Modules over coproducts of rings
George M.
Bergman
1-32
Abstract: Let ${R_0}$ be a skew field, or more generally, a finite product of full matrix rings over skew fields. Let ${({R_\lambda })_{\lambda \in \Lambda }}$ be a family of faithful ${R_0}$rings (associative unitary rings containing $ {R_0}$) and let $ R$ denote the coproduct ("free product") of the $ {R_\lambda }$ as $ {R_0}$-rings. An easy way to obtain an $R$-module $M$ is to choose for each $\lambda \in \Lambda \cup \{ 0\}$ an ${R_\lambda }$-module $ {M_\lambda }$, and put $M = \oplus {M_\lambda }{ \otimes _{{R_\lambda }}}R$. Such an $M$ will be called a ``standard'' $R$-module. (Note that these include the free $R$-modules.) We obtain results on the structure of standard $R$-modules and homomorphisms between them, and hence on the homological properties of $R$. In particular: (1) Every submodule of a standard module is isomorphic to a standard module. (2) If $M$ and $N$ are standard modules, we obtain simple criteria, in terms of the original modules ${M_\lambda },{N_\lambda }$, for $N$ to be a homomorphic image of $ M$, respectively isomorphic to a direct summand of $M$, respectively isomorphic to $M$. (3) We find that r gl$\dim R = {\sup _\Lambda }($r gl$\dim {R_\lambda })$ if this is > 0, and is 0 or 1 in the remaining case.
Coproducts and some universal ring constructions
George M.
Bergman
33-88
Abstract: Let $R$ be an algebra over a field $ k$, and $P,Q$ be two nonzero finitely generated projective $R$-modules. By adjoining further generators and relations to $R$, one can obtain an extension $S$ of $R$ having a universal isomorphism of modules, $i:P{ \otimes _R}S \cong Q{ \otimes _R}S$. We here study this and several similar constuctions, including (given a single finitely generated projective $ R$-module $P$) the extension $S$ of $R$ having a universal idempotent module-endomorphism $ e:P \otimes S \to P \otimes S$, and (given a positive integer $n$) the $k$-algebra $S$ with a universal $k$-algebra homomorphism of $R$ into its $n \times n$ matrix ring, $f:R \to {\mathfrak{m}_n}(S)$. A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring $ {R_0}$ ( $( = k \times k \times k,k \times k$, and $ k$ respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of $S$ is the same as that of $R$ unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation $[P] = [Q]$. These results allow one to construct a large number of unusual examples. We discuss the problem of obtaining similar results for some related constructions: the adjunction to $R$ of a universal inverse to a given homomorphism of finitely generated projective modules, $ f:P \to Q$, and the formation of the factor-ring $R/{T_P}$ by the trace ideal of a given finitely generated projective $R$-module $P$ (in other words, setting $P = 0$). The idea for a category-theoretic generalization of the ideas of the paper is also sketched.
Topological semigroups and representations. I
James C. S.
Wong
89-109
Abstract: Let $S$ be a topological semigroup (separately continuous multiplication) with identity and $W(S)$ the Banach space of all weakly almost periodic functions on $S$. It is well known that if $S = G$ is a locally compact group, then $ W(G)$ always has a (unique) invariant mean. In other words, there exists $m \in W{(G)^ \ast }$ such that $\vert\vert m\vert\vert = m(1) = 1$ and $m({l_s}f) = m({r_s}f) = m(f)$ for any $s \in G,f \in W(G)$ where ${l_s}f(t) = f(st)$ and ${r_s}f(t) = f(ts),t \in S$ The main purpose of this paper is to present several characterisations (functional analytic and algebraic) of the existence of a left (right) invariant mean on $W(S)$ In particular, we prove that $ W(S)$ has a left (right) invariant mean iff a certain compact topological semigroup $p{(S)^\omega }$ (to be defined) associated with $S$ contains a right (left) zero. Other results in this direction are also obtained.
Jordan rings with involution
Seong Nam
Ng
111-139
Abstract: Let $J$ be a Jordan ring with involution $ \ast$ in which $ 2x = 0$ implies $x = 0$ and in which $ 2J = J$. Let the set $ S$ of symmetric elements of $J$ be periodic and let $N$ be the Jacobson radical of $J$. Then ${N^2} = 0$ and $J/N$ is a subdirect sum of $\ast$-simple Jordan rings of the following types (1) a periodic field, (2) a direct sum of two simple periodic Jordan rings with exchange involution, (3) a $3 \times 3$ or $ 4 \times 4$ Jordan matrix algebra over a periodic field, (4) a Jordan algebra of a nondegenerate symmetric bilinear form on a vector space over a periodic field.
