Coordinatization applied to finite Baer * rings
David
Handelman
1-34
Abstract: We clarify and algebraicize the construction of the 'regular rings' of finite Baer $^\ast$ rings. We first determine necessary and sufficient conditions of a finite Baer $^\ast$ ring so that its maximal ring of right quotients is the 'regular ring', coordinating the projection lattice. This is applied to yield significant improvements on previously known results: If R is a finite Baer $^\ast$ ring with right projections $ ^\ast$-equivalent to left projections $({\text{LP}} \sim {\text{RP}})$, and is either of type II or has 4 or more equivalent orthogonal projections adding to 1, then all matrix rings over R are finite Baer $^\ast$ rings, and they also satisfy ${\text{LP}} \sim {\text{RP}}$; if R is a real $A{W^\ast}$ algebra without central abelian projections, then all matrix rings over R are also $ A{W^\ast}$. An alternate approach to the construction of the 'regular ring' is via the Coordinatization Theorem of von Neumann. This is discussed, and it is shown that if a Baer $^\ast$ ring without central abelian projections has a 'regular ring', the 'regular ring' must be the maximal ring of quotients. The following result comes out of this approach: A finite Baer $ ^\ast$ ring satisfying the 'square root' (SR) axiom, and either of type II or possessing 4 or more equivalent projections as above, satisfies $ {\text{LP}} \sim {\text{RP}}$, and so the results above apply. We employ some recent results of J. Lambek on epimorphisms of rings. Some incidental theorems about the existence of faithful epimorphic regular extensions of semihereditary rings also come out.
On the degree of approximation of a function by the partial sums of its Fourier series
Elaine
Cohen
35-74
Abstract: When f is a $ 2\pi$ periodic function with rth order fractional derivative, $r \geqslant 0$, of p-bounded variation, Golubov has obtained estimates of the degree of approximation of f, in the ${L^q}$ norm, $q > p$, by the partial sums of its Fourier series. Here we consider the analogous problem for functions whose fractional derivatives are of $ \Phi$-bounded variation and obtain estimates of the degree of approximation in an Orlicz space norm. In a similar manner we shall extend various results that he obtained on degree of approximation in the sup norm.
Spectral properties of tensor products of linear operators. I
Takashi
Ichinose
75-113
Abstract: The aim of the present paper is to obtain, for tensor products of linear operators, their essential spectra in the sense of F. E. Browder, F. Wolf and M. Schechter and explicit formulae of their nullity, deficiency and index. The theory applies to $A \otimes I + I \otimes B$ and $A \otimes B$.
Injective Banach spaces of continuous functions
John
Wolfe
115-139
Abstract: A description is given of the compact Hausdorff spaces S such that the Banach space $C(S)$ of continuous functions on S is a ${P_\lambda }$-space for $ \lambda < 3$ (under the assumption that S satisfies the countable chain condition). The existence of extension operators from $C({X^\ast}\backslash X)$ to $C({X^\ast})$ is examined under the assumption that $C({X^\ast})$ is injective where $ {X^\ast}$ is some compactification of a locally compact extremally disconnected Hausdorff space X (if $C(S)$ is injective, S is of this form). Some new examples of injective spaces $ C(S)$ are given.
Galois groups and complex multiplication
Michael
Fried
141-163
Abstract: The Schur problem for rational functions is linked to the theory of complex multiplication and thereby solved. These considerations are viewed as a special case of a general problem, prosaically labeled the extension of constants problem. The relation between this paper and a letter of J. Herbrand to E. Noether (published posthumously) is speculatively summarized in a conjecture that may be regarded as an arithmetic version of Riemann's existence theorem.
Real structure in complex $L\sb{1}$-preduals
Daniel E.
Wulbert
165-181
Abstract: Call a complex Banach space selfadjoint if it is isometrically isomorphic to a selfadjoint subspace of a $ C(X,{\mathbf{C}})$-space. B. Hirsberg and A. Lazar proved that if the unit ball of a complex Lindenstrauss space, E, has an extreme point, then E is selfadjoint. Here we will give a characterization of selfadjoint Lindenstrauss spaces, and construct a nonselfadjoint complex Lindenstrauss space.
Ergodic transformations from an interval into itself
Tien Yien
Li;
James A.
Yorke
183-192
Abstract: A class of piecewise continuous, piecewise ${C^1}$ transformations on the interval $J \subset R$ with finitely many discontinuities n are shown to have at most n invariant measures.
