in $N({U^n})$. These results are extensions of the work of J. H. Shapiro and A. L. Shields to $n > 1$.
Linearization of second-order nonlinear oscillation theorems
Man Kam
Kwong;
James S. W.
Wong
705-722
Abstract: The problem of oscillation of super- and sublinear Emden-Fowler equations is studied. Established are a number of oscillation theorems involving comparison with related linear equations. Recent results on linear oscillation can thus be used to obtain interesting oscillation criteria for nonlinear equations.
Nonimmersions and nonembeddings of quaternionic spherical space forms
Teiichi
Kobayashi
723-728
Abstract: We determine the orders of the canonical elements in $KO$-rings of quaternionic spherical space forms $ {S^{4n + 3}}/{Q_k}$ and apply them to prove the nonexistence theorems of immersions and embeddings of ${S^{4n + 3}}/{Q_k}$ in Euclidean spaces.
All three-manifolds are pullbacks of a branched covering $S\sp{3}$ to $S\sp{3}$
Hugh M.
Hilden;
María Teresa
Lozano;
José María
Montesinos
729-735
Abstract: There are two main results in this paper. First, we show that every closed orientable $3$-manifold can be constructed by taking a pair of disjoint bounded orientable surfaces in $ {S^3}$, call them $ {F_1}$ and ${F_2}$; taking three copies of $ {S^3}$; splitting the first along ${F_1}$, the second along ${F_1}$ and ${F_2}$, and the third along ${F_2}$; and then pasting in the natural way. Second, we show that given any closed orientable $ 3$-manifold ${M^3}$ there is a $3$-fold irregular branched covering space, $ p:{M^3} \to {S^3}$, such that $p:{M^3} \to {S^3}$ is the pullback of the $3$-fold irregular branched covering space $q:{S^3} \to {S^3}$ branched over a pair of unknotted unlinked circles.
Invariant subspaces on Riemann surfaces of Parreau-Widom type
Mikihiro
Hayashi
737-757
Abstract: In this paper we generalize Beurling's invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed ( weak$^{\ast}$ closed, if $p = \infty$) $ {H^\infty }(d\chi )$-submodules, say $ \mathfrak{m}$, of ${L^p}(d\chi )$, $1 \leqslant p \leqslant \infty$, where $ d\chi$ is the harmonic measure on the Martin boundary of a Riemann surface $ R$, and ${H^\infty }(d\chi )$ is the set of boundary functions of all bounded analytic functions on $R$. Our main result is stated roughly as follows. Let $R$ be of Parreau-Widom type, that is, the space ${H^\infty }(R,\gamma )$ of bounded analytic sections contains a nonzero element for every complex flat line bundle $\gamma \in \pi {(R)^{\ast}}$. We may assume, without loss of generality, that the Green's function of $R$ vanishes at the infinity. Set $ {m^\infty }(\gamma ) = \sup \{ \vert f({\mathbf{O}})\vert:f \in {H^\infty }(R,\gamma ),\vert f\vert \leqslant 1\}$ for a fixed point $ {\mathbf{O}}$ of $ R$. Then, a necessary and sufficient condition in order that every such an $\mathfrak{m}$ takes either the form $ \mathfrak{m} = {C_E}{L^p}(d\chi )$, where ${C_E}$ is the characteristic function of a set $E$, or the form $\mathfrak{m} = q{H^p}(d\chi ,\gamma )$, where $\vert q\vert = 1$ a.e. and $\gamma$ is some element of $\pi {(R)^{\ast}}$ is that ${m^\infty }(\gamma )$ is continuous for the variable $ \gamma \in \pi {(R)^{\ast}}$.
Spectral properties of a certain class of complex potentials
V.
Guillemin;
A.
Uribe
759-771
Abstract: In this paper we discuss spectral properties of the Schroedinger operator $- \Delta + q$ on compact homogeneous spaces for certain complex valued potentials $q$. We show, for instance, that for these potentials the spectrum of $- \Delta + q$ is identical with the spectrum of $- \Delta$.