Join-irreducible cross product varieties of groups
James J.
Woeppel
141-148
Abstract: Let $ \mathfrak{U},\mathfrak{B}$ be varieties of groups which have finite coprime exponents, let $ \mathfrak{U}$ be metabelian and nilpotent with ``small'' nilpotency class, and let $ \mathfrak{B}$ be abelian. The product variety $\mathfrak{U}\mathfrak{B}$ is shown to be join-irreducible if and only if $ \mathfrak{U}$ is join-irreducible. This is done by obtaining a simple description for the critical groups generating $\mathfrak{U}\mathfrak{B}$ when $\mathfrak{U}$ is join-irreducible and finding a word which is not a law in $ \mathfrak{U}\mathfrak{B}$ but is a law in every proper subvariety of $ \mathfrak{U}\mathfrak{B}$
Extending continuous linear functionals in convergence vector spaces
S. K.
Kranzler;
T. S.
McDermott
149-168
Abstract: Let $(E,\tau )$ be a convergence vector space, $ M$ a subspace of $ E$, and $\varphi$ a linear functional on $ M$ continuous in the induced convergence structure. Sufficient and sometimes necessary conditions are given that (1) $\varphi$ has a continuous linear extension to the $\tau$-adherence $\bar M$ of$M$; (2) $\varphi$ has a continuous linear extension to $E$; (3) $\bar M$ is $\tau$-closed; (4) every $\tau$-closed convex subset of $E$ is
Homeomorphisms between Banach spaces
Roy
Plastock
169-183
Abstract: We consider the problem of finding precise conditions for a map $ F$ between two Banach spaces $X,Y$ to be a global homeomorphism. Using methods from covering space theory we reduce the global homeomorphism problem to one of finding conditions for a local homeomorphism to satisfy the ``line lifting property.'' This property is then shown to be equivalent to a limiting condition which we designate by $ (L)$. Thus we finally show that a local homeomorphism is a global homeomorphism if and only if $(L)$ is satisfied. In particular we show that if a local homeomorphism is (i) proper (Banach-Mazur) or (ii)
Subbundles of the tangent bundle
R. E.
Stong
185-197
Abstract: This paper studies pairs $(M,\xi )$ where $M$ is a closed manifold and $\xi$ is a $k$-dimensional subbundle of the tangent bundle of $ M$ in terms of cobordism.
On certain sequences of integers
K.
Thanigasalam
199-205
Abstract: Let the sequence $\{ {k_i}\}$ satisfy $2 \leqslant {k_1} \leqslant {k_2} \leqslant \cdots$. Then, under certain conditions satisfied by $\{ {k_i}\}$, it is shown that there exists an integer $s$ such that the sequence of integers of the form $ x_1^{{k_1}} + \cdots + x_s^{{k_s}}$ has positive density. Also, some special sequences having positive densities are constructed.
Semigroups of scalar type operators on Banach spaces
Ahméd Ramzy
Sourour
207-232
Abstract: The main result is that if $ \{ T(t):t \geqslant 0\}$ is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space $X$ and if the resolutions of the identity for $T(t)$ are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure $K( \cdot )$ such that $T(t) = \smallint \exp (\lambda t)dK(\lambda )$, for $t \geqslant 0$. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that $ R$ is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.
Linear control problems with total differential equations without convexity
M. B.
Suryanarayana
233-249
Abstract: Neustadt type existence theorems are given for optimal control problems described by Dieudonné-Rashevsky type total differential equations which are linear in the state variable. The multipliers from the corresponding conjugate problem are used to obtain an integral representation for the functional which in turn is used in conjunction with a Lyapunov type theorem on convexity of range of integrals to derive the existence of a usual solution from that of a generalized solution, which thus needs no convexity. Existence of optimal solutions is also proved in certain cases using an implicit function theorem along with the sufficiency of the maximum principle for optimality in the case of linear systems. Bang bang type controls are shown to exist when the system is linear in the control variable also.
Right-bounded factors in an LCM domain
Raymond A.
Beauregard
251-266
Abstract: A right-bounded factor is an element in a ring that generates a right ideal which contains a nonzero two-sided ideal. Right-bounded factors in an LCM domain are considered as a generalization of the theory of two-sided bounded factors in an atomic $2$-fir, that is, a weak Bezout domain satisfying the acc and dcc for left factors. Although some elementary properties are valid in a more general context most of the main results are obtained for an LCM domain satisfying $ ({\text{M}})$ and the dcc for left factors; the condition $({\text{M}})$ is imposed to insure that prime factorizations are unique in an appropriate sense. The right bound ${b^ \ast }$ of a right bounded element $ b$ is considered in general, then in case $b$ is a prime, and finally in case $ b$ is indecomposable. The effect of assuming that right bounds are two-sided is also considered.