A vector lattice topology and function space representation
W. A.
Feldman;
J. F.
Porter
193-204
Abstract: A locally convex topology is defined for a vector lattice having a weak order unit and a certain partition of the weak order unit, analogous to the order unit topology. For such spaces, called ``order partition spaces,'' an extension of the classical Kakutani theorem is obtained: Each order partition space is lattice isomorphic and homeomorphic to a dense subspace of ${C_c}(X)$ containing the constant functions for some locally compact X, and conversely each such $ {C_c}(X)$ is an order partition space. $({C_c}(X)$ denotes all continuous real-valued functions on X with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of ${C_c}(X)$ for X locally compact and realcompact. Conditions for an M-space to be an order partition space are provided.
Lattice-valued Borel measures. II
Surjit Singh
Khurana
205-211
Abstract: Let T be a completely regular Hausdorff space, ${C_b}(T)$ the set of all bounded real-valued continuous functions on T, E a boundedly monotone complete ordered vector space, and $\varphi :{C_b}(T) \to E$ a positive linear map. It is proved that under certain conditions there exist $ \sigma$-additive, $ \tau$-smooth or tight E-valued measures on T which represent $ \varphi$.
Compact manifolds in hyperbolicity
Robert
Brody
213-219
Abstract: In this paper we establish the strongest possible criterion for the hyperbolicity of a compact complex manifold: such a manifold is hyperbolic if and only if it contains no (nontrivial) complex lines. In addition, we study the behavior of such manifolds under deformation and, in particular, answer the two most natural questions about such deformations: Is the space of hyperbolic complex structures on a given $ {C^\infty }$ manifold open in the space of all its complex structures? (Yes.) Is it closed? (Not in general.) These results answer questions first posed by Kobayashi in [4] and [5].
Cardinal Hermite spline interpolation: convergence as the degree tends to infinity
M. J.
Marsden;
S. D.
Riemenschneider
221-244
Abstract: Let ${\mathcal{S}_{2m,r}}$, denote the class of cardinal Hermite splines of degree $2m - 1$ having knots of multiplicity r at the integers. For $f(x) \in {C^{r - 1}}(R)$, the cardinal Hermite spline interpolant to $f(x)$ is the unique element of ${\mathcal{S}_{2m,r}}$ which interpolates $ f(x)$ and its first $ r - 1$ derivatives at the integers. For $y = ({y^0}, \ldots ,{y^{r - 1}})$ an r-tuple of doubly-infinite sequences, the cardinal Hermite spline interpolant to y is the unique $ S(x) \in {\mathcal{S}_{2m,r}}$ satisfying ${S^{(s)}}(\nu) = {y^s},s = 0,1, \ldots ,r - 1$, and $ \nu$ an integer. The following results are proved: If $f(x)$ is a function of exponential type less than $ r\pi$, then the derivatives of the cardinal Hermite spline interpolants to $ f(x)$ converge uniformly to the respective derivatives of $f(x)$ as $ m \to \infty$. For functions from more general, but related, classes, weaker results hold. If y is an r-tuple of $ {l^p}$ sequences, then the cardinal Hermite spline interpolants to y converge to ${W_r}(y)$, a certain generalization of the Whittaker cardinal series which lies in the Sobolev space ${W^{p,r - 1}}(R)$. This convergence is in the Sobolev norm. The class of all such $ {W_r}(y)$ is characterized. For small values of r, the explicit forms of ${W_r}(y)$ are described.
Homotopy and uniform homotopy
Allan
Calder;
Jerrold
Siegel
245-270
Abstract: It is shown that the sets, homotopy and uniform homotopy classes of maps from a finite dimensional normal space to a space of finite type with finite fundamental group, coincide. Applications of this result to the study of remainders of Stone-Cech compactifications, Kan extensions, and other areas are given.
The Diophantine problem for addition and divisibility
L.
Lipshitz
271-283
Abstract: An algorithm is given for deciding existential formulas involving addition and the divisibility relation over the natural numbers.
An existence result on a Volterra equation in a Banach space
Stig-Olof
Londen
285-304
Abstract: Let W be a real reflexive Banach space, dense in a Hilbert space H and with dual $W'$. Let the injection $W \to H$ be continuous and compact. We consider the nonlinear integral equation $u''(t) + Au(t) = f'(t),t \geqslant 0$.
