$C\sp{\ast} $-algebras of multivariable Wiener-Hopf operators
Paul S.
Muhly;
Jean N.
Renault
1-44
Abstract: The ${C^ \ast }$-algebra $ \mathfrak{W}$ generated by the Wiener-Hopf operators defined over a subsemigroup of a locally compact group is shown to be the image of a groupoid ${C^ \ast }$-algebra under a suitable representation. When the subsemigroup is either a polyhedral cone or a homogeneous, self-dual cone in an Euclidean space, this representation may be used to show that $ \mathfrak{W}$ is postliminal and to find a composition series with very explicit subquotients. This yields a concrete parameterization of the spectrum of $ \mathfrak{W}$ and exhibits the topology on it.
The inverse of a totally positive bi-infinite band matrix
Carl
de Boor
45-58
Abstract: It is shown that a bounded bi-infinite banded totally positive matrix $ A$ is boundedly invertible iff there is one and only one bounded sequence mapped by $A$ to the sequence $ ({( - )^i})$. The argument shows that such a matrix has a main diagonal, i.e., the inverse of $A$ is the bounded pointwise limit of inverses of finite sections of $A$ principal with respect to a particular diagonal; hence $ ({( - )^{i + j}}{A^{ - 1}}(i,j))$ or its inverse is again totally positive.
Inverses of infinite sign regular matrices
C.
de Boor;
S.
Friedland;
A.
Pinkus
59-68
Abstract: Let $A$ be an infinite sign regular (sr) matrix which can be viewed as a bounded linear operator from ${l_\infty }$ to itself. It is proved here that if the range of $A$ contains the sequence $( \ldots ,1, - 1,1, - 1, \ldots )$, then $ A$ is onto. If ${A^{ - 1}}$ exists, then $ D{A^{ - 1}}D$ is also sr, where $D$ is the diagonal matrix with diagonal entries alternately $1$ and $- 1$. In case $A$ is totally positive (tp), then $D{A^{ - 1}}D$ is also tp under additional assumptions on $A$.
Conditioning by $\langle $equal, linear$\rangle $
Chii-Ruey
Hwang
69-83
Abstract: We deal with a limit problem of regularity controlled probabilities in metric pattern theory. The probability on the generator space is given by a density function $f(x,y)$ on which some integrability conditions are imposed. Let $T$ denote the integral operator with kernel $ f$. When $n$ i.i.d. generators $({X_k},{Y_k})$ are connected together to form the configuration space $ {\mathcal{C}_n}$ via the regularity $ \left\langle {{\text{EQUAL}},{\text{LINEAR}}} \right\rangle$, i.e. "conditioning" on $ {X_{k + 1}} = {Y_k}$ for $1 \leqslant k < n$, an approximate identity is used to define the regularity controlled probability on $ {\mathcal{C}_n}$. The probabilistic effect induced by the regularity conditions on some fixed subconfiguration of a larger configuration $ {\mathcal{C}_n}$ is described by its corresponding marginal probability within $ {\mathcal{C}_n}$. When $ n$ goes to infinity in a suitable way, the above mentioned marginal probability converges weakly to a limit whose density can be expressed in terms of the largest eigenvalues and the corresponding eigenspaces of $T$ and ${T^ \ast }$. When $f$ is bivariate normal, the eigenvalue problem is solved explicitly. The process determined by the limiting marginal probabilities is strictly stationary and Markovian.
A trace formula for compact manifolds
K. S.
Sarkaria
85-88
Abstract: An integral formula for the Euler characteristic is given, in which the data consists of a finite dimensional transitive vector space $V$ of vector fields and a volume form $\Omega$ supported in a small neighborhood of the origin of $V$.
Faisceaux amples sur les espaces analytiques
Vincenzo
Ancona
89-100
Abstract: The following result is established. Let $f:X \to Y$ be a morphism between two compact complex spaces and $ \mathfrak{L}$ a weakly positive invertible sheaf on $X$; then for suitable $nf{}_ \ast {\mathfrak{L}^n}$ is weakly positive on $Y$. It follows that Moišezon spaces can be characterized via weakly positive coherent sheaves. Moreover, a problem posed by Grauert on the exceptional subspaces of complex spaces can be solved.
Decisive subgroups of analytic groups
T. Christine
Stevens
101-108
Abstract: It is known that every analytic group $(L,\tau )$ contains a closed abelian subgroup $ H$ which is "decisive" in the sense that the Hausdorff topologies for $ L$ which are weaker than $ \tau$ are completely determined by their restrictions to $H$. We show here that $H$ must ordinarily contain the entire center of $ L$ but that the rest of $ H$ can in general be reduced. The proof involves constructing "unusual" topologies for abelian Lie groups.
The Albanese mapping for a punctual Hilbert scheme. II. Symmetrized differentials and singularities
Mark E.