Deficiency sets and bounded information reducibilities
Leonard P.
Sasso
267-290
Abstract: For recursively enumerable sets $A$ and $H$ of natural numbers $H$ is a deficiency set of $A$ if there is a one-one, recursive function $ f$ with $A = \operatorname{Rng} (f)$ and $H = \{ i:(\exists j)[i < j \mathrel{\&} f(j) < f(i)]\}$. The relation between recursively enumerable sets and their deficiency sets under bounded information reducibilities (i.e. weak truth table, truth table, bounded truth table, many-one, and one-one reducibility) is investigated.
The multiplicative Cousin problem and a zero set for the Nevanlinna class in the polydisc
Sergio E.
Zarantonello
291-313
Abstract: Let $\Omega$ be a polydomain in ${{\mathbf{C}}^n}$, the Nevanlinna class $N(\Omega )$ consists of all holomorphic functions $ f$ in $\Omega$ such that ${\log ^ + }\vert f\vert$ has an $n$-harmonic majorant in $\Omega$. Let ${U^n}$ be the open unit polydisc $\{ z \in {{\mathbf{C}}^n}:\vert{z_1}\vert < 1, \cdots ,\vert{z_n}\vert < 1\}$. THEOREM 1. Given an open covering ${({\Omega _\alpha })_{\alpha \in A}}$ of the closure $ {\bar U^n}$ of the polydisc, consisting of polydomains, and for each $\alpha \in A$ a function ${f_\alpha } \in N({\Omega _\alpha } \cap {U^n})$ such that for all $\alpha ,\beta \in A,{f_\alpha }f_\beta ^{ - 1}$ is an invertible element of $ N({\Omega _\alpha } \cap {\Omega _\beta } \cap {U^n})$. There exists a function $ F \in N({U^n})$ such that for all $\alpha \in A,Ff_\alpha ^{ - 1}$ is an invertible element of $ N({\Omega _\alpha } \cap {U^n})$. This result enables us to find the following sufficient condition for the zero sets of $ N({U^n})$: THEOREM 2. Let $f$ be a holomorphic function in ${U^n},n \geqslant 2$. If there exists a constant $ 0 < r < 1$ and a continuous function $ n:[r,1) \to [r,1)$ such that $\displaystyle \vert{z_n}\vert \leqslant n\left( {\frac{{\vert{z_1}\vert + \cdots + \vert{z_{n - 1}}\vert}}{{n - 1}}} \right)$ for all points $({z_1}, \cdots ,{z_n})$ satisfying $\vert{z_1}\vert > r, \cdots ,\vert{z_n}\vert > r$ and $f({z_1}, \cdots ,{z_n}) = 0$, then $f$ has the same zeros as some function $F \in N({U^n})$. In the above if ${\overline {\lim } _{\lambda \to 1}}n(x) < 1$, then $Z(f)$ is a Rudin variety in which case there is a bounded holomorphic function with the same zeros as $ f$.
A local result for systems of Riemann-Hilbert barrier problems
Kevin F.
Clancey
315-325
Abstract: The Riemann-Hilbert barrier problem (for $n$ pairs of functions) $\displaystyle G{\Phi ^ + } = {\Phi ^ - } + g$ is investigated for the square integrable functions on a union of analytic Jordan curves $C$ bounding a domain in the complex plane. In the special case, where at each point ${t_0}$ of $C$ the symbol $G$ has at most two essential cluster values $ {G_1}({t_0}),{G_2}({t_0})$, then the condition $ \det [(1 - \lambda ){G_1}({t_0}) + \lambda {G_2}({t_0})] \ne 0$, for all $ {t_0}$ in $C$ and all $\lambda (0 \leqslant \lambda \leqslant 1)$, implies the Riemann-Hilbert operator is Fredholm. In the case, where for some ${t_0}$ in $C$ and some $ {\lambda _0}(0 \leqslant {\lambda _0} \leqslant 1),\det [(1 - {\lambda _0}){G_1}({t_0}) + {\lambda _0}{G_2}({t_0})] = 0$, the Riemann-Hilbert operator is not Fredholm. An application is given to systems of singular integral equation on ${L^2}(E)$, where $E$ is a measurable subset of $C$.
Bordism invariants of intersections of submanifolds
Allen
Hatcher;
Frank
Quinn
327-344
Abstract: This paper characterizes certain geometric intersection problems in terms of bordism obstructions. These obstructions give a setting in which to study such things as parametrized $ h$-cobordisms (pseudoisotopy), and surgery above the middle dimension and on fibrations, where such intersection problems arise.