Bases for the positive cone of a partially ordered module
W. Russell
Belding
305-313
Abstract: $(R,{R^ + })$ is a partially ordered ring and $(M,{M^ + })$ is a strict $ (R,{R^ + })$-module. So M is a left R-module and $({R^ + }\backslash \{ 0\} )({M^ + }\backslash \{ 0\} ) \subseteq {M^ + }\backslash \{ 0\}$. Let ${M^ + }.B \subseteq {M^ + }$ is an ${R^ + }$-basis for ${M^ + }$ means ${R^ + }B = {M^ + }$ (spanning) and if r is in R, b in B with $0 < 'rb \leqslant 'b$ then $rb \notin {R^ + }(B\backslash \{ b\} )$ (independence). Result: If B and D are $ {R^ + }$-bases for $ {M^ + }$ then card $ B =$ card D and to within a permutation ${b_i} = {u_i}{d_i}$, for units ${u_i}$ of ${R^ + }$.
The centralizer of a Cartan subalgebra of a Jordan algebra
Edgar G.
Goodaire
314-322
Abstract: If L is a diagonable subspace of an associative algebra A over a field $\Phi \;(L$ is spanned by commuting elements and the linear transformations ad $x:a \mapsto x - xa,x \in L$, are simultaneously diagonalizable), then a map $\lambda :L \to \Phi$ is said to be a weight of L on an A-module V if the space ${V_\lambda } = \{ v \in V:vx = \lambda (x)v\;{\text{for}}\;{\text{all}}\;x \in L\}$ is nonzero. It is shown that if A is finite dimensional semisimple and the characteristic of $\Phi$ is zero then the centralizer of L in A is the centralizer of an element $x \in A$ if and only if x distinguishes the weights of L on every irreducible A-module. This theorem can be used to show that for each representative V of an isomorphism class of irreducible A-modules and for each weight $\lambda$ of L on V, the centralizer of L contains the matrix ring ${D_{{n_\lambda }}},D = {\text{End}_A}V,{n_\lambda } = {\dim _D}{V_\lambda }$ and in fact is the direct sum of all such algebras. If J is a finite dimensional simple reduced Jordan algebra, one can determine precisely those x in J whose centralizer in the universal enveloping algebra of J coincides with the centralizer of a Cartan subalgebra. The simple components of such a centralizer can also be found and in fact are listed for the degree $J \geqslant 3$ case.
Topological entropy at an $u$-explosion
Louis
Block
323-330
Abstract: In this paper an example is given of a ${C^2}$ map g from the circle onto itself, which permits an $\Omega$-explosion. It is shown that topological entropy (considered as a map from ${C^2}({S^1},{S^1})$ to the nonnegative real numbers) is continuous at g.
Characteristic numbers of $G$-manifolds and multiplicative induction
Michael
Bix;
Tammo
tom Dieck
331-343
Abstract: We determine those finite groups G for which characteristic numbers determine G-equivariant bordism in the unoriented and unitary cases.
Existence theorems for Warfield groups
Roger
Hunter;
Fred
Richman;
Elbert
Walker
345-362
Abstract: Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.
Some one-relator Hopfian groups
Donald J.
Collins
363-374
Abstract: The group presented by $\displaystyle (a,t;{t^{ - 1}}{a^l}t = {a^m})$ is non-Hopfian if $l,m \ne \pm 1$ and $\pi (l) \ne \pi (m)$, where $\pi (l)$ and $\pi (m)$ denote the sets of prime divisors of l and m. By contrast, we prove that if w is a word of the free group $F({a_1},{a_2})$ which is not primitive and not a proper power, then the group $({a_1},{a_2},t;{t^{ - 1}}{w^l}t = {w^m})$ is Hopfian.
A probabilistic approach to a boundary layer problem
Walter
Vasilsky
375-385
Abstract: An elliptic second order linear operator is approximated by the transition operator of a Markov chain, and the solution to the corresponding approximate boundary value problem is expanded in terms of a small parameter, up to the first order term. In characterizing the boundary values of the first order term in the expansion, a problem of a boundary layer arises, which is treated by probabilistic methods.
An extension of Carlson's theorem for entire functions of exponential type
R.
Gervais;
Q. I.
Rahman
387-394
Abstract: The paper contains some new extensions of the well-known theorem of F. Carlson for entire functions of exponential type.
Rational approximation to $e\sp{-x}$. II
Q. I.
Rahman;
G.
Schmeisser
395-402
Abstract: It is shown that as compared to reciprocals of polynomials of degree n, rational functions of degree n provide an effectively better uniform approximation to the function ${e^{ - x}}$ on $ [0,\infty )$.
Norbert Wiener's ergodic theorem for convex regions
Norberto A.
Fava;
Jorge H.
Nanclares
403-406
Abstract: It is proved that the geometric hypothesis of a theorem which generalizes Norbert Wiener's multiparameter ergodic theorem are satsified in the case of arbitrary convex regions, provided only that they form a substantial family as defined in the introduction.