Uncomplemented $C(X)$-subalgebras of $C(X)$
John Warren
Baker
1-15
Abstract: In this paper, the uncomplemented subalgebras of the Banach algebra $ C(X)$ which are isometrically and algebraically isomorphic to $C(X)$ are investigated. In particular, it is shown that if X is a 0-dimensional compact metric space with its $\omega$th topological derivative ${X^{(\omega )}}$ nonempty, then there is an uncomplemented subalgebra of $C(X)$ isometrically and algebraically isomorphic to $ C(X)$. For each ordinal $\alpha \geq 1$, a class ${\mathcal{C}_\alpha }$ of homeomorphic 0-dimensional uncountable compact metric spaces is introduced. It is shown that each uncountable 0-dimensional compact metric space contains an open-and-closed subset which belongs to some $ {\mathcal{C}_\alpha }$.
The values of exponential polynomials at algebraic points. I
Carlos Julio
Moreno
17-31
Abstract: A strengthening of Siegel's proof of the Hermite-Lindemann Theorem is given. The results are used to obtain lower bounds for the values of exponential polynomials at algebraic points. The question of how well the root of an exponential polynomial can be approximated by algebraic numbers is considered, and a lower bound is obtained for the absolute value of the difference between a root of the exponential polynomial and an algebraic number.
Archimedean-like classes of lattice-ordered groups
Jorge
Martinez
33-49
Abstract: Suppose $\mathcal{C}$ denotes a class of totally ordered groups closed under taking subgroups and quotients by o-homomorphisms. We study the following classes: (1) $ {\text{Res}}(\mathcal{C})$, the class of all lattice-ordered groups which are subdirect products of groups in $\mathcal{C}$; (2) $ {\text{Hyp}}(\mathcal{C})$, the class of lattice-ordered groups in $ {\text{Res}}(\mathcal{C})$ having all their l-homomorphic images in $ {\text{Res}}(\mathcal{C})$; Para $ (\mathcal{C})$, the class of lattice-ordered groups having all their principal convex l-subgroups in $ {\text{Res}}(\mathcal{C})$. If $\mathcal{C}$ is the class of archimedean totally ordered groups then Para $(\mathcal{C})$ is the class of archimedean lattice-ordered groups, $ {\text{Res}}(\mathcal{C})$ is the class of subdirect products of reals, and $ {\text{Hyp}}(\mathcal{C})$ consists of all the hyper archimedean lattice-ordered groups. We show that under an extra (mild) hypothesis, any given representable lattice-ordered group has a unique largest convex l-subgroup in $ {\text{Hyp}}(\mathcal{C})$; this socalled hyper- $ \mathcal{C}$-kernel is a characteristic subgroup. We consider several examples, and investigate properties of the hyper- $\mathcal{C}$-kernels. For any class $\mathcal{C}$ as above we show that the free lattice-ordered group on a set X in the variety generated by $ \mathcal{C}$ is always in $ {\text{Res}}(\mathcal{C})$. We also prove that $ {\text{Res}}(\mathcal{C})$ has free products.
Relative projectivity, the radical and complete reducibility in modular group algebras
D. C.
Khatri
51-63
Abstract: If $H \leq G$ and every G-module is H-projective then (G, H) is a projective pairing. If Rad $FG \subseteq ({\text{Rad}}\;FH)FG$ then (G, H) is said to have property p. A third property considered is that for each irreducible H-module the induced G-module be completely reducible. It is shown that these three are equivalent properties in many interesting cases. Also examples are given to show that they are, in general, independent of each other.
Structure theorems for certain topological rings
James B.
