Closed convex hulls of unitary orbits in von Neumann algebras
Fumio
Hiai;
Yoshihiro
Nakamura
1-38
Abstract: Let $\mathcal{M}$ be a von Neumann algebra. The distance $ \operatorname{dist} (x,\operatorname{co} \mathcal{U}(y))$ between $ x$ and $ \operatorname{co} \mathcal{U}(y)$ for selfadjoint operators $x$, $ y \in \mathcal{M}$ and the distance $ \operatorname{dist} (\varphi ,\operatorname{co} \mathcal{U}(\psi ))$ between $\varphi$ and $\operatorname{co} \mathcal{U}(\psi )$ for selfadjoint elements $\varphi$, $\psi \in {\mathcal {M}_*}$ are exactly estimated, where $\operatorname{co} \mathcal{U}(y)$ and $ \operatorname{co} \mathcal{U}(\psi )$ are the convex hulls of the unitary orbits of $y$ and $\psi$, respectively. This is done separately in the finite factor case, in the infinite semifinite factor case, and in the type III factor case. Simple formulas of distances between two convex hulls of unitary orbits are also given. When $ \mathcal{M}$ is a von Neumann algebra on a separable Hilbert space, the above cases altogether are combined under the direct integral decomposition of $ \mathcal{M}$ into factors. As a result, it is known that if $\mathcal{M}$ is $\sigma$-finite and $ x \in \mathcal{M}$ is selfadjoint, then $\overline {\operatorname{co} } \mathcal{U}(x) = {\overline {\operatorname{co} } ^{\mathbf{w}}}\mathcal{U}(x)$ where $ \overline {\operatorname{co} } \mathcal{U}(x)$ and ${\overline {\operatorname{co} } ^{\mathbf{w}}}\mathcal{U}(x)$ are the closures of $ \operatorname{co} \mathcal{U}(x)$ in norm and in the weak operator topology, respectively.
Computing the topological entropy of general one-dimensional maps
P.
Góra;
A.
Boyarsky
39-49
Abstract: A matrix-theoretic method for computing the topological entropy of continuous, piecewise monotonic maps of the interval is presented. The method results in a constructive procedure which is easily implemented on the computer. Examples for families of unimodal, nonunimodal and discontinuous maps are presented.
Microlocal Holmgren's theorem for a class of hypo-analytic structures
S.
Berhanu
51-64
Abstract: A microlocal version of Holmgren's Theorem is proved for a certain class of the hypo-analytic structures of Baouendi, Chang, and Treves.
A discrete approach to monotonicity of zeros of orthogonal polynomials
Mourad E. H.
Ismail;
Martin E.
Muldoon
65-78
Abstract: We study the monotonicity with respect to a parameter of zeros of orthogonal polynomials. Our method uses the tridiagonal (Jacobi) matrices arising from the three-term recurrence relation for the polynomials. We obtain new results on monotonicity of zeros of associated Laguerre, Al-Salam-Carlitz, Meixner and Pollaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlitz and Meixner polynomials.
Commuting fully invariant congruences on free completely regular semigroups
F.
Pastijn
79-92
Abstract: We show that "almost all" fully invariant congruences on every free completely regular semigroup commute. From this it is shown that the lattice of completely regular semigroup varieties is arguesian.
On Hermite-Fej\'er interpolation in a Jordan domain
Charles K.
Chui;
Xie Chang
Shen
93-109
Abstract: The Hermite-Fejér interpolation problem on a Jordan domain is studied. Under certain mild conditions on the smoothness of the boundary curve, we give both uniform and $ {L^p}$, $0 < p < \infty$, estimates on the rate of convergence. Our estimates are sharp even for the unit disk setting.
Genericity of nontrivial $H$-superrecurrent $H$-cocycles
Karma
Dajani
111-132
Abstract: We prove that most ${\text{H}}$-cocycles for a nonsingular ergodic transformation of type $ {\text{II}}{{\text{I}}_\lambda }$, $0 < \lambda < 1$, are $ {\text{H}}$-superrecurrent. This is done by showing that the set of nontrivial $ {\text{H}}$-superrecurrent ${\text{H}}$-cocycles form a dense ${G_\delta}$ set with respect to the topology of convergence in measure.
