Showing papers in "Pacific Journal of Mathematics in 1978"

On the extension of additive functionals on classes of convex sets

Abstract: Let Sf be a class of sets and let U(&) denote the class of all finite unions of sets from £f This paper concerns itself with the question whether a vector valued function on Sf has an additive extension to U(£f) Several characterizations of functions with such an extension property are presented One of these characterizations establishes a relationship between such extensions and integrals of certain types of simple functions The special cases when Sf is the class of all convex polytopes or the class of all compact convex subsets of a euclidean space are investigated in more detail Some examples are given to show that various extension problems that have been solved previously by methods particularly designed for each individual problem can also be solved by the application of these general results

Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces

Abstract: This paper establishes that the homotopy category of rational differential graded commutative coalgebras is equivalent to the homotopy category of rational differential graded Lie algebras which have a nilpotent completion as homology. This generalizes a result which Quillen proved in the simply connected case. When combined with Sullivan's work on rational homotopy theory, our result shows that the homotopy category of rational differential graded Lie algebras with nilpotent finite type homology is equivalent to the rational homotopy category of nilpotent topological spaces with finite type rational homology. Our results include the construction of minimal Lie algebra models for simply connected spaces, and we show that the rational homotopy groups of a simply connected CW complex may be calculated from a free Lie algebra generated by the cells with a differential given on generators by the attaching maps.

Geometrical implications of upper semi-continuity of the duality mapping on a banach space

Abstract: For the duality mapping on a Banach space the relation between lower semi-continuity and upper semi-continuity properties is explored, upper semi-continuity is characterized in terms of slices of the ball and upper semi-continuity properties are related to geometrical properties which imply that the space is an Asplund space. The duality mapping is a natural set-valued mapping from the unit sphere of a normed linear space into subsets of its dual sphere, and which for an inner product space is the mapping associating an element of the unit sphere with the corresponding continuous linear functional given by the inner product. It is an example of a subdifferential mapping of a continuous convex function (in this case, the norm), which is in turn a special kind of maximal monotone mapping. Cudia [4, p. 298] showed that the duality mapping is always upper semi-continuous when the space has the norm and the dual space has the weak* topology, and Kenderov [10, p. 67] extended this to maximal monotone mappings. Bonsall, Cain, and Schneider [3] used the property to prove the connectedness of the numerical range of an operator on a normed linear space. Along with the activity which culminated in StegalΓs theorem [15] characterizing an Asplund space as one whose dual has the Radon-Nikodym property, there has been some interest in finding geometrical conditions sufficient for a space to be Asplund. A Banach space X is an Asplund space if every continuous convex function defined on an open convex subset of X is strongly differentiate on a dense Gδ subset of its domain. Ekeland and Lebourg [6, p. 204] have shown that a Banach space is Asplund if there exists a strongly differentiate real function on the space with bounded nonempty support, in particular, if the space can be equivalently renormed to have norm strongly differentiable on the unit sphere. Using StegalΓs theorem, a result of Diestel and Faires [5, p. 625] gives that a Banach space is Asplund if the space can be equivalently renormed to be very smooth, that is, to be smooth and to have the single valued duality mapping continuous when the space has the norm and the dual space has the weak topology. Recently Smith and Sullivan [13, Theorem 15] have exhibited a more general condition which is sufficient for a Banach space to be Asplund. We show that

Focal sets of submanifolds.

Abstract: We explain the notation used in the theorem. Under the hypotheses, the principal vectors corresponding to λ form a smooth vdimensional distribution Tλ on M whose leaves are umbilic submanif olds of M. On each leaf, λ is constant. fλ is the map from part of M onto the set of focal points arising from the principal curvature λ. Mi is the space of leaves of Tλ which intersect the domain of fλ. The proof relies on the work of Palais [13] on foliations. This theorem generalizes that of Nomizu [10] who proved a similar result for hypersurfaces in the sphere with constant principal curvatures. Because of the abundance of examples of such hypersurfaces, one can produce (through stereographic projection) many examples of hypersurf aces in Euclidean and hyperbolic space to which our theorem applies (see §3.c). If λ has constant multiplicity one, then fλ(M) is not an (n — 1)dimensional manifold without additional hypotheses. This case is handled by Theorem 3.2 which is a generalization of the classical determination of conditions under which a sheet of the focal set of a surface in E is a curve. (See, for example, Eisenhart [6, p. 310314].) A key ingredient in the proofs of the above results is the computation of the rank of fλ. Our result in this area (Theorem 2.1) applies to submanifolds of arbitrary codimension. The classical version of these theorems was used by Banchoff [1] and Cecil [3] in characterizing taut immersions of surfaces in E. Applications of the results of the present paper to the classification of taut immersions of S x S~ into E may be found in our forthcoming paper [5].

