Showing papers in "Publications of The Research Institute for Mathematical Sciences in 1980"

Cohomology and Extensions of von Neumann Algebras. II

Abstract: We develop a theory of extensions of von Neumann algebras by locally compact groups of automorphisms. The emphasis is on the description (from an algebraic point of view) of those extensions of a given von Neumann algebra by a given group which determine a fixed homomorphism from the group into the outer automorphism classes of the given algebra. Thus the study of such homomorphisms occupies a substantial part of the paper; for a large class of examples we are able to determine when such a homomorphism is split, and give a simple algebraic description of the extensions. We then give necessary and sufficient conditions (of an analytic nature) for an extension to be equivalent to a twisted crossed product extension, and give some applications to the study of representations of certain topological groups, and to approximately finite dimensional von Neumann algebras.

Some Simple C*-Algebras Constructed As Crossed Products with Discrete Outer Automorphism Groups

Abstract: An analogue for C*-algebras is given of the theorem of von Neumann (Theorem VIII of [24]) that the crossed product of a commutative von Neumann algebra by a discrete group acting freely and ergodically is a factor. The method of proof also works for certain noncommutative C*-algebras, and so in these cases one obtains also an analogue of Kallman's noncommutative generalization of von Neumann's theorem (3.3 of [18]).

Duality of Mixed Hodge Structures of Algebraic Varieties

Abstract: which are dual to each other via Poincare pairings (cf. (1.5)). On the other hand, when Z is an algebraic variety (as we assume in the following), Deligne defined in [3] [4] the natural mixed (Q-)Hodge structure on each term of the above sequences, in such a way that the morphisms are those of mixed Hodge structures. The purpose of this article is then to show that the duality mentioned above is also compatible with the mixed Hodge structures under a suitable definition. A result in a sense analogous to ours has been obtained by Herrera and Lieberman in [13] in which they showed that the above duality is compatible with 'infinitesimal Hodge filtrations' of X along Y. Duality of mixed Hodge structure itself was also mentioned in the introduction of [4] as according to N. Katz. However, since there seems no published articles on this subject, it would not be of little use to give a detailed exposition like the present one. In Section 1 a precise statement of the theorem will be given and its proof is reduced to the case where we have to show that the pairing i^y: H*(Y9 Q)x Hy~(X, (?)->(? gives a duality of mixed Hodge structures under the assumption that Yis a divisor with only normal crossings in X. In this case we have

On Complete Kähler Domains with C1-Boundary

Abstract: In the study of several complex variables the problem of characterizing Stein manifolds by their geometric properties has been one of the central topics since Oka showed that every pseudoconvex unramified domain over the complex number space C\" is a Stein manifold. In 1956 Grauert [4] showed that every unramified domain over C\" with real analytic boundary and complete Kahler metric is a Stein manifold. He proved the pseudoconvexity of the domain and applied Oka's theorem. The purpose of the present note is to prove the following theorem. Theorem: Let X be a domain with C-boundary and with a complete Kahler metric in a complex manifold M. Then X is pseudoconvex. To prove the theorem we apply the method of L estimates of d which is summarized in Section 1.

A Weak Equivalence and Topological Entropy

Abstract: In this paper we will investigate the topological entropies of mutually weakly equivalent topological flows. Roughly speaking, any two flows which are weakly equivalent to each other have the same orbits. So the notion of weak equivalence of flows is, in a sense, a generalization of time changes of flows. In [5] Totold investigated time changes of flows from a measure theoretical point of view. Especially he showed that for metrical (=measure theoretical) flows time changes preserve the properties that the metrical entropy is zero, positive, finite or infinite respectively. Here we will be rather concerned with topological flows and their topological entropies. First we will consider flows without fixed point. In this case we obtain a result analogous to Totoki's one. Namely, the properties that the topological entropy is zero, positive, finite or infinite respectively are invariant under weak equivalence of flows without fixed point (Theorem 1 in §3). But this is not the case if the flows have fixed points. Indeed, we will construct a pair of flows with the same orbits and a fixed point one of which has a positive entropy and the other has zero entropy (Theorem 2 in §4). In the proof of these two theorems we will appeal to a measure theoretical method. The point is that the topological entropy is the supremum of metrical entopies with respect to all invariant Borel probability measures (Lemma 2 in §3). The idea of the construction of the example in Section 4 to prove Theorem 2 is as follows. Take a flow with a fixed point such that each orbit visits a neighbourhood of the fixed point infinitely often and that the ratio of the sojourn time in the neighbourhood is uniformly positive. One can construct such a flow with a positive topological entropy. Then, lowering the speed of the flow