Lucke;
Seth
Warner
65-90
Abstract: A Hausdorff topological ring B is called centrally linearly compact if the open left ideals form a fundamental system of neighborhoods of zero and B is a strictly linearly compact module over its center. A topological ring A is called locally centrally linearly compact if it contains an open, centrally linearly compact subring. For example, a totally disconnected (locally) compact ring is (locally) centrally linearly compact, and a Hausdorff finite-dimensional algebra with identity over a local field (a complete topological field whose topology is given by a discrete valuation) is locally centrally linearly compact. Let A be a Hausdorff topological ring with identity such that the connected component c of zero is locally compact, A/c is locally centrally linearly compact, and the center C of A is a topological ring having no proper open ideals. A general structure theorem for A is given that yields, in particular, the following consequences: (1) If the additive order of each element of A is infinite or squarefree, then $A = {A_0} \times D$ where ${A_0}$ is a finite-dimensional real algebra and D is the local direct product of a family $({A_\gamma })$ of topological rings relative to open subrings $ ({B_\gamma })$, where each $({A_\gamma })$ is the cartesian product of finitely many finite-dimensional algebras over local fields. (2) If A has no nonzero nilpotent ideals, each $ {A_\gamma }$ is the cartesian product of finitely many full matrix rings over division rings that are finite dimensional over their centers, which are local fields. (3) If the additive order of each element of A is infinite or squarefree and if C contains an invertible, topologically nilpotent element, then A is the cartesian product of finitely many finite-dimensional algebras over R, C, or local fields.
Extremal structure in operator spaces
M.
Sharir
91-111
Abstract: (a) A characterization of extreme operators (in the unit ball of operators) between ${L^1}$-spaces is given, together with other related properties, (b) A general theorem of Kreĭn-Milman type for the unit ball of operator spaces is proved, and is applied to operators between ${L^1}$-spaces and to operators into C-spaces.
The module decomposition of $I(\bar A/A)$
Klaus G.
Fischer
113-128
Abstract: Let A and B be scalar rings with B an A-algebra. The B-algebra ${D^n}(B/A) = I(B/A)/{I^n}(B/A)$ is universal for n-truncated A-Taylor series on B. In this paper, we consider the $\bar A$ module decomposition of ${D^n}(\bar A/A)$ with a view to classifying the singularity A which is assumed to be the complete local ring of a point on an algebraic curve at a one-branch singularity. We assume that $A/M = k < A$ and that k is algebraically closed with no assumption on the characteristic. We show that ${D^n}(\bar A/A) = I(\bar A/A)$ for n large and that the decomposition of $ I(\bar A/A)$ as a module over the P.I.D. $\bar A$ is completely determined by the multiplicity sequence of A. The decomposition is displayed and a length formula for $ I(\bar A/A)$ developed. If B is another such ring, where $\bar B = \bar A = k[[t]]$, we show that $ I(\bar A/A) \cong I(\bar B/B)$ as $k[[t]]$ modules if and only if the multiplicity sequence of A is equal to the multiplicity sequence of B. If $ A < B < \bar A$, then $ I(\bar A/A) \cong I(\bar B/B)$ as $\bar A = \bar B$ modules if and only if the Arf closure of A and B coincide. This is equivalent to the existence of an algebra isomorphism between $ I(\bar A/A)$ and $I(\bar B/B)$.
Chebyshev constant and Chebyshev points
Susan L.
Friedman
129-139
Abstract: Using $ \lambda$th power means in the case $\lambda \geq 1$, it is proven that the Chebyshev constant for any compact set in $ {R_n}$, real Euclidean n-space, is equal to the radius of the spanning sphere. When $\lambda > 1$, the Chebyshev points of order m for all $m \geq 1$ are unique and coincide with the center of the spanning sphere. For the case $\lambda = 1$, it is established that Chebyshev points of order m for a compact set E in $ {R_2}$ are unique if and only if the cardinality of the intersection of E with its spanning circle is greater than or equal to three.
Complex Lindenstrauss spaces with extreme points
B.
Hirsberg;
A. J.
Lazar
141-150
Abstract: We prove that a complex Lindenstrauss space whose unit ball has at least one extreme point is isometric to the space of complex valued continuous affine functions on a Choquet simplex. If X is a compact Hausdorff space and $A \subset {C_{\text{C}}}(X)$ is a function space then A is a Lindenstrauss space iff A is selfadjoint and Re A is a real Lindenstrauss space.
Critical point theory for nonlinear eigenvalue problems with indefinite principal part
Melvyn S.
Berger
151-169
Abstract: A study of the nontrivial solutions of the operator equation
Ergodicity of the Cartesian product
Elias G.
Flytzanis
171-176
Abstract: ${h_1}$ is an ergodic conservative transformation on a $\sigma$-finite measure space and ${h_2}$ is an ergodic measure preserving transformation on a finite measure space. We study the point spectrum properties of ${h_1} \times {h_2}$. In particular we show ${h_1} \times {h_2}$ is ergodic if and only if ${h_1} \times {h_2}$ have no eigenvalues in common other than the eigenvalue 1. The conditions on ${h_1},{h_2}$ stated above are in a sense the most general for the validity of this result.