Involutions of seifert fiber spaces

Abstract: A Seifert fiber space M is a compact 3-manifold which decomposes into a collection &~ of disjoint simple closed curves, called fibers, such that each fiber has a tubular neighborhood which consists of fibers and is a *'standard fibered solid torus." We consider the question, given a PL involution h of M, can the fiber structure _^~ be chosen in such a way that h will be fiber-preserving? We give an affirmative answer for the case when M is orientable, irreducible, and either ΘMΦ 0 or M contains an incompressibl e fibered torus. THEOREM. Let h be a PL involution of the orientable, irreducible Seifert fiber space M. If the orbit-surface (Zerlegungsfiache) of the fiber structure is a 2-sphere, assume in addition that there exist at least four exceptional fibers. Then there exists a Seifert fiber structure on M with respect to which h is fiber-preserving. This theorem touches on two earier results. In [6] Montesinos considers the following problem. Given any orientable Seifert fiber space M, determine whether M is homeomorphic to a 2-fold cyclic covering of S 3 branched over a link. He shows that all orientable Seifert fiber spaces with orbit-surface either a 2-sphere or a nonorientable surface are such 3-manifolds. For those with an orientable orbit-surface of positive genus, he compiles a list of all those which are 2-fold branched cyclic covering spaces of S3 with fiber-preserving covering transformations. We can now conclude that this list is complete since it follows from our theorem that all the PL involutions involved as covering transformations can be viewed as fiberpreserving involutions. In [1] it is shown that if an irreducible, orientable, sufficiently large 3-manifold M is covered by a compact Seifert fiber space then M is either a Seifert fiber space or the union of two twisted line bundles over a closed nonorientable surface. It is not clear whether the union of these two twisted line bundles admits a Seifert fiber structure, but there exists a two-sheeted covering space W of the union which is a Seifert fiber space. Thus, if one could show that M always contains an incompressibl e fibered torus then it would follow that M is a Seifert fiber space. Let us describe how one goes about constructing a fiber structure which is preserved by an involution. Let h be a PL involution of the Seifert fiber space M. If M is closed we construct a fibered

On joint numerical ranges

Abstract: The joint numerical status of commuting bounded operators Ai and A2 on a Hubert space is defined as {{φiA^y φ(A2)) such that φ is a state on the C*-algebra generated by Ax and A2}. It is shown that if At and A2 are commuting normal operators then their joint numerical status equals the closure of their joint numerical range. It is also shown that certain points in the boundary of the joint numerical range are joint approximate reducing eigenvalues. The joint numerical range of Ax and A2 denoted by w(Alf A2) is jX, x), (A2x, x)) such that xeH and ||g|| = 1}. Thus w(Alf A2) is a bounded subset of C 2. It is not known whether this set is convex, Dash [4, 6]. In this note, we shall show that there is faithful * representation of the C*-algebra generated by Ax and Ai9 C*(Aif A2), under which the joint numerical range of At and A2 is convex. Following Berberian and Orland [1], we study the joint numerical status of Ax and A2, Σ(Aλ, A2) = {{φ(Ax)9 Φ(A2)) such that φ is a state on C*(A19 A2)}. If Ax and A2 are commuting normal operators then Σ(Aif A2) — w(A19 A2). We also show that certain points in the boundary of w(A19 A2) are joint approximate reducing eigenvalues. For the sake of notational convenience, all the results are being stated for two commuting operators. However, the results hold for any finite family of commuting operators. Let B(H) denote the algebra of all bounded linear operators on