Commuting and topological densities and liftings
Trevor J.
McMinn
1-22
Abstract: For fairly general conditions on a measure space, a group of bijections on the space and a topology on the space, densities and lifttings commuting with members of the group and with topologies finer than the given topology are obtained.
A generalization of Jarn\'\i k's theorem on Diophantine approximations to Ridout type numbers
I.
Borosh;
A. S.
Fraenkel
23-38
Abstract: Let s be a positive integer, $c > 1,{\mu _0}, \ldots ,{\mu _s}$ reals in [0, 1], $ \sigma = \Sigma _{i = 0}^s\;{\mu _i}$, and t the number of nonzero $ {\mu _i}$. Let ${\Pi _i}\;(i = 0, \ldots ,s)$ be $s + 1$ disjoint sets of primes and S the set of all $(s + 1)$-tuples of integers $({p_0}, \ldots ,{p_s})$ satisfying $\vert p_i^\ast\vert \leq c\vert{p_i}{\vert^{{\mu _i}}}$, and all prime factors of $ {p'_i}$ are in ${\Pi _i},i = 0, \ldots ,s$. Let $\lambda > 0$ if $t = 0,\lambda > \sigma /\min (s,t)$ otherwise, ${E_\lambda }$ the set of all real s-tuples $ ({\alpha _1}, \ldots ,{\alpha _s})$ satisfying $ \vert{\alpha _i} - {p_i}/{p_0}\vert < p_0^{ - \lambda }\;(i = 1, \ldots ,s)$ for an infinite number of $({p_0}, \ldots ,{p_s}) \in S$. The main result is that the Hausdorff dimension of $ {E_\lambda }$ is $\sigma /\lambda$. Related results are obtained when also lower bounds are placed on the $p_i^\ast$. The case $s = 1$ was settled previously (Proc. London Math. Soc. 15 (1965), 458-470). The case $ {\mu _i} = 1\;(i = 0, \ldots ,s)$ gives a well-known theorem of Jarník (Math. Z. 33 (1931), 505-543).
The structure of inseparable field extensions
William C.
Waterhouse
39-56
Abstract: The goal of this paper is to introduce some structural ideas into the hitherto chaotic subject of infinite inseparable field extensions. The basic discovery is that the theory is closely related to the well-developed study of primary abelian groups. This analogy undoubtedly has implications beyond those included here. We consider only modular extensions, which are the inseparable equivalent of galois extensions. §§2 and 3 develop the theory of pure independence, basic subfields, and tensor products of simple extensions. The following sections are devoted to Ulm invariants and their computation; the existence of nonzero invariants of arbitrary index is proved by means of a theorem which furnishes an actual connection between primary groups and inseparable fields. The final section displays some complications in the field extensions not occurring in abelian groups.
Topological properties of subanalytic sets
Robert M.
Hardt
57-70
Abstract: The stratification of a semianalytic or subanalytic set (that is, a set which locally is the proper analytic image of some semianalytic set) leads easily, by consecutive projections in Euclidean space, to a CW decomposition. In the category of subanalytic sets and continuous maps with subanalytic graphs, theories of slicing, intersection, and homology result through use of the topological chains defined by subanalytic sets.
On differentials of the first kind and theta constants for certain congruence subgroups
A. J.
Crisalli
71-84
Abstract: Let $\Gamma (8)$ denote the principal congruence subgroup of level 8 and let $ \Gamma (16,32)$ denote the subgroup of $ \Gamma (16)$ satisfying $ cd \equiv ab \equiv 0\;(\bmod 32)$. We are dealing only with the elliptic modular case. Consider the spaces of cusp forms of weight 2 (differentials of the first kind) with respect to these groups. It is proved that these spaces are generated by certain monomials of theta constants of degree 4.
Extensions and liftings of positive linear mappings on Banach lattices
Heinrich P.
Lotz
85-100
Abstract: Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into ${L^1}(\mu )$ or $C(X)$ for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.
Submersive and unipotent group quotients among schemes of a countable type over a field $k$
Paul
Cherenack
101-112
Abstract: An algebraic group G is called submersive if every quotient in affine schemes ${c^G}:{\text{Spec}}\;A \to {\text{Spec}}\;{A^G}$ which is surjective is also submersive. We prove that every unipotent group is submersive. Suppose G is submersive. We show that if ${c^G}({\text{Spec}}\;A)$ is open in ${\text{Spec}}\;{A^G}$ or if some restrictions on the action of G on A are made, $ {c^G}$ is a topological quotient. A criterion for semisimplicity of points is extended to the case where G is unipotent. Finally, applications of the theory are provided.