Stability of Newton boundaries of a family of real analytic singularities
Masahiko
Suzuki
133-150
Abstract: Let ${f_t}(x,y)$ be a real analytic $t$-parameter family of real analytic functions defined in a neighborhood of the origin in $ {\mathbb{R}^2}$. Suppose that ${f_t}(x,y)$ admits a blow analytic trivilaization along the parameter $t$ (see the definition in $\S1$ of this paper). Under this condition, we prove that there is a real analytic $t$-parameter family ${\sigma _t}(x,y)$ with ${\sigma _0}(x,y)=(x,y)$ and ${\sigma _t}(0,0)=(0,0)$ of local coordinates in which the Newton boundaries of $ {f_t}(x,y)$ are stable. This fact claims that the blow analytic equivalence among real analytic singularities is a fruitful relationship since the Newton boundaries of singularities contains a lot of informations on them.
Lie supergroup actions on supermanifolds
Charles P.
Boyer;
O. A.
Sánchez-Valenzuela
151-175
Abstract: Lie supergroups are here understood as group objects in the category of supermanifolds (as in [$2$, $5$, and $15$]). Actions of Lie supergroups in supermanifolds are defined by means of diagrams of supermanifold morphisms. Examples of such actions are given. Among them emerge the linear actions discussed in [$2$, $5$, and $12$] and the natural actions on the Grassmannian supermanifolds studied in [$6$-$9$ and $13$]. The nature of the isotropy subsupergroup associated to an action is fully elucidated; it is exhibited as an embedded subsupergroup in the spirit of the theory of smooth manifolds and Lie groups and with no need for the Lie-Hopf algebraic approach of Kostant in [$3$]. The notion of orbit is also discussed. Explicit calculations of isotropy subsupergroups are included. Also, an alternative proof of the fact that the structural sheaf of a Lie supergroup is isomorphic to the sheaf of sections of a trivial exterior algebra bundle is given, based on the triviality of its supertangent bundle.
Hypersurface variations are maximal. II
James A.
Carlson
177-196
Abstract: We show that certain variations of Hodge structure defined by sufficiently ample hypersurfaces are maximal integral manifolds of Griffiths' horizontal distribution.
A characterization of the complemented translation-invariant subspaces of $H\sp 1({\bf R})$
Dale E.
Alspach
197-207
Abstract: The purpose of this paper is to characterize the complemented translation-invariant subspaces of ${H^1}({\mathbf{R}})$ in terms of the zero set of the Fourier transform. It is shown that if $X$ is such a subspace then $X = I(A)$ where $A$ is in the ring generated by arithmetic progressions and lacunary sequences and $A$ is $ \varepsilon$-separated for some $\varepsilon > 0$. This proves a conjecture of the author and D. Ullrich.
The hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type
A.
Boggess;
R.
Dwilewicz;
A.
Nagel
209-232
Abstract: We show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in ${C^n}$ contains an open set in ${C^n}$ which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball. In addition, we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions.
Finite group actions on the moduli space of self-dual connections. I
Yong Seung
Cho
233-261
Abstract: Let $M$ be a smooth simply connected closed $4$-manifold with positive definite intersection form. Suppose a finite group $G$ acts smoothly on $M$. Let $ \pi :E \to M$ be the instanton number one quaternion line bundle over $ M$ with a smooth $ G$-action such that $ \pi$ is an equivariant map. We first show that there exists a Baire set in the $ G$-invariant metrics on $ M$ such that the moduli space $\mathcal{M}_ * ^G$ of $G$-invariant irreducible self-dual connections is a manifold. By utilizing the $ G$-transversality theory of T. Petrie, we then identify cohomology obstructions to globally perturbing the full space ${\mathcal{M}_ * }$ of irreducible self-dual connections to a $G$-manifold when $G = {{\mathbf{Z}}_2}$ and the fixed point set of the $ {\mathbf{Z}}_2$ action on $ M$ is a nonempty collection of isolated points and Riemann surfaces.
Sobolev interpolation inequalities with weights
Cristian E.
Gutiérrez;
Richard L.
Wheeden
263-281
Abstract: We study weighted local Sobolev interpolation inequalities of the form \begin{displaymath}\begin{gathered}\frac{1} {{{w_2}(B)}}{\int\limits_B {\vert u(... ...\vert u(x){\vert^p}v(x)dx} } \right), \end{gathered} \end{displaymath} , where $1 < p < \infty,h > 1, B$ is a ball in $ {{\mathbf{R}}^n}$, and $ v$ ,${w_1}$, and ${w_2}$ are weight functions. The case $p = 2$ is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case $u$ has compact support in $B$, and we derive global Landau inequalities ${\left\Vert {\nabla u} \right\Vert _{L_w^q}} \leq c\left\Vert {\nabla u} \righ... ...eft\Vert {{\nabla ^2}u} \right\Vert _{L_v^p}^a,0 < a < 1,1 < p \leq q < \infty $, when $u$ has compact support.