The constrained coefficient problem for typically real functions
George B.
Leeman
177-189
Abstract: Let $- 2 \leq c \leq 2$. In this paper we find the precise upper and lower bounds on the nth Taylor coefficient ${a_n}$ of functions $ f(z) = z + c{z^2} + \Sigma _{k = 3}^\infty {a_k}{z^k}$ typically real in the unit disk for $n = 3,4, \cdots $. In addition all the extremal functions are identified.
Symmetric completions and products of symmetric matrices
Morris
Newman
191-201
Abstract: We show that any vector of n relatively prime coordinates from a principal ideal ring R may be completed to a symmetric matrix of $ {\text{SL}}(n,R)$, provided that $n \geq 4$. The result is also true for $n = 3$ if R is the ring of integers Z. This implies for example that if F is a field, any matrix of ${\text{SL}}(n,F)$ is the product of a fixed number of symmetric matrices of $ {\text{SL}}(n,F)$ except when $n = 2$, $F = {\text{GF}}(3)$, which is a genuine exception.
The Veech structure theorem
Robert
Ellis
203-218
Abstract: The main result is the proof of the Veech structure theorem for point-distal flows without the assumption that the distal points form a residual set. This allows one to conclude that, in the case of metrizable flows, if there is one distal point then there is a residual set of such points.
Denjoy-type flows on orientable $2$-manifolds of higher genus
Carl S.
Hartzman
219-227
Abstract: The author generalizes A. Denjoy's theory of flows on a torus to compact orientable 2-manifolds of higher genus. Natural extensions of A. Denjoy's hypotheses are made and necessary conditions that a flow satisfy the new hypotheses are given.
Extending cell-like maps on manifolds
B. J.
Ball;
R. B.
Sher
229-246
Abstract: Let X be a closed subset of a manifold M and ${G_0}$ be a cell-like upper semicontinuous decomposition of X. We consider the problem of extending ${G_0}$ to a cell-like upper semicontinuous decomposition G of M such that $M/G \approx M$. Under fairly weak restrictions (which vanish if $M = {E^n}$ or ${S^n}$ and $n \ne 4$ we show that such a G exists if and only if the trivial extension of ${G_0}$, obtained by adjoining to $ {G_0}$ the singletons of $ M - X$, has the desired property. In particular, the nondegenerate elements of Bing's dogbone decomposition of ${E^3}$ are not elements of any cell-like upper semicontinuous decomposition G of ${E^3}$ such that ${E^3}/G \approx {E^3}$. Call a cell-like upper semicontinuous decomposition G of a metric space X simple if $ X/G \approx X$ and say that the closed set Y is simply embedded in X if each simple decomposition of Y extends trivially to a simple decomposition of X. We show that tame manifolds in ${E^3}$ are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat k-manifold in an m-manifold $(k,m \ne 4)$. Examples are given of an everywhere wild simply embedded simple closed curve in $ {E^3}$ and of a compact absolute retract which embeds in ${E^3}$ yet has no simple embedding in ${E^3}$.
Weighted Grothendieck subspaces
Jo ao B.
Prolla;
Silvio
Machado
247-258
Abstract: Let V be a family of nonnegative upper semicontinuous functions on a completely regular Hausdorff space X. For a locally convex Hausdorff space E, let $C{V_\infty }(X;E)$ be the corresponding Nachbin space, that is, the vector space of all continuous functions f from X into E such that vf vanishes at infinity for all $v \in V$, endowed with the topology given by the seminorms of the type $f\vert \to \sup \{ v(x)p(f(x));x \in X\}$, where $v \in V$ and p is a continuous seminorm on E. Given a vector subspace L of $C{V_\infty }(X;E)$, the set of all pairs $ x,y \in X$ such that either $0 = {\delta _x}\vert L = {\delta _y}\vert L$ or there is $t \in R,t \ne 0$, such that $0 \ne {\delta _x}\vert L = t{\delta _y}\vert L$, is an equivalence relation, denoted by $ {G_L}$, and we define for $ (x,y) \in {G_L},g(x,y) = 0$ or t, accordingly. The subsets $K{S_L}$, resp. $W{S_L}$, where $g(x,y) \geq 0$, resp. $g(x,y) \in \{ 0,1\}$, are likewise equivalence relations. The G-hull (resp. KS-hull, WS-hull) of L is the vector subspace $\{ f \in C{V_\infty }(X;E);f(x) = g(x,y)f(y)$ for all $(x,y) \in {G_L}\;({\text{resp}}.\;K{S_L},W{S_L})\}$ and L is a G-space (resp. KS-space, WS-space) if its G-hull (resp. KS-hull, WS-hull) is contained in its closure. This paper is devoted to the characterization, by invariance properties, of the G-spaces resp. KS-spaces and WS-spaces of a given Nachbin space $C{V_\infty }(X;E)$. As an application we derive an infinite-dimensional Weierstrass polynomial approximation theorem, and a Tietze extension theorem for Banach space valued compact mappings.
Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics
Thomas G.
Kurtz
259-272
Abstract: Let ${A_1},{A_2}, \cdots$ be dissipative sets that generate semigroups of nonlinear contractions ${T_1}(t),{T_2}(t) \cdots $ Conditions are given on $\{ {A_n}\}$ which imply the existence of a limiting semigroup T(t). The results include types of convergence besides strong convergence. As an application, it is shown that solutions of the pair of equations $\displaystyle {u_t} = - \alpha {u_x} + {\alpha ^2}({v^2} - {u^2})$ and $\displaystyle {v_t} = \alpha {v_x} + {\alpha ^2}({u^2} - {v^2}),$ $\alpha$ a constant, approximate the solutions of $\displaystyle {u_t} = 1/4({d^2}/d{x^2})\,\log \,u$ as $ \alpha$ goes to infinity.
Joint measures and cross-covariance operators
Charles R.
Baker
273-289
Abstract: Let ${H_1}$ (resp., ${H_2}$) be a real and separable Hilbert space with Borel $\sigma$-field $ {\Gamma _1}$ (resp., ${\Gamma _2}$), and let $ ({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$ be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on $({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$, i.e., joint measures, and the projections of such measures on $ ({H_1},{\Gamma _1})$ and $({H_2},{\Gamma _2})$. In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are obtained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized.
Zero-one laws for Gaussian measures on Banach space
Charles R.
Baker
291-308
Abstract: Let $\mathcal{B}$ be a real separable Banach space, $ \mu$ a Gaussian measure on the Borel $\sigma$-field of $ \mathcal{B}$, and ${B_\mu }[\mathcal{B}]$ the completion of the Borel $\sigma$-field under $\mu$. If $G \in {B_\mu }[\mathcal{B}]$ is a subgroup, we show that $ \mu (G) = 0$ or 1, a result essentially due to Kallianpur and Jain. Necessary and sufficient conditions are given for $\mu (G) = 1$ for the case where G is the range of a bounded linear operator. These results are then applied to obtain a number of 0-1 statements for the sample function properties of a Gaussian stochastic process. The zero-one law is then extended to a class of non-Gaussian measures, and applications are given to some non-Gaussian stochastic processes.
A Kurosh subgroup theorem for free pro-$\mathcal{C}$-products of pro-$\mathcal{C}$-groups
Dion
Gildenhuys;
Luis
Ribes
309-329
Abstract: Let $\mathcal{C}$ be a class of finite groups, closed under finite products, subgroups and homomorphic images. In this paper we define and study free pro- $\mathcal{C}$-products of pro- $\mathcal{C}$-groups indexed by a pointed topological space. Our main result is a structure theorem for open subgroups of such free products along the lines of the Kurosh subgroup theorem for abstract groups. As a consequence we obtain that open subgroups of free pro- $\mathcal{C}$-groups on a pointed topological space, are free pro- $ \mathcal{C}$-groups on (compact, totally disconnected) pointed topological spaces.
Asymptotic stability and spiraling properties for solutions of stochastic equations
Avner
Friedman;
Mark A.
Pinsky
331-358
Abstract: We consider a system of Itô equations in a domain in ${R^d}$. The boundary consists of points and closed surfaces. The coefficients are such that, starting for the exterior of the domain, the process stays in the exterior. We give sufficient conditions to ensure that the process converges to the boundary when $t \to \infty $. In the case of plane domains, we give conditions to ensure that the process ``spirals"; the angle obeys the strong law of large numbers.
