Linear operators on $L\sb{p}$ for $0<p<1$
N. J.
Kalton
319-355
Abstract: If $0\, < \,p\, < \,1$ we classify completely the linear operators $ T:\,{L_p}\, \to \,X$ where X is a p-convex symmetric quasi-Banach function space. We also show that if $T:\,{L_p}\, \to \,{L_0}$ is a nonzero linear operator, then for $p\, < \,q\, \leqslant \,2$ there is a subspace Z of ${L_p}$, isomorphic to ${L_q}$, such that the restriction of T to Z is an isomorphism. On the other hand, we show that if $ p\, < \,q\, < \,\infty$, the Lorentz space $L(p,\,q)$ is a quotient of ${L_p}$ which contains no copy of ${l_p}$.
Convergence and Cauchy structures on lattice ordered groups
Richard N.
Ball
357-392
Abstract: This paper employs the machinery of convergence and Cauchy structures in the task of obtaining completion results for lattice ordered groups. §§1 and 2 concern l-convergence and l-Cauchy structures in general. §4 takes up the order convergence structure; the resulting completion is shown to be the Dedekind-MacNeille completion. §5 concerns the polar convergence structure; the corresponding completion has the property of lateral completeness, among others. A simple theory of subset types routinizes the adjoining of suprema in §3. This procedure, nevertheless, is shown to be sufficiently general to prove the existence and uniqueness of both the Dedekind-MacNeille completion in §4 and the lateral completion in §5. A proof of the existence and uniqueness of a proper class of similar completions comes free. The principal new hull obtained by the techniques of adjoining suprema is the type $\mathcal{Y}$ hull, strictly larger than the lateral completion in general.
Riemann surfaces and bounded holomorphic functions
Walter
Pranger
393-400
Abstract: The principal result of this article asserts the equivalence of the following four conditions on a hyperbolic Riemann surface X: (a) the following set $ z\vert\,\vert f(z)\vert\, \leqslant \,{\text{sup}}\,\vert f\vert$ on K for every bounded holomorphic section f of $\xi$ is compact for every unitary vector bundle $ \xi$ and every compact set K; (b) every unitary line bundle has nontrivial bounded holomorphic sections and the condition in (a) holds for $ \xi \, = \,{i_d}$; (c) every unitary line bundle has nontrivial bounded holomorphic sections and X is regular for potential theory; (d) every unitary line bundle has nontrivial bounded holomorphic sections and X is its own B-envelope of holomorphy. If X is a subset of C, these are also equivalent to the following: (e) for every unitary line bundle $\xi$ the bounded holomorphic sections are dense in the holomorphic sections.
The von Neumann kernel and minimally almost periodic groups
Sheldon
Rothman
401-421
Abstract: We calculate the von Neumann kernel $n(G)$ of an arbitrary connected Lie group. As a consequence we see that the closed characteristic subgroup $n(G)$ is also connected. It is shown that any Levi factor of a connected Lie group is closed. Then, various characterizations of minimal almost periodicity for a connected Lie group are given. Among them is the following. A connected Lie group G with radical R is minimally almost periodic (m.a.p.) if and only if $G/R$ is semisimple without compact factors and $G\, = \,{[G,\,G]^ - }$. In the special case where R is also simply connected it is proven that $ G\, = \,[G,\,G]$. This has the corollary that if the radical of a connected m.a.p. Lie group is simply connected then it is nilpotent. Next we prove that a connected m.a.p. Lie group has no nontrivial automorphisms of bounded displacement. As a consequence, if G is a m.a.p. connected Lie group, H is a closed subgroup of G such that $G/H$ has finite volume, and $\alpha$ is an automorphism of G with ${\text{disp}}(\alpha ,\,H)$ bounded, then $ \alpha$ is trivial. Using projective limits of Lie groups we extend most of our results on the characterization of m.a.p. connected Lie groups to arbitrary locally compact connected topological groups, and finally get a new and relatively simple proof of the Freudenthal-Weil theorem.
Central Fourier-Stieltjes transforms with an isolated value
Alan
Armstrong
423-437
Abstract: Let $\mu$ be a central Borel measure on a compact, connected group G. If 0 is isolated in the range of $ {\hat \mu }$, then there exists a closed, normal subgroup H of G such that ${\pi _H}\mu$, the restriction of $ \mu$ to the cosets of H, is the convolution of an invertible measure with a nonzero idempotent measure. This result extends I. Glicksberg's result for LCA groups. An example is given which shows that this result is false in general for disconnected groups.
Some curvature properties of locally conformal K\"ahler manifolds
Izu
Vaisman
439-447
Abstract: Curvature identities and holomorphic sectional curvature of locally conformal Kähler manifolds are investigated. Particularly, sufficient conditions for such manifolds to be globally conformal Kähler are derived.