Puret\'e, rigidit\'e, et morphismes entiers
Gabriel
Picavet
283-313
Abstract: Bousfield and Kan have shown that a ring morphism with domain ${\mathbf{Z}}$ is rigid; we say that a ring morphism is rigid if it admits a factorization by an epimorphism, followed by a pure morphism. A ring $ A$ is said to be rigid if every morphism with domain $A$ is a rigid one. Our principal results are: the rigid domains are the Prüferian rings $ A$, with $\operatorname{Dim} (A) \leq 1$, and the Noetherian rigid rings are the Z.P.I. rings. The quasi-compact open sets of an affine rigid scheme, having as underlying ring a domain or a Noetherian ring, are affine and schematically dense if they contain the assassin of the ring. Every injective integral ring morphism with rigid domain is a pure morphism. We give two criteria of purity for integral injective morphisms. As a consequence of these results we obtain the following properties: if $A$ is a normal ring, containing the field of rationals, or is a regular ring, containing a field, every injective integral morphism with domain $A$ is a pure one. For a reduced ring, we define the category of reduced modules and show that any injective integral morphism is pure with respect to the category of the reduced modules.
Structure locale de l'espace des r\'etractions d'une surface
Robert
Cauty
315-334
Abstract: Let $\Sigma$ be a compact connected $ 2$-manifold, and $\mathcal{R}(\Sigma )$ the space of retractions of $ \Sigma$. We prove that $ \mathcal{R}(\Sigma )$ is an ${l^2}$-manifold if the boundary of $\Sigma$ is not empty, and is the union of an ${l^2}$-manifold and an isolated point $ {\text{i}}{{\text{d}}_\Sigma }$ if $\Sigma$ is closed.
Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces
O.
Blasco;
A.
Pełczyński
335-367
Abstract: We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype $2$) which includes all Banach lattices of cotype $2$, all Banach spaces whose dual is of type $ 2$ and also the preduals of ${C^ * }$-algebras. For the trace class $ {S_1}$ and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from $ {H^1}$ into ${l^1}$; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type $ > 1$ (Bourgain).
Indecomposable Cohen-Macaulay modules and their multiplicities
Dorin
Popescu
369-387
Abstract: The main aim of this paper is to find a large class of rings for which there are indecomposable maximal Cohen-Macaulay modules of arbitrary high multiplicity (or rank in the case of domains).
On collectionwise normality of locally compact, normal spaces
Zoltán T.
Balogh
389-411
Abstract: We prove that by adjoining supercompact many Cohen or random reals to a model of ZFC set theory, in the resulting model, every normal locally compact space is collectionwise normal. In the same models, countably paracompact, locally compact $ {T_3}$-spaces are expandable. Local compactness in the above theorems can be weakened to being of point-countable type, a condition that is implied by both Čech-completeness and first countability.
Univalent functions which map onto regions of given transfinite diameter
P. L.
Duren;
M. M.
Schiffer
413-428
Abstract: By a variational method, the sharp upper bound is obtained for the second coefficients of normalized univalent functions which map the unit disk onto regions of prescribed transfinite diameter, or logarithmic capacity.
Inner functions and cyclic vectors in the Bloch space
J. M.
Anderson;
J. L.
Fernández;
A. L.
Shields
429-448
Abstract: In this paper we construct a singular inner function whose polynomial multiples are dense in the little Bloch space ${\mathcal{B}_0}$. To do this we construct a singular measure on the unit circle with "best possible" control of both the first and second differences.
Circuit partitions and the homfly polynomial of closed braids
François
Jaeger
449-463
Abstract: We present an expansion of the homfly polynomial $P(D,z,a)$ of a braid diagram $D$ in terms of its circuit partitions. Another aspect of this result is an expression of $ P(D,z,a)$ as the trace of a matrix associated to $D$ in a simple way. We show how certain degree properties of the homfly polynomial can be derived easily from this model. In particular we obtain that if $ D$ is a positive braid diagram on $n$ strings with $w$ crossings, the maximum degree of $P(D,z,a)$ in the variable $a$ equals $n - 1 - w$. Nous présentons une expansion pour le polynôme homfly $P(D,z,a)$ d'un diagramme de tresse $ D$ en termes de ses partitions en circuits. Un autre aspect de ce résultat consiste en une expression de $P(D,z,a)$ comme trace d'une matrice associee de façon simple à $D$. Nous montrons comment certaines propriétés de degré du polynôme homfly dérivent simplement de ce modèle. En particulier nous obtenons que pour un diagramme de tresse positif $ D$ à $n$ brins et $w$ croisements, le degré maximum de $ P(D,z,a)$ en la variable $ a$ est égal à $ n - 1- w$.