Produit Croise D'une Algebre de von Neumann par Une Algebre de Kac, II

Abstract: La construction du produit croise d'une algebre de von Neumann par un groupe continu d'automorphismes est a Forigine des resultats recents [2], [20]) dans Fetude de la structure des facteurs. Soit G un groupe localement compact. M. Landstad a remarque dans [12] l'interet des proprietes d'algebre de Hopf-von Neumann de L°°(G) et <3tt(G) (respectivement algebre des fonctions mesurables essentiellement bornees sur G et algebre de la representation reguliere gauche de G) pour traiter la construction des produits croises. La construction qu'il utilise s'interprete en fait comme une \"action\" de JM(G) (considere comme substitut dans le cas non-abelien du groupe dual de G) sur une algebre de von Neumann. Les algebres de Kac ([5], [17]) dont L°°(G) et <3M(G) sont des cas particuliers duaux l'un de Fautre, apparaissent alors comme devant fournir un cadre bien adapte ä une generalisation de la notion de produit croise. Celle-ci contenant a la fois la construction du produit croise par un groupe [20], et par un \"dual de groupe\" [13], [14]. Une premiere partie de ce travail a dejä ete faite [4], mais eile ne fournit pas tous les resultats dont on pouvait esperer la generalisation, notamment en ce qui concerne la dualite. Les obstacles techniques etant desormais leves, nous donnons ci-apres une theorie plus complete, qui permet de generaliser au cas des algebres de Kac le theoreme de dualite de M. Takesaki [20], le theoreme de commutation de T. Digernes [3] et la caracterisation des produits croises, due ä M. Landstad [12].

Some Theorems about Formal Functions

Abstract: It is well-known that in algebraic geometry there are very few nontrivial problems about continuation of objects from open subsets, which is a big difference to analytic geometry. So we may expect that in the algebraic version of analytic geometry, formal geometry, there are some interesting problems of this kind, and this assumption is supported by the following paper. In there we take a local ring which is complete in some */-adic topology and consider the formal scheme obtained by completing the spectrum of our local ring along the closed subscheme defined by */. As all interesting formal objects on this formal scheme stem from objects on the usual spectrum of our local ring, we remove a closed subset. After that there may exist nonalgebraic formal objects, and such objects are characterized by the fact that there is no continuation of them to the whole formal scheme. The objects we are interested in are formal meromorphic functions and formal subsheaves of algebraic formal sheaves. As it is the case in analytic geometry these two cases are connected, and any result about one of them implies a corresponding result about the other. As application of our theory we prove certain results about connectedness, including a generalization of Zariski's connectedness-theorem. I express my gratitude to Professor Hironaka for some very stimulating discussion, and to the Deutsche Forschungsgemeinschaft for support during the last year.

Some Calculations in the Unstable Adams-Novikov Spectral Sequence

Abstract: The unstable Adams-Novikov spectral sequence for a space X is a sequence of groups {Er(X)}9 r = 2, 3,..., which converge to the homotopy groups of X, and whose £2"term depends on the complex cobordism groups of X. We investigate this spectral sequence when X is the infinite special unitary group SU, or one of the finite groups SU(n), or when X is an odd sphere S2"+1. The reader is referred to [2] for the construction and properties of the unstable Adams-Novikov spectral sequence. For some purposes, it is convenient to localize at a prime p, in which case the complex cobordism homology theory, based on the spectrum MU, is replaced by Brown-Peterson homology, based on the spectrum BP. We then have a useful spectral sequence with many of the properties of the stable Adams-Novikov spectral sequence. Namely, the nitrations are less than or equal to the filtrations in the unstable Adams spectral sequence based on mod-p homology. When X is a space for which H*(X; BP) is free over the coefficient ring n*(BP) and cofree as a coalgebra, then the E2-term is isomorphic to an Ext group in an abelian category (see §2; also [2, § 7]). Furthermore, this Ext group may be computed as the homology of an unstable cobar complex which we describe explicitly in Section 2. In particular, these considerations apply to the cases X = SU, X = SU(ri), or X = S2n+1. We first consider the situation where X is a p-local H-space with torsion-free homotopy and torsion-free homology. The results of Wilson [10] and the

