Free products of topological groups with central amalgamation. I
M. S.
Khan;
Sidney A.
Morris
405-416
Abstract: It is proved that the amalgamated free product of any two Hausdorff topological groups exists and is Hausdorff, providing the subgroup which is being amalgamated is closed and central.
Free products of topological groups with central amalgamation. II
M. S.
Khan;
Sidney A.
Morris
417-432
Abstract: In Free products of topological groups with central amalgamation. I, we introduced the notion of amalgamated free product of topological groups and showed that if $ A$ is a common central closed subgroup of Hausdorff topological groups $ G$ and $H$, then the amalgamated free product $G{\coprod _A}H$ exists and is Hausdorff. In this paper, we give an alternative much shorter (but less informative) proof of this result. We then proceed to describe the properties of $G{\coprod _A}H$. In particular, we find necessary and sufficient conditions for $G{\coprod _A}H$ to be a locally compact Hausdorff group, a complete metric group, and a maximally almost periodic group. Properties such as being a Baire space and connectedness are also investigated. In the case that $G$, $H$ and $A$ are $ {k_\omega }$-groups, the topology of $ G{\coprod _A}H$ is fully described. A consequence of this description is that for ${k_\omega }$-groups $ G{\coprod _A}H$ is homeomorphic to $(G{ \times _A}H) \times F(G/A\Lambda H/A)$, where $G{ \times _A}H$ is the direct product of $G$ and $H$ with $A$ amalgamated, and $F(G/A\Lambda H/A)$ is the free topological group on the smash product of $G/A$ and $H/A$.
Lattices over orders: finitely presented functors and preprojective partitions
M.
Auslander;
S. O.
Smalø
433-446
Abstract: Suppose $ R$ is a commutative noetherian equidimensional Gorenstein ring and $ \Lambda$ an $ R$-algebra which is finitely generated as an $R$-module. A $\Lambda$-module $M$ is a lattice if ${M_{\underline{\underline p} }}$ is ${\Lambda _{\underline{\underline p} }}$-projective and $ {\text{Ho}}{{\text{m}}_R}{(M,R)_{\underline{\underline p} }}$ is $ \Lambda _{\underline{\underline p} }^{{\text{op}}}$-projective for all nonmaximal prime ideals $\underline{\underline p} $ in $R$. We assume that $\Lambda$ is an $R$-order in the sense that $\Lambda$ is a lattice when viewed as a $ \Lambda$-module. The first main result is to show that simple contravariant functors from lattices to abelian groups are finitely presented. This is then applied to showing that if $ R$ is also local and complete, then the category of lattices has a preprojective partition. This generalizes previous results of the authors in the cases $R$ is artinian or a discrete valuation ring.
Asymptotic analysis of Gaussian integrals. I. Isolated minimum points
Richard S.
Ellis;
Jay S.
Rosen
447-481
Abstract: This paper derives the asymptotic expansions of a wide class of Gaussian function space integrals under the assumption that the minimum points of the action are isolated. Degenerate as well as nondegenerate minimum points are allowed. This paper also derives limit theorems for related probability measures which correspond roughly to the law of large numbers and the central limit theorem. In the degenerate case, the limits are non-Gaussian.
Reproducing kernels and bilinear sums for $q$-Racah and $q$-Wilson polynomials
Mizan
Rahman
483-508
Abstract: A five-parameter family of reproducing kernels is constructed for $ q$-Racah polynomials. Special cases for $q$-Hahn and little $q$-Jacobi polynomials are considered by selecting the parameters appropriately. Corresponding bilinear sums are also obtained for a whole range of $q$-orthogonal polynomials. As a special case, some product formulas are obtained for $ q$-Racah and $ q$-Wilson polynomials.
Irreducible representations of $A\sb{n}$ with a $1$-dimensional weight space
D. J.
Britten;
F. W.
Lemire
509-540
Abstract: In this paper we classify all irreducible linear representations of the simple Lie algebra ${A_n}$ which admit a one-dimensional weight space with respect to some Cartan subalgebra $H$ of ${A_n}$. We first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the Cartan subalgebra $H$ in the universal enveloping algebra of ${A_n}$ to the base field. We construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of $ {A_n}$.
A general principle for limit theorems in finitely additive probability
Rajeeva L.
Karandikar
541-550
Abstract: In this paper we formulate and prove a general principle which enables us to deduce limit theorems for sequences of independent random variables in a finitely additive setting from their analogues in the conventional countably additive setting.
Transfinite duals of quasireflexive Banach spaces
Steven F.
