Showing papers in "Tohoku Mathematical Journal in 1967"

Weighted sums of certain dependent random variables

Abstract: 1. Let be a probability space and,be an increasing family of sub o'-fields of(we put(c) Let (xn)n=1, 2, •c be a sequence of bounded martingale differences on , that is,xn(ƒÖ) is bounded almost surely (a.s.) ands. for n =1, 2,.... It is easily seen that this sequence has the following properties[G] and [M], which have been introduced by Y. S. Chow ([1]) in an analogous form and by G. Alexits ([4]), respectively, and may be of independent interest.

On some extention properties of von neumann algebras

Abstract: 1. In the theory of the structure of von Neumann algebras, a great deal of discussions has been devoted to the algebraic invariants. Recently, J. Schwartz [6], has shown a new behavior of the hyperfinite factor by introducing a new property, called property P, and proved the existence of new algebraic type of continuous finite factor. Very recently, one of the authors has proved that property P is an algebraic invariant [2] and it makes us to have some interest that the main results of the paper [6], especially the key results [6: Cor. 6 and Lemma 7], can be deducsd only from the point of view of the existence of a projection mapping of norm one. Thus we shall investigate in the following the algebraic version of property P as an extension property of the commutant of a given von Neumann algebra. We shall also study this extension property as the property of the commutant itself. These properties will turn out to be algebraic invariants and it is proved that our extension property can be defined space-freely in a form very similar to the usual extension property of Banach space. Relationships between tensor products of von Neumann algebras and these properties are also studied.

Certain hypersurfaces of an odd dimensional sphere

Abstract: Using thisresult, we can also prove the latter assertion of the theorem.BIBLIOGRAPHY[ 1 ] S. KOBAYASHI AND K. NOMIZU, Foundations of Differential Geometry, InterscienceTracts, John Wiley and Sons, New York, 1963.[ 2 ] S. MYERS, Riemannian manifolds with positive mean curvature, Duke Math. Journ, 8(1941) 401-404.[ 3 ] M. OKUMURA, Certain almost contact hypersurfaces in Euclidean spaces, Kδdai Math.Sem. Rep., 16(1964), 44-54.4) S.Myers [2].5) J.L.Synge [51.6) For example S.Kobayashi and K. Nomizu [1], p. 294.[4] S.Sasaki AND Y. Hatakeyama, On differentiable manifolds with contact metric structure,Journ, Math. Soc. of Japan, 14(1962) , 249-271.

On c-harmonic forms in a compact sasakian space

Abstract: Introduction. It is well known that in a 2m-dimensional compact Kahlerian space, any harmonic ^>-form (pt===. nt) can be written uniquely in terms of effective harmonic forms and the fundamental 2-form of the space. When we consider the analogy in a compact Sasakian space, it is insignificant as far as we are concerned about harmonic forms, because any harmonic form is effective. S. Tachibana [1] has introduced the notion of C-harmonic forms in a compact Sasakian space, which is wider than that of harmonic forms, and succeeded to prove the analogy of the decomposition theorem for C-harmonic forms. In this paper we try to make the definition of C-harmonic forms a little looser than that of Tachibana's original one. On the other hand, S. Tanno has drawn the relation of Betti numbers between the base space and the bundle space in the fibering of a regular i^-contact Riemannian space. It is shown that a ^>-form (p^ m) on the bundle space is C-harmonic if and only if it is induced from a harmonic ^>-form on the base space. Thus we can obtain the theorem of Tanno again. Lastly we investigate the C*-harmonic forms which are dual to the C-harmonic forms, and in connection with them, we observe Killing forms and give one of its example on a Sasakian space. Manifolds are assumed to bs connected and the differentiable structures on them are assumed to be of class C°°. I should like to express my hearty gratitude to Prof. S. Tachibana for his kind suggestions and many valuable advices. Contentes are as follows: 1. Preliminaries 2. C-harmonic forms 3. Decomposition theorem 4. Regular Sasakian structure 5. C*-harmonic forms