Conversion from nonstandard to standard measure spaces and applications in probability theory
Peter A.
Loeb
113-122
Abstract: Let $(X,\mathcal{A},\nu )$ be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard $\sigma$-algebra $ \mathfrak{M}$ containing the algebra $ \mathcal{A}$, where the extended real-valued measure $\mu$ on $ \mathfrak{M}$ is generated by the standard part of $\nu$. If f is $ \mathcal{A}$-measurable, then its standard part $^0f$ is $ \mathfrak{M}$-measurable on X. If f and $\mu$ are finite, then the $\nu $-integral of f is infinitely close to the $\mu$-integral of $^0f$. Applications include coin tossing and Poisson processes. In particular, there is an elementary proof of the strong Markov property for the stopping time of the jth event and a construction of standard sample functions for Poisson processes.
Results on sums of continued fractions
James L.
Hlavka
123-134
Abstract: Let $F(m)$ be the (Cantor) set of infinite continued fractions with partial quotients no greater than m and let $ F(m) + F(n) = \{ \alpha + \beta :\alpha \in F(m),\beta \in F(n)\}$. We show that $F(3) + F(4)$ is an interval of length 1.14 ... so every real number is the sum of an integer, an element of $F(3)$ and an element of $F(4)$. Similar results are given for $F(2) + F(7),F(2) + F(2) + F(4),F(2) + F(3) + F(3)$ and $ F(2) + F(2) + F(2) + F(2)$. The techniques used are applicable to any Cantor sets in R for which certain parameters can be evaluated.
On the Calkin algebra and the covering homotopy property
John B.
Conway
135-142
Abstract: Let $\mathcal{H}$ be a separable Hilbert space, $\mathcal{B}(\mathcal{H})$ the bounded operators on $\mathcal{H},\mathcal{K}$ the ideal of compact operators, and $\pi$ the natural map from $\mathcal{B}(\mathcal{H})$ onto the Calkin algebra $ \mathcal{B}(\mathcal{H})/\mathcal{K}$. Suppose X is a compact metric space and $\Phi :C(X) \times [0,1] \to \mathcal{B}(\mathcal{H})/\mathcal{K}$ is a continuous function such that $\Phi ( \cdot ,t)$ is a $ \ast$-isomorphism for each t and such that there is a $ \ast$-isomorphism $\psi :C(X) \to \mathcal{B}(\mathcal{H})$ with $\pi \psi ( \cdot ) = \Phi ( \cdot ,0)$. It is shown in this paper that if X is a simple Jordan curve, a simple closed Jordan curve, or a totally disconnected metric space then there is a continuous map $\Psi :C(X) \times [0,1] \to \mathcal{B}(\mathcal{H})$ such that $ \pi \Psi = \Phi$ and $ \Psi ( \cdot ,0) = \psi ( \cdot )$. Furthermore if X is the disjoint union of two spaces that both have this property, then X itself has this property.
On imbedding finite-dimensional metric spaces
Stephen Leon
Lipscomb
143-160
Abstract: The classical imbedding theorem in dimension theory gives a nice topological characterization of separable metric spaces of finite covering dimension. The longstanding problem of obtaining an analogous theorem for the nonseparable case is solved.
Commutative regular rings with integral closure
L.
Lipshitz
161-170
Abstract: First order conditions are given which are necessary for a commutative regular ring to have a prime integrally closed extension. If the ring is countable these conditions are also sufficient.
Immersions of complex hypersurfaces
Stanley R.
Samsky
171-184
Abstract: The varieties ${V^n}(d) = \{ [{z_0}, \ldots ,{z_n}] \in C{P^n}:z_0^d + \cdots + z_n^d = 0,d > 0\}$ form a class of manifolds containing the complex projective spaces. Maps from ${V^n}(d)$ to ${V^k}(e)$ are partially characterized by a ``degree". We prove some nonimmersion results which are phrased in terms of this degree, and which generalize the results of S. Feder [4] on complex projective spaces.
