Crossed products of continuous-trace $C\sp \ast$-algebras by smooth actions
Iain
Raeburn;
Jonathan
Rosenberg
1-45
Abstract: We study in detail the structure of $ {C^{\ast}}$-crossed products of the form $ A \rtimes {}_\alpha G$, where $A$ is a continuous-trace algebra and $\alpha$ is an action of a locally compact abelian group $G$ on $A$, especially in the case where the action of $ G$ on $\hat A$ has a Hausdorff quotient and only one orbit type. Under mild conditions, the crossed product has continuous trace, and we are often able to compute its spectrum and Dixmier-Douady class. The formulae for these are remarkably interesting even when $G$ is the real line.
New combinatorial interpretations of Ramanujan's partition congruences mod $5,7$ and $11$
F. G.
Garvan
47-77
Abstract: Let $p(n)$ denote the number of unrestricted partitions of $n$. The congruences referred to in the title are $ p(5n + 4)$, $p(7n + 5)$ and $p(11n + 6) \equiv 0$ ($\bmod 5$, $7$ and $11$, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences $\bmod 5$ and $7$. These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence $\bmod 11$. In this paper we give a crank which not only gives a combinatorial interpretation of the congruence $\bmod 11$ but also gives new combinatorial interpretations of the congruences $\bmod 5$ and $7$. However, our crank is not quite what Dyson asked for; it is in terms of certain restricted triples of partitions, rather than in terms of ordinary partitions alone. Our results and those of Dyson, Atkin and Swinnerton-Dyer are closely related to two unproved identities that appear in Ramanujan's "lost" notebook. We prove the first identity and show how the second is equivalent to the main theorem in Atkin and Swinnerton-Dyer's paper. We note that all of Dyson's conjectures $\bmod 5$ are encapsulated in this second identity. We give a number of relations for the crank of vector partitions $\bmod 5$ and $7$, as well as some new inequalities for the rank of ordinary partitions $\bmod 5$ and $7$. Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.
A lifting theorem and uniform algebras
Takahiko
Nakazi;
Takanori
Yamamoto
79-94
Abstract: In this paper we discuss the possible generalizations of a lifting theorem of a $2 \times 2$ matrix to uniform algebras. These have applications to Hankel operators, weighted norm inequalities for conjugation operators and Toeplitz operators on uniform algebras. For example, the Helson-Szegö theorems for general uniform algebras follow.
Sharp distortion theorems for quasiconformal mappings
G. D.
Anderson;
M. K.
Vamanamurthy;
M.
Vuorinen
95-111
Abstract: Continuing their earlier work on distortion theory, the authors prove some dimension-free distortion theorems for $ K$-quasiconformal mappings in ${R^n}$. For example, one of the present results is the following sharp variant of the Schwarz lemma: If $ f$ is a $ K$-quasiconformal self-mapping of the unit ball ${B^n}$, $ n \geqslant 2$, with $ f(0) = 0$, then ${4^{1 - {K^2}}}\vert x{\vert^K} \leqslant \vert f(x)\vert \leqslant {4^{1 - 1/{K^2}}}\vert x{\vert^{1/K}}$ for all $x$ in ${B^n}$.
Euler-Poincar\'e characteristic and higher order sectional curvature. I
Chuan-Chih
Hsiung;
Kenneth Michael
Shiskowski
113-128
Abstract: The following long-standing conjecture of H. Hopf is well known. Let $ M$ be a compact orientable Riemannian manifold of even dimension $n \geqslant 2$. If $M$ has nonnegative sectional curvature, then the Euler-Poincaré characteristic $ \chi (M)$ is nonnegative. If $M$ has nonpositive sectional curvature, then $ \chi (M)$ is nonnegative or nonpositive according as $n \equiv 0$ or $ 2\bmod 4$. This conjecture for $n = 4$ was proved first by J. W. Milnor and then by S. S. Chern by a different method. The main object of this paper is to prove this conjecture for a general $ n$ under an extra condition on higher order sectional curvature, which holds automatically for $n = 4$. Similar results are obtained for Kähler manifolds by using holomorphic sectional curvature, and F. Schur's theorem about the constancy of sectional curvature on a Riemannian manifold is extended.