The space of conjugacy classes of a topological group
Dennis
Daluge
345-353
Abstract: The space $ {G^\char93 }$ of conjugacy classes of a topological group $G$ is the orbit space of the action of $ G$ on itself by inner automorphisms. For a class of connected and locally connected groups which includes all analytic $[Z]$-groups, the universal covering space of ${G^\char93 }$ may be obtained as the space of conjugacy classes of a group which is locally isomorphic with $G$, and the Poincaré group of ${G^\char93 }$ is found to be isomorphic with that of $G/G'$, the commutator quotient group. In particular, it is shown that the space ${G^\char93 }$ of a compact analytic group $ G$ is simply connected if and only if $G$ is semisimple. The proof of this fact has not appeared in the literature, even though more specialized methods are available for this case.
The $p$-class in a dual $B\sp{\ast} $-algebra
Pak Ken
Wong
355-368
Abstract: In this paper, we introduce and study the class ${A_p}(0 < p \leqslant \infty )$ in a dual ${B^ \ast }$-algebra $A$. We show that, for $1 \leqslant p \leqslant \infty ,{A_p}$ is a dual ${A^ \ast }$-algebra which is a dense two-sided ideal of $A$. If $1 < p < \infty$, we obtain that $ {A_p}$ is uniformly convex and hence reflexive. We also identify the conjugate space of $ {A_p}(1 \leqslant p < \infty )$. This is a generalization of the class $ {C_p}$ of compact operators on a Hilbert space.
A continuity property with applications to the topology of $2$-manifolds
Neal R.
Wagner
369-393
Abstract: A continuity property is proved for variable simply connected domains with locally connected boundaries. This theorem provides a link between limits of conformal mappings and of retractions. Applications are given to the space of retractions of a compact $2$-manifold ${M^2}$, where it is shown that the space of deformations retractions is contractible and the space of nullhomotopic retrac tions has the same homotopy type as ${M^2}$. Other applications include a proof that the space of retracts of ${M^2}$ (with a natural quotient topology) is an absolute neighborhood retract, and a type of global solution to the Dirichlet problem.
Existence and stability for partial functional differential equations
C. C.
Travis;
G. F.
Webb
395-418
Abstract: The existence and stability properties of a class of partial functional differential equations are investigated. The problem is formulated as an abstract ordinary functional differential equation of the form $du(t)/dt = Au(t) + F({u_t})$, where $ A$ is the infinitesimal generator of a strongly continuous semigroup of linear operators $ T(t),t \geqslant 0$, on a Banach space $X$ and $F$ is a Lipschitz operator from $C = C([ - r,0];X)$ to $X$. The solutions are studied as a semigroup of linear or nonlinear operators on $C$. In the case that $F$ has Lipschitz constant $L$ and $\vert T(t)\vert \leqslant {e^{\omega t}}$, then the asymptotic stability of the solutions is demonstrated when $\omega + L < 0$. Exact regions of stability are determined for some equations where $ F$ is linear.
Interpolation polynomials which give best order of approximation among continuously differentiable functions of arbitrary fixed order on $[-1,\,+1]$
A. K.
Varma
419-426
Abstract: The object of this paper is to show that there exists a polynomial $ {P_n}(x)$ of degree $\leqslant 2n - 1$ which interpolates a given function exactly at the zeros of $n$th Tchebycheff polynomial and for which $\vert\vert f - {P_n}\vert\vert \leqslant {C_k}{w_k}(1/n,f)$ where $ {w_k}(1/n,f)$ is the modulus of continuity of $f$ of $k$th order.
Sets of multiplicity and differentiable functions. II
Robert
Kaufman
427-435
Abstract: The stability of certain sets of multiplicity is studied with reference to special classes of differentiable functions. Kronecker sets are produced as examples of instability. The most difficult theorem uses probability theory and an estimation of Kolmogoroff's $ \varepsilon$-entropy in a certain space of functions.
Spectral order preserving matrices and Muirhead's theorem
Kong Ming
Chong
437-444
Abstract: In this paper, a characterization is given for matrices which preserve the Hardy-Littlewood-Pólya spectral order relation $ \prec$ for $n$-vectors in ${R^n}$. With this characterization, a new proof is given for the classical Muirhead theorem and some Muirhead-type inequalities are obtained. Moreover, sufficient conditions are also given for matrices which preserve the Hardy-Littlewood-Pólya weak spectral order relation $ \prec\prec$.
Erratum to ``On the uniform convergence of quasiconformal mappings''
Bruce
Palka
445-445