Duality theory for locally compact groups with precompact conjugacy classes. I. The character space
Terje
Sund
185-202
Abstract: Let G be a locally compact group, and let $\mathcal{X}(G)$ consist of the nonzero extreme points of the set of continuous, G-invariant, positive definite functions f on G such that $f(e) \leq 1$. $ \mathcal{X}(G)$ is called the character space, and is given the topology of uniform convergence on compacta. The purpose of the present paper is to extend the main results from the duality theory of abelian groups and [Z] groups to the class of ${[FC]^ - }$ groups (i.e., groups with precompact conjugacy classes), letting $\mathcal{X}(G)$ play the role of the character group in the abelian theory. Some of our theorems are only proved for the class ${[FD]^ - }\;( \subset {[FC]^ - })$. If $G \in {[FC]^ - }$ then $\mathcal{X}(G) \approx \mathcal{X}(H)$ where H is a certain $ {[FIA]^ - }$ quotient group. Hence there is no loss of generality to study character spaces of $ {[FIA]^ - }$ groups.
Extensions of maps as fibrations and cofibrations
Frank
Quinn
203-208
Abstract: Suppose $ f:X \to Y$ is a map of 1-connected spaces. In the ``stable'' range, roughly where the connectivity of Y exceeds the homology, or homotopy, dimension of X, it is well known that f can be extended as a cofibration $C \to X \to Y$, or respectively a fibration $X \to Y \to B$. A criterion is given for the existence of such extensions in a less restrictive ``metastable'' range. A main result is that if f is at least 2-connected and 2 con $Y \geq \dim Y - 1,\dim X$, then f extends as a cofibration if and only if the map $ (1 \times f)\Delta :X \to (X \times Y)/X$ factors through f.
Strongly prime rings
David
Handelman;
John
Lawrence
209-223
Abstract: A ring R is (right) strongly prime (SP) if every nonzero twosided ideal contains a finite set whose right annihilator is zero. Examples are domains, prime Goldie rings and simple rings; however, this notion is asymmetric and a right but not left SP ring is exhibited. All SP rings are prime, and every prime ring may be embedded in an SP ring. SP rings are nonsingular, and a regular SP ring is simple; since faithful rings of quotients of SP rings are SP, the complete ring of quotients of an SP ring is simple. All SP rings are coefficient rings for some primitive group ring (a generalization of a result proved for domains by Formanek), and this was the initial motivation for their study. If the group ring RG is SP, then R is SP and G contains no nontrivial locally finite normal subgroups. Coincidentally, SP rings coincide with the ATF rings of Rubin, and so every SP ring has a unique maximal proper torsion theory, and (0) and R are the only torsion ideals.($^{1}$) A list of questions is appended.
Endomorphism rings and direct sums of torsion free abelian groups
D. M.
Arnold;
E. L.
Lady
225-237
Abstract: Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.
Fixed points in representations of categories
J.
Adámek;
J.
Reiterman
239-247
Abstract: Fixed points of endomorphisms of representations, i.e. functors into the category of sets, are investigated. A necessary and sufficient condition on a category K is given for each of its indecomposable representations to have the fixed point property. The condition appears to be the same as that found by Isbell and Mitchell for Colim: ${\text{Ab}^K} \to {\text{Ab}}$ to be exact. A well-known theorem on mappings of Katětov and Kenyon is extended to transformations of functors.
On coverings and hyperalgebras of affine algebraic groups
Mitsuhiro
Takeuchi
249-275
Abstract: Over an algebraically closed field of characteristic zero, the universal group covering of a connected affine algebraic group, if such exists, can be constructed canonically from its Lie algebra only. In particular the isomorphism classes of simply connected affine algebraic groups are in 1-1 correspondence with the isomorphism classes of finite dimensional Lie algebras of some sort. We shall consider the counterpart of these results (due to Hochschild) in case of a positive characteristic, replacing the Lie algebra by the ``hyperalgebra". We show that the universal group covering of a connected affine algebraic group scheme can be constructed canonically from its hyperalgebra only and hence, in particular, that the category of simply connected affine algebraic group schemes is equivalent to a subcategory of the category of hyperalgebras of finite type which contains all the semisimple hyperalgebras.
On the two sheeted coverings of conics by elliptic curves
R. E.
MacRae
277-287
Abstract: Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form ${y^2} = f(x)$ where $f(x)$ is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field $L = k(u,v)$ of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.
Unions of Hilbert cubes
Raymond Y. T.
Wong;
Nelly
Kroonenberg
289-297
Abstract: This paper gives a partial solution to the problem whether the union of two Hilbert cubes is a Hilbert cube if the intersection is a Hilbert cube and a Z-set in one of them. Our results imply West's Intermediate Sum Theorem on Hilbert cube factors. Also a technique is developed to obtain Z-sets as limits of Z-sets.