Infinite rank Butler groups
Manfred
Dugas;
K. M.
Rangaswamy
129-142
Abstract: A torsion-free abelian group $G$ is said to be a Butler group if $\operatorname{Bext} (G,\,T)$ for all torsion groups $T$. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group $G$ satisfies the T.E.P. over a pure subgroup $ H$ if and only if $ H$ is decent in $ G$ in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called ${B_2}$-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ${B_2}$-group. We show under $(V = L)$ that this is indeed the case for Butler groups of rank $ {\aleph _1}$. On the other hand it is shown that, under ZFC, it is undecidable whether a group $B$ for which $\operatorname{Bext} (B,\,T) = 0$ for all countable torsion groups $T$ is indeed a ${B_2}$-group.
Coexistence theorems of steady states for predator-prey interacting systems
Lige
Li
143-166
Abstract: In this paper we give necessary and sufficient conditions for the existence of positive solutions of steady states for predator-prey systems under Dirichlet boundary conditions on $ \Omega \Subset {{\mathbf{R}}^n}$. We show that the positive coexistence of predatorprey densities is completely determined by the "marginal density," the unique density of prey or predator while the other one is absent, i.e. the $({u_0},\,0)$ or $ (0,\,{\nu _0})$. More specifically, the situation of coexistence is determined by the spectral behavior of certain operators related to these marginal densities and is also completely determined by the stability properties of these marginal densities. The main results are Theorems 1 and 4.2.
Ensembles de Riesz
Valérie
Tardivel
167-174
Abstract: Let $G$ be an abelian countable discrete group. We show that there exists no positive characterization of Riesz subsets of $G$, by proving that the Riesz subsets of $ G$ form a coanalytic non-Borel subset of ${2^G}$.
Longtime dynamics of a conductive fluid in the presence of a strong magnetic field
C.
Bardos;
C.
Sulem;
P.-L.
Sulem
175-191
Abstract: We prove existence in the large of localized solutions to the MHD equations for an ideal conducting fluid subject to a strong magnetic field. We show that, for large time, the dynamics may reduce to linear Alfven waves.
Isometries between function spaces
Krzysztof
Jarosz;
Vijay D.
Pathak
193-206
Abstract: Surjective isometries between some classical function spaces are investigated. We give a simple technical scheme which verifies whether any such isometry is given by a homeomorphism between corresponding Hausdorff compact spaces. In particular the answer is positive for the ${C^1}(X)$, $\operatorname{AC} [0,1]$, ${\operatorname{Lip} _\alpha }(X)$ and $ {\operatorname{lip} _\alpha }(X)$ spaces provided with various natural norms.
On the behavior of harmonic functions near a boundary point
Wade
Ramey;
David
Ullrich
207-220
Abstract: Several results on the behavior of harmonic functions at an individual boundary point are obtained. The results apply to positive harmonic functions as well as to Poisson integrals of functions in BMO.
Local projective resolutions and translation functors for Kac-Moody algebras
Wayne
Neidhardt
221-245
Abstract: Let $\mathfrak{g}$ be a Kac-Moody algebra defined by a not necessarily symmetrizable generalized Cartan matrix. We define translation functors and use them to show that the multiplicities $(M({w_1} \cdot \lambda ):L({w_2} \cdot \lambda ))$ are independent of the dominant integral weight $ \lambda$, depending only on the elements of the Weyl group. In order to define the translation functors, we introduce the notion of local projective resolutions and use them to develop the machinery of homological algebra in certain categories of $\mathfrak{g}$-modules.
Representations of hyperharmonic cones
Sirkka-Liisa
Eriksson
247-262
Abstract: Hyperharmonic cones are ordered convex cones possessing order properties similar to those of hyperharmonic functions on harmonic spaces. The dual of a hyperharmonic cone is defined to be the set of extended real-valued additive and left order-continuous mappings $(\not \equiv \infty )$. The second dual gives a representation of certain hyperharmonic cones in which suprema of upward directed families are pointwise suprema, although infima of pairs of functions are not generally pointwise infima. We obtain necessary and sufficient conditions for the existence of a representation of a hyperharmonic cone in which suprema of upward directed families are pointwise suprema and infima of pairs of functions are pointwise infima.
Convergence acceleration for generalized continued fractions
Paul
Levrie;
Lisa
Jacobsen
263-275
Abstract: The main result in this paper is the proof of convergence acceleration for a suitable modification (as defined by de Bruin and Jacobsen) in the case of an $n$-fraction for which the underlying recurrence relation is of Perron-Kreuser type. It is assumed that the characteristic equations for this recurrence relation have only simple roots with differing absolute values.
Complementation in Kre\u\i n spaces
Louis
de Branges
277-291
Abstract: A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product.