Dirichlet problem for degenerate elliptic equations
Avner
Friedman;
Mark A.
Pinsky
359-383
Abstract: Let ${L_0}$ be a degenerate second order elliptic operator with no zeroth order term in an m-dimensional domain G, and let $L = {L_0} + c$. One divides the boundary of G into disjoint sets ${\Sigma _1},{\Sigma _2},{\Sigma _3};{\Sigma _3}$ is the noncharacteristic part, and on ${\Sigma _2}$ the ``drift'' is outward. When c is negative, the following Dirichlet problem has been considered in the literature: $Lu = 0$ in G, u is prescribed on $ {\Sigma _2} \cup {\Sigma _3}$. In the present work it is assume that $c \leq 0$. Assuming additional boundary conditions on a certain finite number of points of ${\Sigma _1}$, a unique solution of the Dirichlet problem is established.
On the structure of semigroups which are unions of groups
R. J.
Warne
385-401
Abstract: We characterize semigroups S which are unions of groups as generalized Schreier products of groups, semilattices of right zero semigroups, and semilattices of left zero semigroups. We then give several specializations of this result utilizing Schreier products, semidirect products, and direct products.
Conditions under which disks are $P$-liftable
Edythe P.
Woodruff
403-418
Abstract: A generalization of the concept of lifting of an n-cell is studied. In the usual upper semicontinuous decomposition terminology, let S be a space, $S/G$ be the decomposition space, and the projection mapping be $ P:S \to S/G$ . A set $X \subset S/G$ if $X'$ is homeomorphic to X and $ P(X')$ is X. Examples are given in which the union of two P-liftable sets does not P-lift. We prove that if compact 2-manifolds A and B each P-lift, their union is a disk in ${E^3}/G$, their intersection misses the singular points of the projection mapping, and the intersection of the singular points with the union of A and B is a 0-dimensional set, then the union of A and B does P-lift. Even if a disk D does not P-lift, it is proven that for $ \epsilon > 0$ there is a P-liftable disk $\epsilon$-homeomorphic to D, provided that the given decomposition of ${E^3}$ is either definable by 3-cells, or the set of nondegenerate elements is countable and $ {E^3}/G$ is homeomorphic to ${E^3}$. With further restrictions on the decomposition, this P-liftable disk can be chosen in such a manner that it agrees with D on the singular points of D.
Abstract homotopy theory and generalized sheaf cohomology
Kenneth S.
Brown
419-458
Abstract: Cohomology groups ${H^q}(X,E)$ are defined, where X is a topological space and E is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the ``homotopical algebra'' of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen's, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized cohomology theory) and the ``local to global'' spectral sequence (in sheaf cohomology theory).
Central idempotent measures on compact groups
Daniel
Rider
459-479
Abstract: Let G be a compact group with dual object $\Gamma = \Gamma (G)$ and let $M(G)$ be the convolution algebra of regular finite Borel measures on G. The author has characterized the central idempotent measures on certain G, including the unitary groups, in terms of the hypercoset structure of $\Gamma$. The characterization also says that, on certain G, a central idempotent measure is a sum of such measures each of which is absolutely continuous with respect to the Haar measure of a closed normal subgroup. The main result of this paper is an extension of this characterization to products of certain groups. The known structure of connected groups and a recent result of Ragozin on connected simple Lie groups will then show that the characterization is valid for connected groups. On the other hand, a simple example will show it is false in general for non-connected groups. This characterization was done by Cohen for abelian groups and the proof borrows extensively from Amemiya and Itô's simplified proof of Cohen's result.
On the existence of invariant measures for piecewise monotonic transformations
A.
Lasota;
James A.
Yorke
481-488
Abstract: A class of piecewise continuous, piecewise ${C^1}$ transformations on the interval [0, 1] is shown to have absolutely continuous invariant measures.
Diffeomorphisms homotopic to the identity
Edward C.
Turner
489-498
Abstract: In this paper, an inductive procedure for describing the group of isotopy classes of null homotopic diffeomorphisms of a manifold is developed --this process depends on the handlebody structure of the manifold. This group is also shown to be finitely generated and in many cases abelian.