Quotients by ${\bf C}\sp{\ast} $ and ${\rm SL}(2,{\bf C})$ actions
Andrzej
Białynicki-Birula;
Andrew John
Sommese
773-800
Abstract: Let $ \rho :{{\mathbf{C}}^{\ast}} \times X \to X$ be a meromorphic action of $ {{\mathbf{C}}^{\ast}}$ on a compact normal analytic space. We completely classify $ {{\mathbf{C}}^{\ast}}$-invariant open $ U \subseteq X$ with a compact analytic space $U/T$ as a geometric quotient for a wide variety of actions, including all algebraic actions. As one application, we settle affirmatively a conjecture of D. Mumford on compact geometric quotients by $ {\text{SL(2}},{\mathbf{C}})$ of Zariski open sets of $ {({\mathbf{P}}_{\mathbf{C}}^1)^n}$.
On a question of Quillen
S. M.
Bhatwadekar;
R. A.
Rao
801-810
Abstract: Let $R$ be a regular local ring, and $ f$ a regular parameter of $ R$. Quillen asked whether every projective ${R_f}$-module is free. We settle this question when $ R$ is a regular local ring of an affine algebra over a field $k$. Further, if $R$ has infinite residue field, we show that projective modules over Laurent polynomial extensions of ${R_f}$ are also free.
Preorderings compatible with probability measures
Rolando
Chuaqui;
Jerome
Malitz
811-824
Abstract: The main theorem proved in this paper is: Let $B$ be a $\sigma$-complete Boolean algebra and $\succcurlyeq a$ binary relation with field $ B$ such that: (i) Every finite subalgebra $ B^{\prime}$ admits a probability measure $ \mu^{\prime}$ such that for $p,q \in B^{\prime},p \succcurlyeq q\;iff\mu 'p \geqslant \mu 'q$. (ii) If for every $i,{p_i},q \in B$ and $ {p_i} \subseteq {p_{i + 1}} \preccurlyeq q$, then ${ \cup_{i < \infty }}{p_i} \preccurlyeq q$. Under these conditions there is a $\sigma $-additive probability measure $\mu$ on $B$ such that: (a) If there is $a\;p \in B$, such that for every $q \subseteq p$ there is a $q^{\prime} \subseteq q$ with $ q^{\prime} \preccurlyeq q,q^{\prime} \npreceq 0$, and $q \npreceq q^{\prime}$, then we have that for every $p,q \in B,\mu \,p \geqslant \mu \,q\,iff\,p \succcurlyeq q$. (b) If for every $p \in B$, there is $a\;q \subseteq p$ such that $q^{\prime} \subseteq q$ implies $q \preccurlyeq q^{\prime}\;or\;q^{\prime} \preccurlyeq 0$, then we have that for every $p,q \in B,p \succcurlyeq q$ implies $\mu p \geqslant \mu q$.
The invariant subspace structure of nonselfadjoint crossed products
Baruch
Solel
825-840
Abstract: Let $\mathcal{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra $ M$ and a trace preserving $ ^{\ast}$-automorphism $ \alpha$ of $M$. We study the invariant subspace structure of the subalgebra $ {\mathcal{L}_ + }$ of $\mathcal{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group on $\mathcal{L}$ is nonnegative. We investigate the conditions for two invariant subspaces ${\mathcal{M}_1}$, and ${\mathcal{M}_2}$ (with ${Q_{1}},{Q_2}$ the corresponding orthogonal projections) to satisfy ${Q_1} = {R_\upsilon }\,{Q_2}\,R_\upsilon^{\ast}$ for some partial isometry ${R_{\upsilon }}$ in $\mathcal{L}^{\prime}$. We use this analysis to find the general form of a $\sigma$-weakly closed subalgebra of $\mathcal{L}$ that contains ${\mathcal{L}_ + }$.
The properties $\sp{\ast} $-regularity and uniqueness of $C\sp{\ast} $-norm in a general $\sp{\ast} $-algebra
Bruce A.
Barnes
841-859
Abstract: In this paper two properties of a $^{\ast}$-algebra $A$ are considered which are concerned with the relationship between the algebra and its ${C^{\ast}}$-enveloping algebra. These properties are that $A$ have a unique $ {C^{\ast}}$-norm, and that $ A$ be $ ^{\ast}$-regular. Both of these concepts are closely involved with the representation theory of the algebra.