Locally free affine group actions
J. F.
Plante
449-456
Abstract: Differentiable actions by the nonabelian 2-dimensional Lie group on compact manifolds are considered. When the action is locally free and the orbits have codimension one it is shown that there are at most finitely many minimal sets each containing a countably infinite number of cylindrical orbits. Examples are given to show that various codimension, differentiability, and minimality restrictions are necessary.
On linear algebraic semigroups
Mohan S.
Putcha
457-469
Abstract: Let K be an algebraically closed field. By an algebraic semigroup we mean a Zariski closed subset of ${K^n}$ along with a polynomially defined associative operation. Let S be an algebraic semigroup. We show that S has ideals $ {I_0},\, \ldots \,,\,{I_t}$ such that $S\, = \,{I_t}\, \supseteq \, \cdots \, \supseteq \,{I_0}$, ${I_0}$ is the completely simple kernel of S and each Rees factor semigroup ${I_k}/{I_{k - 1}}$ is either nil or completely 0-simple $ (k\, = \,1,\, \ldots \,,\,t)$. We say that S is connected if the underlying set is irreducible. We prove the following theorems (among others) for a connected algebraic semigroup S with idempotent set $E(S)$. (1) If $E(S)$ is a subsemigroup, then S is a semilattice of nil extensions of rectangular groups. (2) If all the subgroups of S are abelian and if for all $a\, \in \,S$, there exists $e\, \in \,E(S)$ such that $ea\, = \,ae\, = a$, then S is a semilattice of nil extensions of completely simple semigroups. (3) If all subgroups of S are abelian and if S is regular, then S is a subdirect product of completely simple and completely 0-simple semigroups. (4) S has only trivial subgroups if and only if S is a nil extension of a rectangular band.
On linear algebraic semigroups. II
Mohan S.
Putcha
471-491
Abstract: We continue from [11] the study of linear algebraic semigroups. Let S be a connected algebraic semigroup defined over an algebraically closed field K. Let $\mathcal{U}(S)$ be the partially ordered set of regular $ \mathcal{J}$-classes of S and let $E(S)$ be the set of idempotents of S. The following theorems (among others) are proved. (1) $\mathcal{U}(S)$ is a finite lattice. (2) If S is regular and the kernel of S is a group, then the maximal semilattice image of S is isomorphic to the center of $E(S)$. (3) If S is a Clifford semigroup and $f\, \in \,E(S)$, then the set $\{ \,e\,\vert\,e\, \in \,E(S),\,e\, \geqslant \,f\}$ is finite. (4) If S is a Clifford semigroup, then there is a commutative connected closed Clifford subsemigroup T of S with zero such that T intersects each $ \mathcal{J}$-class of S. (5) If S is a Clifford semigroup with zero, then S is commutative and is in fact embeddable in $({K^n},\, \cdot )$ for some $n\, \in \,{\textbf{Z}^ + }$. (6) If ${\text{ch}}\, \cdot \,K\, = \,0$ and S is a commutative Clifford semigroup, then S is isomorphic to a direct product of an abelian connected unipotent group and a closed connected subsemigroup of $({K^n},\, \cdot )$ for some $n\, \in \,{\textbf{Z}^ + }$. (7) If S is a regular semigroup and $ {\text{dim}}\, \cdot \,S\, \leqslant \,2$, then $\left\vert {\mathcal{U}(S)} \right\vert\, \leqslant \,4$. (8) If S is a Clifford semigroup with zero and ${\text{dim}}\, \cdot \,S\, = \,3$, then $\left\vert {E(S)} \right\vert\, = \,\left\vert {\mathcal{U}(S)} \right\vert$ can be any even number $ \geqslant \,8$. (9) If S is a Clifford semigroup then $\mathcal{U}(S)$ is a relatively complemented lattice and all maximal chains in $\mathcal{U}(S)$ have the same number of elements.
Some categorical equivalences for $E$-unitary inverse semigroups
Mario
Petrich
493-503
Abstract: The structure of E-unitary inverse semigroups has been described by McAlister and by Reilly and the author. The parameters in the first structure theorem may be made into a category, and the same holds for the parameters in the second structure theorem. We prove that each of these categories is equivalent to the category of E-unitary inverse semigroups and their homomorphisms. We also provide functors between the two first-mentioned categories which are naturally equivalent to the composition of the functors figuring in the categorical equivalence referred to above.
An application of homological algebra to the homotopy classification of two-dimensional CW-complexes
Micheal N.