The spinor genus of quaternion orders
Gordon L.
Nipp
299-309
Abstract: Let D be a global domain whose quotient field F does not have characteristic 2, let $ \mathfrak{A}$ be a quaternion algebra over F, and let $\mathfrak{D}$ be an order on $\mathfrak{A}$ over D. A right $\mathfrak{D}$-module M which is simultaneously a lattice on $ \mathfrak{A}$ over D is said to be right $ \mathfrak{D}$-generic if there exists $\alpha \in \mathfrak{A},N(\alpha ) \ne 0$, such that ${\alpha ^{ - 1}}M \in {\operatorname{gen}}\;\mathfrak{D}$. Our main result is that every right $\mathfrak{D}$-generic module is cyclic if and only if every class in the spinor genus of $\mathfrak{D}$ represents a unit in D. One consequence is that $ \mathfrak{D}$ is in a spinor genus of one class if and only if $\mathfrak{D}$-generic modules are cyclic and $\mathfrak{D}$ represents every unit represented by its spinor genus. In addition, it is shown that a necessary and sufficient condition that an integral ternary lattice L be in a spinor genus of one class is that every right $ {\mathfrak{D}_L}$-generic pair be equivalent to a two-sided $ {\mathfrak{D}_L}$-generic pair, where $ {\mathfrak{D}_L}$ is the quaternion order associated with L.
Some open mapping theorems for marginals
Larry Q.
Eifler
311-319
Abstract: Let S and T be compact Hausdorff spaces and let $ P(S),P(T)$ and $P(S \times T)$ denote the collection of probability measures on S, T and $S \times T$, respectively. Given a probability measure $\mu$ on $S \times T$, set $\pi \mu = (\alpha ,\beta )$ where $\alpha$ and $\beta$ are the marginals of $\mu$ on S and T. We prove that the mapping $\pi :P(S \times T) \to P(S) \times P(T)$ is norm open and $ {\text{weak}^\ast}$ open. An analogous result for ${L_1}(X \times Y,\mu \times \nu )$ where $ (X,\mu )$ and $ (Y,\nu )$ are probability spaces is established.
Semifree actions on homotopy spheres
Kai
Wang
321-337
Abstract: In this paper, we study the semifree ${Z_m}$ actions on homotopy sphere pairs. We show that in some cases the equivariant normal bundle to the fixed point set is equivariantly stably trivial. We compute the rank of the torsion free part of the group of semifree actions on homotopy sphere pairs in some cases. We also show that there exist infinitely many semifree ${Z_{4s}}$ actions on even dimensional homotopy sphere pairs.
The oscillation of an operator on $L\sp{p}$
George R.
Barnes;
Robert
Whitley
339-351
Abstract: We introduce and discuss the oscillation of an operator T mapping ${L^p}(S,\Sigma ,\mu )$ into a Banach space. We establish results relating the oscillation, a ``local norm", to the norm of the operator. Also using the oscillation we define a generalization of the Fredholm operators T with index $\kappa (T) < \infty$ and a corresponding perturbation class which contains the compact operators.
Mutual existence of product integrals in normed rings
Jon C.