$k$-dimensional regularity classifications for $s$-fractals
Miguel Ángel
Martín;
Pertti
Mattila
293-315
Abstract: We study subsets $ E$ of ${{\mathbf{R}}^n}$ which are ${H^s}$ measurable and have $0 < {H^s}(E) < \infty$, where $ {H^s}$ is the $ s$-dimensional Hausdorff measure. Given an integer $k$, $ s \leqslant k \leqslant n$, we consider six ($s$, $k$) regularity definitions for $ E$ in terms of $ k$-dimensional subspaces or surfaces of $ {{\mathbf{R}}^n}$. If $ s = k$, they all agree with the (${H^k}$, $k$) rectifiability in the sense of Federer, but in the case $s < k$ we show that only two of them are equivalent. We also study sets with positive lower density, and projection properties in connection with these regularity definitions.
Counting semiregular permutations which are products of a full cycle and an involution
D. M.
Jackson
317-331
Abstract: Character theoretic methods and the group algebra of the symmetric group are used to derive properties of the number of permutations, with only $p$-cycles, for an arbitrary but fixed $ p$, which are expressible as the product of a full cycle and a fixed point free involution. This problem has application to single face embeddings of $p$-regular graphs on surfaces of given genus.
The ideal structure of certain nonselfadjoint operator algebras
Justin
Peters
333-352
Abstract: Let $(X,\,\phi )$ be a locally compact dynamical system, and ${{\mathbf{Z}}^ + }{ \times _\phi }\,{C_0}(X)$ the norm-closed subalgebra of the crossed product $Z{ \times _\phi }{C_0}(X)$ generated by the nonnegative powers of $\phi$ in case $\phi$ is a homeomorphism. If $ \phi$ is just a continuous map, ${{\mathbf{Z}}^ + }{ \times _\phi }{C_0}$ can still be defined by a crossed product type construction. The ideal structure of these algebras is determined in case $\phi$ acts freely. A class of strictly transitive Banach modules is described, indicating that for the nonselfadjoint operator algebras considered here, not all irreducible representations are on Hilbert space. Finally in a special case, the family of all invariant maximal right ideals is given.
Operator theoretical realization of some geometric notions
Qing
Lin
353-367
Abstract: This paper studies the realization of certain geometric constructions in Cowen-Douglas operator class. Through this realization, some operator theoretical phenomena are easily seen from the corresponding geometric phenomena. In particular, we use this technique to solve the first-order equivalence problem and introduce a new operation among certain operators.
The action of a solvable group on an infinite set never has a unique invariant mean
Stefan
Krasa
369-376
Abstract: Theorem 1 of the paper proves a conjecture of J. Rosenblatt on nonuniqueness of invariant means for the action of a solvable group $G$ on an infinite set $X$. The same methods used in this proof yield even a more general result: Nonuniqueness still holds if $ G$ is an amenable group containing a solvable subgroup $H$ such that $ \operatorname{card} (G/H) \leqslant \operatorname{card} (H)$.
A problem in convexity leading to the analysis of two functional equations
John V.
Ryff
377-396
Abstract: Transformation semigroups can often be studied effectively by examining their orbit structure. If the class of transformations has a special quality, such as convexity, it is generally reflected in the orbits. This work is concerned with such a circumstance. The goal is to examine the behavior of transformations on extreme points of orbits through the construction of a class of extreme operators. The construction leads naturally to the study of two functional equations which are analyzed in detail. Information about solutions is obtained through different $ {L^2}$-methods depending on whether or not two basic parameters are rational or irrational. In two cases all solutions are classified. In a third an example of a spanning set of solutions is obtained. Techniques of harmonic analysis and ergodic theory are used to study the functional equations.
Interpolation of Besov spaces
Ronald A.
DeVore;
Vasil A.
Popov
397-414
Abstract: We investigate Besov spaces and their connection with dyadic spline approximation in $ {L_p}(\Omega )$, $0 < p \leqslant \infty$. Our main results are: the determination of the interpolation spaces between a pair of Besov spaces; an atomic decomposition for functions in a Besov space; the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines.
Finite order solutions of second order linear differential equations
Gary G.
Gundersen
415-429
Abstract: We consider the differential equation $f\not \equiv 0$ of the equation will have infinite order. We will also find conditions on $ A(z)$ and $B(z)$ which will guarantee that any finite order solution $ f\not \equiv 0$ of the equation will not have zero as a Borel exceptional value. We will also show that if $A(z)$ and $B(z)$ satisfy certain growth conditions, then any finite order solution of the equation will satisfy certain other growth conditions. Related results are also proven. Several examples are given to complement the theory.