Sur les Singularités Essentielles et Isolées des Applications Holomorphes à Valeurs dans une Surface Complexe

Abstract: Introduction. Chapitre I — Quelques proprietes generales de l'ensemble limite (ciuster set) /(O; M) 463 1. Definition de/(0)=/(0; M) (Theoreme 1) 463 2. Rapport avec la pseudo-distance de Kobayashi (Theoreme 2) 464 3. Cas ou /(O) est une courbe analytique (Theoreme 3) 465 Chapitre II — Cas ou /(O) est une courbe analytique compacte dans une surface complexe 469 4. Courbes de valeurs exceptionnelles et l(P) (Theoreme 4) 469 5. Ensemble limites d'une suite d'images analytiques du disque ferme J : |z|^l 471 6. Surfaces de recouvrement d'Ahlfors 479 7. Liste des types de/(0) compacts compris dans une courbe de valeurs exceptionnelles (Theoreme 5) 484 Chapitre III — Singularites essentielles des isomorphismes analytiques le long d'une courbe compacte 488 8.

A Product Formula and Its Application to the Schrödinger Equation

Abstract: A product formula is given which represents the unitary group for the form sum of a pair of nonnegative selfadjoint operators in a Hilbert space.

On the Green Functions of 1-Dimensional Diffusion Processes

Abstract: For every a>0, G# is always well-defined, but G0 does not exist in general, as is seen in the case of 1or 2-dimensional Brownian motions. However, in many cases, G0/can be defined to be the limit of G^/as a-»0 if we restrict /to be in a certain class of functions, called the domain of the potential operator. There have been a lot of works on potential operators for Markov processes, e.g. G. Hunt [1], J. Takeuch,i T. Yamada and S. Watanabe [2], G. Kemeny-J.L. Snell [3], R. Kondo [4], K. Yosida [5], K. Sato [6] and T. Arakawa and J. Takeuchi [7]. An important class of recurrent Markov processes for which we can define potential operators is the following: Suppose that Ga(x, dy) (a>0) has density ga(x, y) with respect to a measure v(dy) of the following form:

Interval Logics with Applications to Study of Tense and Aspect in English

Abstract: Chapter 1: Philosophical Motivation 417 §11: Introduction 417 §12: Temporal Dualism 418 Chapter 2: The Prepositional Logic I 420 §21: Formal Language LI 420 §22: Semantics for I 421 §23: Dl-sequence 423 §24: Formal System I 425 §25: The Soundness and Completeness of I 427 Chapter 3: The Prepositional Logic IM 435 §31: Formal Language LIM 435 §32: Semantics for IM 436 §33: DMI-sequence and DIM-sequence 441 §34: Formal System IM 445 §35: The Soundness and Completeness of IM 450 Chapter 4: Some Applications 450 § 41: Some Interesting Tense Operators Definable within IM 450 §42: Tense and Aspect within IM 455

Gentzen Reduction Revisited

Abstract: The major objective of this paper is to advocate Gentzen's first work on the consistency of arithmetic., which appeared in 1936 ([!]). He later published a new version of the consistency proof, and it is the reduction method of the latter which has been mostly used for various consistency proofs. His first work should be studied, however, more than it has been, not only for the strangely beautiful flavor it contains, but also for that it gives a general reduction method for arbitrary derivations (not just for supposedly contradictory ones), hence giving a systematic means to investigate the structure of formal systems of arithmetic. We first present a reformulation of Gentzen's reduction of first order arithmetic in [1] in the style of [2] (§1). As an outright application of the result in Section 1, we obtain a form of quantifier-free interpretation of formal derivations of arithmetic (§2) and an interpretation of derivable formulas (§3) in terms of ordinal recursive functions. Variations of the preceding results will be given in Section 4 (first order systems) and in Section 5 (second order systems). Many relevant studies have been made: those are seen, for example, in several papers of Kreisel, Schiitte's book and Tait's paper (cf. [6]~[12] and [14]). We wish to make the following points here. With our formulation, various known results follow systematically, without individual adjustments. It is also our point not to interpret a given system in a formal system with the (constructive) cy-rule, for such a manoeuver seems to lose some delicate nature of the reducts of derivations.