Helton
353-363
Abstract: Definitions and integrals are of the subdivision-refinement type, and functions are from $R \times R$ to N, where R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element represented by 1 and a norm $\vert \cdot \vert$ with respect to which N is complete and $\vert 1\vert = 1$. If G is a function from $R \times R$ to N, then $G \in O{M^\ast}$ on [a, b] only if (i) $_x{\Pi ^y}(1 + G)$ exists for $a \leq x < y \leq b$ and (ii) if $\varepsilon > 0$, then there exists a subdivision D of [a, b] such that, if $\{ {x_i}\} _{i = 0}^n$ is a refinement of D and $0 \leq p < q \leq n$, then $\displaystyle \left\vert{}_{x_{p}}\prod ^{x_q} (1 + G) - \prod\limits_{i = p + 1}^q {(1 + {G_i})} \right\vert < \varepsilon ;$ and $G \in O{M^ \circ }$ on [a, b] only if (i) $_x{\Pi ^y}(1 + G)$ exists for $a \leq x < y \leq b$ and (ii) the integral $ \smallint _a^b\vert 1 + G - \Pi (1 + G)\vert$ exists and is zero. Further, $G \in O{P^ \circ }$ on [a, b] only if there exist a-subdivision D of [a, b] and a number B such that, if $\{ {x_i}\} _{i = 0}^n$ is a refinement of D and $0 < p \leq q \leq n$, then $\vert\Pi _{i = p}^q(1 + {G_i})\vert < B$. If F and G are functions from $R \times R$ to N, $F \in O{P^ \circ }$ on [a, b], each of $ {\lim _{x,y \to {p^ + }}}F(x,y)$ and ${\lim _{x,y \to {p^ - }}}F(x,y)$ exists and is zero for $p \in [a,b]$, each of ${\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)$ and $ {\lim _{x \to {p^ - }}}G(x,p)$ exists for $ p \in [a,b]$, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) $F + G \in OM^\ast$ on [a, b], (2) $F \in OM^\ast$ on [a, b], and (3) $G \in OM^\ast$ on [a, b]. In addition, with the same restrictions on F and G, any two of the following statements imply the other: (1) $F + G \in OM^\circ$ on [a, b], (2) $F \in OM^\circ$ on [a, b], and (3) $G \in OM^\circ$ on [a, b]. The results in this paper generalize a theorem contained in a previous paper by the author [Proc. Amer. Math. Soc. 42 (1974), 96-103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297-322].
Characteristic principal bundles
Harvey A.
Smith
365-375
Abstract: Characteristic principal bundles are the duals of commutative twisted group algebras. A principal bundle with locally compact second countable (Abelian) group and base space is characteristic iff it supports a continuous eigenfunction for almost every character measurably in the characters, also iff it is the quotient by Z of a principal E-bundle for every E in $ {\operatorname{Ext}}(G,Z)$ and a measurability condition holds. If a bundle is locally trivial, n.a.s.c. for it to be such a quotient are given in terms of the local transformations and Čech cohomology of the base space. Although characteristic G-bundles need not be locally trivial, the class of characteristic G-bundles is a homotopy invariant of the base space. The isomorphism classes of commutative twisted group algebras over G with values in a given commutative $ {C^\ast}$-algebra A are classified by the extensions of G by the integer first Čech cohomology group of the maximal ideal space of A.
On subnormal operators
Mehdi
Radjabalipour
377-389
Abstract: Let T be the adjoint of a subnormal operator defined on a Hilbert space H. For any closed set $\delta$, let ${X_T}(\delta ) = \{ x \in H$: there exists an analytic function ${f_x}:{\text{C}}\backslash \delta \to H$ such that $ (z - T){f_x}(z) \equiv x\}$. It is shown that T is decomposable (resp. normal) if ${X_T}(\partial {G_\alpha })$ is closed (resp. if ${X_T}(\partial {G_\alpha }) = \{ 0\} )$ for a certain family $ \{ {G_\alpha }\}$ of open sets. Some of the results are extended to the case that T is the adjoint of the restriction of a spectral or decomposable operator to an invariant subspace.
On K\=omura's closed-graph theorem
Michael H.
Powell
391-426
Abstract: Let $(\alpha )$ be a property of separated locally convex spaces. Call a locally convex space $E[\mathcal{J}]$ an $ (\bar \alpha )$-space if $\mathcal{J}$ is the final topology defined by ${\{ {u_i}:{E_i}[{\mathcal{J}_i}] \to E\} _{i \in I}}$, where each ${E_i}[{\mathcal{J}_i}]$ is an $(\alpha )$-space. Then, for each locally convex space $ E[\mathcal{J}]$, there is a weakest $(\bar \alpha )$-topology on E stronger that $ \mathcal{J}$, denoted $ {\mathcal{J}^{\bar \alpha }}$. Kōmura's closed-graph theorem states that the following statements about a locally convex space $ E[\mathcal{J}]$ are equivalent: (1) For every $(\alpha )$-space F and every closed linear map $ u: F \to E[\mathcal{J}]$, u is continuous. (2) For every separated locally convex topology $ {\mathcal{J}_0}$ on E, weaker than $ \mathcal{J}$, we have $\mathcal{J} \subset \mathcal{J}_0^{\bar \alpha }$. Much of this paper is devoted to amplifying Kōmura's theorem in special cases, some well-known, others not. An entire class of special cases, generalizing Adasch's theory of infra-(s) spaces, is established by considering a certain class of functors, defined on the category of locally convex spaces, each functor yielding various notions of ``completeness'' in the dual space.