Dyer
505-514
Abstract: Let $\pi$ be ${Z_m}\, \times \,{Z_n}$. In this paper the homotopy types of finite connected two dimensional CW-complexes with fundamental group $\pi$ are shown to depend only on the Euler characteristic. The basic method is to study the structure of the group ${\text{Ext}}_{Z\pi }^1(I{\pi ^2},\,Z)$ as a principal ${\text{End(}}I{\pi ^2}{\text{)}}$-module.
Branched extensions of curves in orientable surfaces
Cloyd L.
Ezell;
Morris L.
Marx
515-532
Abstract: Given a set of regular curves ${f_1}\,,\,\ldots,\,{f_\rho }$ in an orientable surface N, we are concerned with the existence and structure of all sense-preserving maps $F:\,M\, \to \,N$ where (a) M is a bordered orientable surface with $\rho$ boundary components ${K_1},\ldots,\,{K_\rho }$, (b) $ F\vert{K_i}\, = \,{f_i},\,i\, = \,1,\,\ldots,\,\rho$, (c) at each interior point of M, there is an integer n such that F is locally topologically equivalent to the complex map $ w\, = \,{z^n}$.
Branched extensions of curves in compact surfaces
Cloyd L.
Ezell
533-546
Abstract: A polymersion is a map $ F:\,M\, \to \,N$ where M and N are compact surfaces, orientable or nonorientable, M a surface with boundary, where (a) At each interior point of M, there is an integer $ n\, \geqslant \,1$ such that F is topologically equivalent to the complex map ${z^n}$ in a neighborhood about the point. (b) At each point x in the boundary of M, $ \delta M$, there is a neighborhood U containing x such that U is homeomorphic to F(U). A normal polymersion is one where $F(\delta M)$ is a normal set of curves in N. We are concerned with establishing a combinatorial representation for normal polymersions which map to arbitrary compact surfaces.
Quasilinear evolution equations in Banach spaces
Michael G.
Murphy
547-557
Abstract: This paper is concerned with the quasi-linear evolution equation $[0,\,T],\,u(0)\, = \,{x_0}$ in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a ``limit solution.'' The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed.
Effective $p$-adic bounds for solutions of homogeneous linear differential equations
B.
Dwork;
P.
Robba
559-577
Abstract: We consider a finite set of power series in one variable with coefficients in a field of characteristic zero having a chosen nonarchimedean valuation. We study the growth of these series near the boundary of their common ``open'' disk of convergence. Our results are definitive when the wronskian is bounded. The main application involves local solutions of ordinary linear differential equations with analytic coefficients. The effective determination of the common radius of convergence remains open (and is not treated here).
Affine connections and defining functions of real hypersurfaces in ${\bf C}\sp{n}$
Hing Sun
Luk
579-588
Abstract: The affine connection and curvature introduced by Tanaka on a strongly pseudoconvex real hypersurface are computed explicitly in terms of its defining function. If Fefferman's defining function is used, then the Ricci form is shown to be a function multiple of the Levi form. The factor is computable by Fefferman's algorithm and its positivity implies the vanishing of certain cohomology groups (of the $ {\bar \partial _b}$ complex) in the compact case.
Complementary series for $p$-adic groups. I
Allan J.
Silberger
589-598
Abstract: Let $\Omega$ be a nonarchimedean local field, G the group of $\Omega$-points of a connected reductive algebraic group defined over $\Omega$. This paper establishes that to each zero of the Plancherel measure of G one can associate complementary series. Our result is the analogue for p-adic groups of a similar statement, announced separately by Knapp-Stein and Harish-Chandra, for real groups.
Approximately finite-dimensional $C\sp{\ast} $-algebras and Bratteli diagrams
A. J.
Lazar;
D. C.
Taylor
599-619
Abstract: We determine properties of an AF algebra by observing the characteristics of its diagram. In particular, we characterize AF algebras that are liminal, postliminal, antiliminal and with continuous trace; moreover, we characterize liminal AF algebras with Hausdorff spectrum. Some elementary examples of AF algebras with certain desired properties are constructed by using these characterizations.
Hermite-Birkhoff interpolation in the $n$th roots of unity
A. S.
Cavaretta;
A.
Sharma;
R. S.
Varga
621-628
Abstract: Consider, as nodes for polynomial interpolation, the nth roots of unity. For a sufficiently smooth function $ f(z)$, we require a polynomial $p(z)$ to interpolate f and certain of its derivatives at each node. It is shown that the so-called Pólya conditions, which are necessary for unique interpolation, are in this setting also sufficient.
Flows on fibre bundles
J. L.
Noakes
629-635
Abstract: Conditions are given under which a fibrewise flow on a fibre bundle must have a nonempty catastrophe space.
Erratum to: ``On parabolic measures and subparabolic functions''
Jang Mei G.
Wu
636-636