Bellenot
551-577
Abstract: The transfinite duals of a space with a neighborly basis are constructed until they become nonseparable. Let $ s(X)$ be the first ordinal $ \alpha$ so that ${X^\alpha }$ is nonseparable. It is shown that if $X$ is nonreflexive, $s(X) \leqslant {\omega ^2} + 1$ (this is best possible) and that $ \{ s(X):X{\text{separable quasireflexive of order one}}\} = \{ \omega + 1,\omega + 2,2\omega + 1,2\omega + 2,{\omega ^2} + 1\}$. A quasireflexive space $X$ is constructed so that ${X^\omega }$ is isomorphic to $X \oplus {c_0}$ and no basic sequence in $ X$ is equivalent to a neighborly basis. It is shown that the ${\omega ^2}$th dual of James space and James function space are isomorphic to subspaces of one another. Also, perhaps of interest on its own, a reflexive space with a subsymmetric basis is constructed whose inversion spans a nonreflexive space.
Fr\'echet spaces with nuclear K\"othe quotients
Steven F.
Bellenot;
Ed
Dubinsky
579-594
Abstract: Each separable Fréchet non-Banach space $X$ with a continuous norm is shown to have a quotient $ Y$ with a continuous norm and a basis. If, in addition, $Y$ can be chosen to be nuclear, we say that $X$ has a nuclear Köthe quotient. We obtain a (slightly technical) characterization of those separable Fréchet spaces with nuclear Köthe quotients. In particular, separable reflexive Fréchet spaces which are not Banach (and thus Fréchet Montel spaces) have nuclear Köthe quotients.
A characterization of Fourier and Radon transforms on Euclidean space
Alexander
Hertle
595-607
Abstract: We show that a continuous operator behaving under rotations, positive dilations, and translations like the Fourier or the Radon transform on $ {{\mathbf{R}}^n}$ must be a constant multiple of one of these transforms. We prove this characterization for various function spaces, e.g. we characterize the Fourier transform as an operator acting on spaces between $ \mathfrak{D}({{\mathbf{R}}^n})$ and $ \mathfrak{D}({{\mathbf{R}}^n})$ to $ \mathfrak{D}({{\mathbf{R}}^n})$ and ${S^{n - 1}} \times {\mathbf{R}}$. In the special case $n = 1$, our methods sharpen results of J. L. B. Cooper and H. Kober, who characterize the Fourier transform as an operator from ${L^p}({\mathbf{R}})$ into $ {L^p}^\prime ({\mathbf{R}}),1 \leqslant p \leqslant 2$.
Rational homotopy of the space of sections of a nilpotent bundle
André
Haefliger
609-620
Abstract: We show that an algebraic construction proposed by Sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle.
Asymptotic Toeplitz operators
José
Barría;
P. R.
Halmos
621-630
Abstract: An asymptotic Toeplitz is an operator $T$ such the sequence $\{ {U^{ \ast n}}T{U^n}\} $ is strongly convergent, where $U$ is the unilateral shift. Every element of the norm-closed algebra generated by all Toeplitz and Hankel opertors together is an asymptotic Toeplitz operator. The authors study the relations among this Hankel algebra, the classical Toeplitz algebra, the set of all asymptotic Toeplitz operators, and the essential commutant of the unilateral shift. They offer several examples of operators in some of these classes but not in others, and they raise several open questions.
The structure of pseudo-inverse semigroups
F.
Pastijn
631-655
Abstract: A regular semigroup $ S$ is called a pseudo-inverse semigroup if $eSe$ is an inverse semigroup for each $e = {e^2} \in S$. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice. We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms. The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory.
Weak $P$-points in \v Cech-Stone compactifications
Jan
van Mill
657-678
Abstract: Let $X$ be a nonpseudocompact space which is either nowhere ccc or nowhere of weight $\leqslant {2^\omega }$. Then $\beta X - X$ contains a point $x$ which is a weak $P$-point of $\beta X$, i.e. if $F \subset \beta X - \{ x\} $ is countable, then $x \notin \bar F$. In addition, under MA, if $ X$ is any nonpseudocompact space, then $ \beta X - X$ contains a point $x$ such that whenever $F \subset \beta X - \{ x\} $ is countable and nowhere dense, then $ x \notin \bar F$.
The Fourier expansion of Eisenstein series for ${\rm GL}(3,\,{\bf Z})$
K.
Imai;
A.