Huibregtse
109-140
Abstract: Let $f:X \to A$ be the canonical mapping from the irreducible and nonsingular surface $ X$ to its Albanese variety $ A$, $X(n)$ the $n$-fold symmetric product of $X$, and $H_X^n$ the punctual Hilbert scheme parameterizing 0-dimensional closed subschemes of length $ n$ on $X$. The latter is an irreducible and nonsingular variety of dimension $2n$, and the "Hilbert-Chow" morphism ${\sigma _n}:H_X^n \to X(n)$ is a birational map which desingularizes $ X(n)$. Let ${f_n}:X(n) \to A$ denote the map induced by $ f$ by addition on $ A$. This paper studies the singularities of the composite morphism $\displaystyle {\varphi _n}:H_X^n\mathop \to \limits^{{\sigma _n}} X(n)\mathop \to \limits^{{f_n}} A,$ which is a natural generalization of the mapping $C(n) \to J$, where $C$ is an irreducible and nonsingular curve and $J$ is its Jacobian. Unlike the latter, however, ${\varphi _n}$ need not be smooth for $n \gg 0$. We prove that ${\varphi _n}$ is smooth for $n \gg 0$ only if $f:X \to A$ is smooth (Theorem 3), and over ${\mathbf{C}}$ we prove the converse (Theorem 4). In case $X = A$ is an abelian surface, we show ${\varphi _n}$ is smooth for $n$ prime to the characteristic (Theorem 5), and give a counterexample to smoothness for all $n$ (Theorem 6). We exhibit a connection (over ${\mathbf{C}}$) between singularities of ${\varphi _n}$ and generalized Weierstrass points of $X$ (Theorem 9). Our method is as follows: We first show that the singularities of ${\varphi _n}$ are the zeros of certain holomorphic $ 1$-forms on $H_X^n$ which are the "symmetrizations" of holomorphic $1$-forms on $X$. We then study "symmetrized differentials" and their zeros on $H_X^n$ (Theorems 1,2). Our method works for curves $ C$ as well; we give an alternative proof of a result of Mattuck and Mayer [10, p. 226] which shows that the zeros of symmetrized differentials on $C(n)$ represent (for $C$ complete nonsingular) the special divisors of degree $n$ on $C$.
Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups
Ching
Chou
141-157
Abstract: A noncompact locally compact group $G$ is called an Eberlein group if $W(G) = B{(G)^ - }$ where $W(G)$ is the algebra of continuous weakly almost periodic functions on $G$ and $B{(G)^ - }$ is the uniform closure of the Fourier-Stieltjes algebra of $G$. We show that if $G$ is a noncompact $[IN]$-group or a noncompact nilpotent group then $W(G)/B{(G)^ - }$ contains a linear isometric copy of $ {l^\infty }$. In particular, $G$ is not an Eberlein group. On the other hand, finite direct products of Euclidean motion groups and, by a result of W. Veech, noncompact semisimple analytic groups with finite centers are Eberlein groups.
Regularizing effects for $u\sb{t}=\Delta \varphi (u)$
Michael G.
Crandall;
Michel
Pierre
159-168
Abstract: One expression of the fact that a nonnegative solution of the initial-value problem $\displaystyle ({\text{IVP}})\quad \left\{ {\begin{array}{*{20}{c}} {{u_t} - \De... ...\right.\quad \begin{array}{*{20}{c}} {t > 0,x \in {R^N},} {} \end{array}$ where $m > 0$, is more regular for $t > 0$ than a rough initial datum ${u_0}$ is the remarkable pointwise inequality ${u_t} = \Delta {u^m} \geqslant - (N/(N(m - 1) + 2)t)u$ obtained by Aronson and Bénilan for $ t > 0$ and $m > \max ((N - 2)/N,0)$. This inequality was used by Friedman and Caffarelli in proving that solutions of (IVP) are continuous for $ t > 0$. The main results of this paper generalize the Aronson-Bénilan inequality and show the extended inequality is valid for a much broader class of equations of the form ${u_t} = \Delta \varphi (u)$. In particular, the results apply to the Stefan problem which is modeled by $ \varphi (r) = {(r - 1)^ + }$ and imply ${({(u - 1)^ + })_t} \geqslant - ({(u - 1)^ + } + N/2)/t$ in this case.
Equilibrium states of grid functions
Nelson G.
Markley;
Michael E.
Paul
169-191
Abstract: It is well known that locally constant functions on symbolic spaces have unique equilibrium states. In this paper we investigate the nature of equilibrium states for a type of continuous function which need not have a finite range. Although most of these functions have a unique equilibrium state, phase transitions or multiple equilibrium states do occur and can be analyzed.