On the Convergence of the Initial-Value Adjusting Method for Nonlinear Boundary Value Problems

Abstract: Here x and X(t, x) are n -dimensional vectors, x(t) is considered as a function J = [a, fc]-»J?" and / is an rc-dimensional vector-valued functional on some subset of Q7). Recently Ojika and Kasue [5] have proposed a numerical procedure called "initialvalue adjusting method" to solve the multi-point boundary value problem, which is considered to be a powerful algorithm for rather complex constraining conditions. And Ojika [4] has given a proof for convergence of their method. The present paper is devoted to an analysis of the method by the different way from his and to obtaining sufficient conditions for convergence of the iteration. Roughly speaking, for the initial-value adjusting method, which can be regarded as a systematical version of the shooting methods, the convergence holds when X and / are sufficiently smooth and the starting value of iteration is taken sufficiently close to the isolated exact one. Moreover, it will be shown that the inverse of the "adjusting matrix" is a good example of the contractor that has been introduced by Altman [1].

Selfpolar Norms on an Indefinite Inner Product Space

Abstract: A vector space equipped with an indefinite inner product is investigated. Selfpolar norms on the space are studied and an operator description for quadratic selfpolar norms is developed when the space allows a Hilbert space topology making the indefinite inner product continuous. The selfpolar norms corresponding to a quasi-decomposition of the space are characterised in terms of the operator description and sufficient conditions for topological equivalence are given. §

The Initial-Value Adjusting Method for Problems of the Least Squares Type of Ordinary Differential Equations

Abstract: ^ j=i where tj are given points on [a, fo], a = tl

On the Uniqueness of Dyer-Lashof Operations on the Bott Periodicity Spaces

Abstract: shows that each of these spaces is an infinite loop space. Dyer-Lashof operations Q (r>0) are defined on the mod p homology (p prime) of any infinite loop space, depending on its infinite loop structure. If p=2, they are natural homology operations of degree r such that Q(x^)=x%9 G^J—O if r

On Parabolic Functions of One-Dimensional Quasidiffusions

Abstract: In this paper we will deal with quasidiffusions X=(Xt)t^0 on [0, 1) assuming that 0 is a reflecting regular boundary, 1 is an accessible or entrance boundary and X is killed as soon as it hits the boundary 1. We shall give a Martin representation of space-time-excessive functions for X. This includes in particular a representation of all parabolic functions f(x, t) satisfying a certain integrability condition by minimal ones (see Theorem 2 below). We shall show that the set of minimal points of space-time Martin boundary is homeomorphic to (0, oo[; in particular the minimal parabolic functions form a one-parameter family (kt) (re(0, oo]). If z

A Difference Scheme for Solving Two Phase Stefan Problem of Heat Equation

Abstract: On revise la demonstration, sur certains points, d'un schema aux differences pour la resolution du probleme de Stefan a deux phases relatif a l'equation de la chaleur (voir ibidem 1980, vol. 16, no 2, pp. 313-341)

On the Diameter of Compact Homogeneous Riemannian Manifolds

Abstract: Let M be a compact Riemannian manifold. The diameter d(M) of M is defined to be the maximum of d(p, q) p, q e M, where d( , ) denotes the distance function on M induced by the Riemannian metric. The main purpose of this paper is to find a positive constant d such that the diameter d(M)^d when the sectional curvature K^l. In this paper we consider the case that the manifold M is homogeneous. In [3] the author proved that d = n/2 if the manifold has a big isotropy subgroup. It has been left to study the case that the isotropy subgroup is finite. Hence we shall mainly study invariant metrics on a Lie group and prove that the number rf>0.23 if the sectional curvature K=£Q (Theorem 5.1).

