Electronic Edition Vol. 31, No. 3, 2005

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


Manfred Dugas, Department of Mathematics, Baylor University, Waco, Texas 76798 (Manfred_Dugas@baylor.edu) and Shalom Feigelstock, Department of Mathematics, Bar Ilan University, Ramat Gan 52600, Israel (feigel@macs.biu.ac.il).
Co-minimal Abelian Groups, pp. 637-648.
ABSTRACT. An abelian group A was called minimal in [1], if A is isomorphic to all its subgroups of finite index. We study the dual notion and call A co-minimal if A is isomorphic to A/K for all finite subgroups K of A . We will see that minimal and co-minimal groups exhibit a similar behavior in some cases, but there are several differences. While a reduced p -group A is minimal if and only if A/pω is minimal, this no longer holds for co-minimal p -groups. We show that a separable p -group A is co-minimal if and only if A is minimal. This does not hold for p -groups with elements of infinite height. We find necessary conditions for co-minimal p -groups in terms of their Ulm-Kaplansky invariants, which are also sufficient for totally projective p -groups. If A is a mixed group with a knice system, also known as Axiom 3 modules, then A is co-minimal if and only if t(A), the torsion part of A, is co-minimal. We construct an example of a mixed group A such that t(A) is a totally projective p -group of length ω+1 such that t(A) is co-minimal but A is not co-minimal. Moreover, we construct p -groups G of length ω+1 such that all Ulm-Kaplansky invariants of G are infinite, i.e. G is minimal, but G is not co-minimal.

Reference: [1] B. Goldsmith and S.OhOgain, On torsion and mixed minimal abelian groups, Rocky Mountain J. Math. 32 (2002), 1451-1465.

B. Fine, Department of Mathematics, Fairfield University, Fairfield, CT 06430 and Anthony M. Gaglione, Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402 (amg@usna.edu) and D. Spellman, Department of Mathematics, Temple University, Philadelphia, PA 19122.
Discriminating and Squarelike Groups II: Examples, pp. 649-674.
ABSTRACT. Discriminating groups were invented by G. Baumslag, A.G. Myasnikov and V.N. Remeslennikov. These groups arose out of consideration by the above authors of their freshly minted theory of algebraic geometry over groups. Although unforeseen at first, the notion admits beautiful and exotic examples some of which appeared earlier in diverse contexts. Squarelike groups were invented by the authors in a previous paper and may be viewed as nonstandard discriminating groups. In this paper we give examples of groups which are discriminating or squarelike as well as groups which are not. The main result is the existence of a finitely generated squarelike group which is not discriminating.

Charles Megibben, Department of Mathematics, Vanderbilt University, Nashville Tennessee 37240 (charles.k.megibben@vanderbilt.edu) and William Ullery, Department of Mathematics, Auburn University, Auburn, Alabama 36849 (ullery@math.auburn.edu).
On global abelian k-groups, pp. 675-692.
ABSTRACT. The class of global k-groups is an abundant class of abelian groups that contains, but is not restricted to, all torsion groups, torsion-free separable groups, and mixed groups with decomposition bases. Our main theorem is that a global k-group of cardinality Àn (for some nonnegative integer n) has sequentially pure projective dimension at most n. This was known previously only in the special case where n≤1

Luigi Salce, Dipartimento di Matematica Pura e Applicata, Università di Padova, Italy (salce@math.unipd.it).
On the Minimal Injective Cogenerator over Almost Perfect Domains, pp. 693-705.
ABSTRACT. Let R be a local almost perfect domain with maximal ideal P. The minimal injective cogenerator E(R/P) and its endomorphism ring A are investigated. It is shown that, if R is non-Noetherian and the square of P is open in the Prüfer (i.e., finitely embedded) topology, then A strictly contains the completion of R, which coincides with its center, hence A is non-commutative. The new class of P-chained domains is introduced. These rings are local almost perfect and pseudo-valuation domains. It is proved that in the Noetherian case they coincide with the pseudo-valuation domains, and in the non-Noetherian case they satisfy the condition which ensures that the ring A is non-commutative.

Mohammad Saleh, Mathematics Department, Birzeit University, West Bank, Palestine, (msaleh@birzeit.edu).
Serre class and the direct sums of modules, pp. 707-720.
ABSTRACT. The purpose of this paper is to further the study of weakly injective and weakly tight modules a generalization of injective modules. For a Serre class K of modules, we study when direct sums of modules from K satisfies a property P in K. In particular, we get characterization of locally q.f.d. modules in terms of weak tightness.

Kazuhiro Ichihara, Nara Women's University, Nara 630-8506, Japan (ichihara@vivaldi.ics.nara-wu.ac.jp) and Shin Satoh, Chiba University, Chiba 263-8522, Japan (satoh@math.s.chiba-u.ac.jp).
Liftability for double coverings of immersions of non-orientable surfaces into 3-space, pp. 721-741.
ABSTRACT. We prove that the double covering map of any generic immersion of a projective plane into 3-space does not lift to an embedding into 4-space. To prove this, we give a criterion of the liftability for double coverings of immersions of any non-orientable surfaces.