Clifford module invariants of spin bundles
Jacques
Allard;
Anthony
Bahri
193-202
Abstract: In this paper, we study $KO$-theory invariants of Spin bundles obtained by the $ \alpha$-construction from Clifford module representations of the Spinor group. We begin by describing their elementary properties including various Whitney sum formulae and their relation with the $d$-invariant for vector bundles over spheres. We next observe an important difference between the two half-Spin representations and then proceed to investigate the fiber homotopy properties of the invariants. We conclude with some applications.
Eilenberg-Moore models for fibrations
J.-C.
Thomas
203-225
Abstract: E. M. model is a new invariant in rational homotopy theory which gives us both a Künneth object and a Tate-Josefiak resolution. With the E. M. model, we study relations between formality of base, total space and fibre of a Serre fibration, obstructions to $ {\mathbf{k}}$-realizability of a cohomology equivalence between two continuous maps and formalizable maps.
Dunford-Pettis operators on Banach lattices
C. D.
Aliprantis;
O.
Burkinshaw
227-238
Abstract: Consider a Banach lattice $E$ and two positive operators $S,T:E \to E$ that satisfy $0 \leqslant S \leqslant T$. In $[{\mathbf{2,3}}]$ we examined the case when $ T$ is a compact (or weakly compact) operator and studied what effect this had on an operator (such as $S$) dominated by $T$. In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator $T$, to ensure that $S$ (or some power of $S$) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters. Theorem. Let $E$ be a Banach lattice, and let $S,T:E \to E$ be two positive operators such that $ 0 \leqslant S \leqslant T$. If $T$ is compact then (1) $ {S^3}$ is a compact operator (although ${S^2}$ need not be compact); (2) $ {S^2}$ is a Dunford-Pettis and weakly compact operator ( although $ S$ need not be ); (3) $S$ is a weak Dunford-Pettis operator. In another direction, our techniques and results will be related to the lattice stracture of the Dunford-Pettis operators. For instance, it will be shown that under certain conditions the Dunford-Pettis operators form a band.
The space of positive definite matrices and Gromov's invariant
Richard P.
Savage
239-263
Abstract: The space $ X_d^n{\text{of}}n \times n$ positive definite matrices with ${\text{determinant}} = 1$ is considered as a subset of ${{\mathbf{R}}^{n(n + 1)/2}}$ with isometries given by $X \to AX{A^t}$ where the determinant of $A = 1$ and $X_d^n$ is given its invariant Riemannian metric. This space has a collection of simplices which are preserved by the isometries and formed by projecting geometric simplices in ${{\mathbf{R}}^{n(n + 1)/2}}$ to the hypersurface $X_d^n$. The main result of this paper is that for each $n$ the volume of all top dimensional simplices of this type has a uniform upper bound. This result has applications to Gromov's Invariant as defined in William P. Thurston's notes, The geometry and topology of $3$-manifolds. The result implies that the Gromov Invariant of the fundamental class of compact manifolds which are formed as quotients of $ X_d^n$ by discrete subgroups of the isometries is nonzero. This gives the first nontrivial examples of manifolds that have a nontrivial Gromov Invariant but do not have strictly negative curvature or nonvanishing characteristic numbers.
Bordered Klein surfaces with maximal symmetry
Newcomb
Greenleaf;
Coy L.
May
265-283
Abstract: A compact bordered Klein surface of (algebraic) genus $g \geqslant 2$ is said to have maximal symmetry if its automorphism group is of order $12(g - 1)$, the largest possible. In this paper we study the bordered surfaces with maximal symmetry and their automorphism groups, the $ {M^\ast}$-groups. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated $ {M^\ast}$-group. We begin by classifying the bordered surfaces with maximal symmetry of low topological genus. We next show that a bordered surface with maximal symmetry is a full covering of another surface with primitive maximal symmetry. A surface has primitive maximal symmetry if its automorphism group is ${M^\ast}$-simple, that is, if its automorphism group has no proper ${M^\ast}$-quotient group. Our results yield an approach to the problem of classifying the bordered Klein surfaces with maximal symmetry. Next we obtain several constructions of full covers of a bordered surface. These constructions give numerous infinite families of surfaces with maximal symmetry. We also prove that only two of the ${M^\ast}$-simple groups are solvable, and we exhibit infinitely many nonsolvable ones. Finally we show that there is a correspondence between bordered Klein surfaces with maximal symmetry and regular triangulations of surfaces.
Krull and global dimensions of semiprime Noetherian PI-rings
Richard
Resco;
Lance W.
Small;
J. T.
Stafford
285-295
Abstract: In this paper it is shown that if $R$ is a semiprime Noetherian PI-ring of finite global dimension, then the Krull dimension of $ R$ is less than or equal to its global dimension. The proof depends upon two preliminary results on arbitrary Noetherian PI-rings, which are of independent interest: (i) any height two prime ideal of $R$ contains infinitely many height one prime ideals; (ii) the localization of the polynomial ring $ R[x]$ at its set of monic elements is a Jacobson ring.