On the Synthesis Problems of the Semimodular State Chart Theory : I. Distributive Chart

Abstract: We normally classify switching circuits as either synchronous or asynchronous depending on whether or not the signals in the circuit are synchronized with some source of fundamental frequency (or clock) which regulates the entire circuit It is possible to predict the state of a synchronous circuit for any given clock signal if one knows the initial state of the circuit and its logical characteristics However from knowledge of the logical characteristics of the circuit alone, it is impossible in an asynchronous circuit to predict the next state from the present one The state may also depend upon the relative speeds of some of the logical elements which comprise the circuit One of the main objectives of asynchronous theory is to describe the properties of circuits in which their ultimate behavior does not depend on the relative speeds of their elements The semimodular circuit theory introduced by D E Muller and W S Bartkey had the purpose of developing the techniques for designing asynchronous circuits [1, 2, 3] In a series of papers [4, 5, 6], it was reorganized as a theory of H-dimensional space of lattice points In this theory a state of a circuit is specified by a n-tuple z = (zl5, zn), Zje{0, l,,p}, and from a sequence of states z-^-*^->••-, we can construct the sequence of ??-tuples M-»M-* M-»---, such that the i-th component MJ of M=(M\, MJ,, Af*), is a nonnegative integer representing the total number of changes of f-th component of zf-»z->z-» >zj during the state change from z to z For instance, if 2 = (00), z = (01), z = (ll), z = (10),, then M^OO), M = (10),

On a Limit Theorem for Branching One-Dimensional Diffusion Processes

Abstract: Let M be a simply connected complete Riemannian manifold with constant sectional curvature, and consider a branching Brownian motion Y=(yt9Py) having M as the underlying state space ([5]), In the case of M = S, the ddimensional sphere, one can apply the results of Watanabe [16], [17] and Asmussen-Hering [1] to obtain a limit theorem on the number of particles in a domain for the process Y. If M= R, then, although M is not compact, the process F belongs to the class considered by Watanabe [17], and his argument works well. But, if M has constant negative sectional curvature — fc, then Y is not necessarily contained in the scheme of [17] and a new phenomenon appears. In this case, the Laplace-Beltrami operator is given by

An Equisingular Deformation Theory via Embedded Resolution of Singularities

Abstract: In this paper we shall give a definition of equisingular deformations of isolated singularities and prove some results about it such as the existence of semi-universal deformations. Roughly speaking, equisingularity means in this paper the existence of a simultaneous embedded resolution. Of course, in the case of plane curve singularities the definition coincides with Zariski's classical one. In Section 1 we shall define equisingular deformation and prove some elementary properties of it. In Section 2 we shall study the deformation of locally product type of the pair obtained by the embedded resolution of singularity. In Section 3 we shall prove the existence theorem of semi-universal deformations. The author would like to express his heartfelt thanks to Professor S. litaka, the conversations with whom were very fruitful and encouraging, and to Pf. M. Merle who pointed out to the author Example 2.

On the Homotopy Types of the Groups of Equivariant Diffeomorphisms

Abstract: The purpose of this paper is to study the homotopy type of the group of the equivariant diffeomorphisms of a closed connected smooth G-manifold M, when G is a compact Lie group and the orbit space M/G is homeomorphic to a unit interval [0, 1]. Let Diffg (M)0 denote the group of equivariant C°° diffeomorphisms of the G-manifold M which are G-isotopic to the identity, endowed with C°° topology. If M/G is homeomorphic to [0, 1], then M has two or three orbit types G/H, G/K0 and GjKl. We can choose the isotropy subgroups H, K0, K^ satisfying HdK0nK1. Moreover the G-manifold structure of M is determined by an element r\\ of a factor group N(H)/H9 where N(H) is the normalizer of H in G (see §1). Let Q(N(H)/H; (N(H) fl N(K0))IH, (N(H) n N(rjK1ri~ y)IH)0 denote the connected component of the identity of the space of paths a: [0, l]-+N(H)IH satisfying a(0)e(N(H) n N(X0))/H and a(l)e(N(H) n

Quasi-Invariance of Measures on an Infinite Dimensional Vector Space and the Continuity of the Characteristic Functions

Abstract: Since the Bochner theorem was extended to the infinite dimensional case by Minlos [1] and Sazonov [2], the continuity of a characteristic function has been discussed mainly in connection with the carrier of the corresponding measure. However, the study of the relation between the continuity of a characteristic function and the quasi-invariance of the corresponding measure has been rather neglected. In this paper we shall discuss this problem. Our main results are as follows. Let £ be a vector space, £' be its algebraical dual space, ^ be a finite measure on £', and x be the characteristic function of \\i defined on E. Consider the weakest vector topology on E that makes x continuous, and denote it with T^. Let 7J, be the set of all translations on E' under which \\i is quasi-invariant. T^ is regarded as a subset of E' by identifying any translation x-»x + a on E' with a. Then we have the following