Terras
679-694
Abstract: The Fourier expansions of Eisenstein series for $ {\text{GL}}(3,{\mathbf{Z}})$ are obtained by two methods--one analogous to the classical method used by many number theorists, including Weber, in his derivation of the Kronecker limit formula. The other method is analogous to that used by Siegel to obtain Fourier expansions of Eisenstein series for the Siegel modular group. The expansions involve matrix argument $K$-Bessel functions recently studied by Tom Bengtson. These $K$-Bessel functions are natural generalizations of the ordinary $K$-Bessel function which arise when considering harmonic analysis on the symmetric space of the general linear group using a certain system of coordinates.
Extending free cyclic actions on spheres
John
Ewing
695-703
Abstract: Connolly and Geist have reduced the problem of determining which free cyclic actions on spheres extend to free actions of specified supergroups to a problem involving a certain transfer map. In this note we develop some algebraic tools for calculating the transfer and show that some cyclic actions do not extend to certain supergroups.
$C(\alpha )$ preserving operators on $C(K)$ spaces
John
Wolfe
705-719
Abstract: Let $A:C(K) \to X$ be a bounded linear operator where $K$ is a compact Hausdorff space and $ X$ is a separable Banach space. Sufficient conditions are given for $A$ to be an isomorphism (into) when restricted to a subspace $Y$ of $C(K)$, such that $Y$ is isometrically isomorphic to a space $C(\alpha )$ of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal $\alpha$.
The asymptotic number of convex polyhedra
L. B.
Richmond;
N. C.
Wormald
721-735
Abstract: We obtain an asymptotic formula for the number of combinatorially distinct convex polyhedra with $n$ edges.
Odd primary Steenrod operations in first-quadrant spectral sequences
John
Sawka
737-752
Abstract: This paper defines two kinds of Steenrod operations in the spectral sequence of a bisimplical $\operatorname{mod} p$ coalgebra and shows them to be a complete list of all such possible Steenrod operations. These operations are compatible with the differentials and with Steenrod operations on the total complex. A general rule is given for computing the operations on ${E_2}$. A generalization of the Kudo transgression theorem is also proved, placing it in a larger and more natural setting.
Solvability of quasilinear elliptic equations with nonlinear boundary conditions
Gary M.
Lieberman
753-765
Abstract: On an $ n$-dimensional domain $ \Omega$, we consider the boundary value problem $\displaystyle (\ast)\quad Qu = 0\;{\text{in}}\Omega {\text{,}}\quad Nu = 0\;{\text{on}}\;\partial \Omega$ where $Q$ is a quasilinear elliptic second-order differential operator and $N$ is a nonlinear first order differential operator satisfying an Agmon-Douglis-Nirenberg consistency condition. If the coefficients of $ Q$ and $N$ satisfy additional hypotheses (such as sufficient smoothness), Fiorenza was able to reduce the solvability of $(\ast)$ to the establishment of a priori bounds for solutions of a related family of boundary value problems. We simplify Fiorenza's argument, obtaining the reduction under more general hypotheses and requiring a priori bounds only for solutions of $ Qu = f$, $Nu = g$ where $f$ and $g$ range over suitable function spaces. As an example, classical solutions of the capillary problem are shown to exist without using the method of elliptic regularization.
A generalization of a theorem of Maximoff and applications
S. J.
Agronsky
767-779
Abstract: Many classes of functions can be characterized in terms of their associated sets. Maximoff gave another type of characterization for the approximately continuous functions. In this paper, we give the conditions under which the two types of characterizations are equivalent. We then show that many classes of functions defined or characterized in terms of their associated sets also admit Maximoff-type characterizations.
The logarithm of the Poisson kernel of a $C\sp{1}$ domain has vanishing mean oscillation
David S.
Jerison;
Carlos E.
Kenig
781-794
Abstract: Let $D$ be a ${C^1}$ domain in $ {{\mathbf{R}}^n}$, and $ \omega$ the harmonic measure of $\partial D$, with respect to a fixed pole in $D$. Then, $ d\omega = kd\sigma$, where $k$ is the Poisson kernel of $D$. We show that log $k$ has vanishing mean oscillation of $\partial D$.
$C\sp{\ast} $-algebra fibre bundles
Maw Ding
Jean
795-801
Abstract: It will be shown in this paper that for any ${C^\ast}$-algebra fibre bundle with base space $ X$ and fibre $ A$, a $ {C^\ast}$-algebra, the Jacobson spectrum of the ${C^\ast}$-algebra of sections of the fibre bundle can be identified as a topological fibre bundle with the same base space $X$ and fibre the Jacobson spectrum of $A$.