Free boundary convergence in the homogenization of the one-phase Stefan problem
José-Francisco
Rodrigues
297-305
Abstract: We consider the one phase Stefan problem in a "granular" medium, i.e., with nonconstant thermal diffusity, and we study the asymptotic behaviour of the free boundary with respect to homogenization. We prove the convergence of the coincidence set in measure and in the Hausdorff metric. We apply this result to the free boundary and we obtain the convergence in mean for the star-shaped case and the uniform convergence for the one-dimensional case, respectively. This gives an answer to a problem posed by J. L. Lions in [L].
Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries
Julian
Gevirtz
307-318
Abstract: A mapping $ f$ of an open subset $ U$ of a Banach space $ X$ into another Banach space $Y$ is said to be $(m,M)$-isometric if it is a local homeomorphism for which $M \geqslant {D^ + }f(x)$ and $m \leqslant {D^ - }f(x)$ for all $x \in U$, where ${D^ + }f(x)$ and $ {D^ - }f(x)$ are, respectively, the upper and lower limits of $\vert f(y) - f(x)\vert/\vert y - x\vert\;{\text{as}}\;y \to x$. For $0 < \rho \leqslant 1$ we find a number $\mu (\rho ) > 1$ which has the following property: Let $X$ and $Y$ be Banach spaces and let $U$ be an open convex subset of $ X$ containing a ball of radius $r$ and contained in the concentric ball of radius $ R$. Then all $ (m,M)$-isometric mappings of $U$ into $Y$ are injective if $ M/m < \mu (r/R)$. We also derive similar injectivity criteria for a more general class of connected open sets $ U$. The basic tool used is an approximate version of the Mazur-Ulam theorem on the linearity of distance preserving transformations between normed linear spaces.
On the division of distributions by analytic functions in locally convex spaces
Denis
Chansolme
319-325
Abstract: Although the division of distributions by real polynomials and real analytic functions (which are nonzero) is always possible in finite dimensional spaces (from classical results of Hörmander and Lojasiewicz respectively), we show that this is not always possible in infinite dimensional locally convex spaces. In particular, we characterize those locally convex spaces where division is always possible.
The preprojective partition for hereditary Artin algebras
D.
Zacharia
327-343
Abstract: The purpose of this paper is to study the preprojective partition of a hereditary artin algebra. For a hereditary algebra of finite representation type, we give some numerical invariants in terms of the length of chains of irreducible maps, also in terms of the length of the maximal indecomposable module, and the orientation of the quiver of the algebra. Similar results are given for algebras stably equivalent to hereditary artin algebras.
Borsuk-Ulam theorems for arbitrary $S\sp{1}$ actions and applications
E. R.
Fadell;
S. Y.
Husseini;
P. H.
Rabinowitz
345-360
Abstract: An ${S^1}$ version of the Borsuk-Ulam Theorem is proved for a situation where Fix ${S^1}$ may be nontrivial. The proof is accomplished with the aid of a new relative index theory. Applications are given to intersection theorems and the existence of multiple critical points is established for a class of functional invariant under an $ {S^1}$ symmetry.
Ampleness and connectedness in complex $G/P$
Norman
Goldstein
361-373
Abstract: This paper determines the "ampleness" of the tangent bundle of the complex homogeneous space, $G/P$, by calculating the maximal fibre dimension of the desingularization of a nilpotent subvariety of the Lie algebra of $G$.
Hardy spaces and Jensen measures
Takahiko
Nakazi
375-378
Abstract: Suppose $ A$ is a subalgebra of ${L^\infty }(m)$ on which $m$ is multiplicative. In this paper, we show that if $ m$ is a Jensen measure and $A + \overline A $ is dense in $ {L^2}(m)$, then $A + \overline A$ is weak-* dense in ${L^\infty }(m)$, that is, $A$ is a weak-* Dirichlet algebra. As a consequence of it, it follows that if $ A + \overline A$ is dense in ${L^4}(m)$, then $A$ is a weak-* Dirichlet algebra. (It is known that even if $ A + \overline A$ is dense in ${L^3}(m)$, $A$ is not a weak-* Dirichlet algebra.) As another consequence, it follows that if $B$ is a subalgebra of the classical Hardy space $ {H^\infty }$ containing the constants and dense in ${H^2}$, then $B$ is weak-* dense in $ {H^\infty }$.
Distal compactifications of semigroups
H. D.
Junghenn
379-397
Abstract: Properties of distal functions on semitopological semigroups are studied via the device of right topological semigroup compactification. Algebras of distal functions are used to construct the universal right simple, left simple, and group compactifications of a semigroup, and these compactifications are in turn systematically employed to obtain information about distal functions. Applications are made to semidirect products and flows.