Bang-Yen Chen, Department of Mathematics, Michigan State University, East Lansing, MI 48824--1027, USA (bychen@math.msu.edu) and Ion Mihai, Faculty of Mathematics, University of Bucharest, Str. Academiei 14,70109 Bucharest, Romania (imihai@math.math.unibuc.ro).
Isometric Immersions of contact Riemannian Manifolds in real space forms, pp. 743-764.
ABSTRACT. In this paper we define some contact Riemannian invariants for almost contact metric manifolds which are analogues to those invariants originally defined for general Riemannian manifolds introduced by the first author. We then establish sharp inequalities between these contact Riemannian invariants and the squared mean curvature for almost contact Riemannian manifolds in a Riemannian manifold of constant curvature. We also investigate almost contact Riemannian submanifolds which verify the equality case of the inequalities. Moreover, examples of contact Riemannian submanifolds satisfying the equality case are provided as well.

Changrim Jang, Mathematics Department, Wichita State University, Wichita KS 67260-0033, USA (crjang@mail.ulsan.ac.kr) (permanent address: Department of Mathematics, College of Natural Sciences, University of Ulsan, Ulsan 680-749, Republic of Korea), Phillip E. Parker, Mathematics Department, Wichita State University, Wichita KS 67260-0033, USA (phil@math.wichita.edu) (http://www.math.wichita.edu/~pparker/), and Keun Park, Department of Mathematics, College of Natural Sciences, University of Ulsan, Ulsan 680-749, Republic of Korea (kpark@mail.ulsan.ac.kr}.
Pseudo H-type 2-step Nilpotent Lie Groups, pp. 765-786.
ABSTRACT. Pseudo H-type is a natural generalization of H-type to geometries with indefinite metric tensors. We give a complete determination of the conjugate locus including multiplicities. We also obtain a partial characterization in terms of the abundance of totally geodesic, 3-dimensional submanifolds.

Ivansic, Ivan, Department of Mathematics, University of Zagreb, Unska 3, P.O. Box 148, 10001 Zagreb, Croatia (ivan.ivansic@fer.hr), and Rubin, Leonard R., Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA (lrubin@ou.edu).
Limit Theorem for Semi-sequences in Extension Theory, pp. 787-807.
ABSTRACT. Given an inverse sequence X = (Xi,pii+1) of topological spaces along with subspaces Ti of Xi and an infinite subset N* of the set of positive integers N, we define the concept of a semi-sequence T=(Ti,N*) and its semi-limit T=semi-lim T which is a subset of X=lim X. A notion of stability of T in X is defined along with extendability of X with respect to a given CW-complex K and T. We show that if the terms Xi of the inverse sequence are stratifiable, Ti is closed in Xi, and stability and extendability apply, then K is an absolute extensor for T. All previous extension theoretic limit theorems about inverse sequences are corollaries to this result.

Yan-Kui Song, Department of Mathematics, Nanjing Normal University, Nanjing 210097, China (songyankui@njnu.edu.cn).
Conditions which imply Lindelöfness in star-Lindelöf spaces, pp. 809-813.
ABSTRACT. In this paper we prove that every regular star-Lindelöf space is Lindelöf if and only if every increasing open cover Uα, α< τ, admits an increasing open cover Vα such that the closures of the Vα are contained in the Uα.

Bonami, Aline Université d'Orléans, BP 6759, F 45067 Orléans Cédex 2, France (aline.bonami@univ-orleans.fr) and Luo Luo, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China (lluo@ustc.edu.cn).
On Hankel Operators between Bergman Spaces on the Unit Ball, pp. 815-828.
ABSTRACT. We study the boundedness of a (small) Hankel operator between different Bergman spaces on the unit ball B in Cn. We give conditions on its symbol which are necessary and/or sufficient for the continuity of the corresponding operator from Ap(B) into Aq(B), for all finite p,q >0.

Kenneth J. Dykema and Roger R. Smith, Department of Mathematics, Texas A&M University, College Station TX 77843--3368, USA (kdykema@math.tamu.edu), (rsmith@math.tamu.edu).
The completely bounded approximation property for extended Cuntz--Pimsner algebras, pp. 829-840.
ABSTRACT. The extended Cuntz-Pimsner algebra E(H), introduced by Pimsner, is constructed from a Hilbert B,B-bimodule H over a C*-algebra B. In this paper we investigate the Haagerup Lambda invariant for these algebras, the main result being that the value of this invariant for E(H) equals that for B whenever H is full over B. In particular, E(H) has the completely bounded approximation property if and only if the same is true for B.

D.E. Edmunds, Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9RF, UK (D.E.Edmunds@sussex.ac.uk) and E. Shargorodsky, {Department of Mathematics, King's College London, Strand, London, WC2R 2LS, UK} (eugene.shargorodsky@kcl.ac.uk)
The inner variation of an operator and the essential norms of pointwise multipliers in function spaces, pp. 841-855.
ABSTRACT. We show that useful estimates for the essential norm of pointwise multipliers acting on a function space may be easily obtained from certain functional-analytic facts related to the notions of inner variation and the measure of noncompactness. Applications to Besov and Lizorkin-Triebel spaces are given.

Zhiguo Hu, University of Windsor, Windsor, Ontario, N9B 3P4, Canada (zhiguohu@uwindsor.ca).
Maximally Decomposable von Neumann Algebras on Locally Compact Groups and Duality, pp. 857-881.
ABSTRACT. We present a decomposition of the abelian von Neumann algebra L(G) of a locally compact group as an inductive union of certain maximally decomposable translation invariant sub von Neumann algebras. As an application of the decomposition, for all locally compact groups G, we precisely express the weight and the cardinality of the spectrum of L(G) in terms of the character and the compact covering number of G. Using decomposability numbers of von Neumann algebras, we provide a unified formulation of the decomposition of L(G) and its dual version on VN(G) in the setting of Kac algebras. A concept of Kakutani-Kodaira numbers for locally compact groups and general Kac algebras is introduced. It is used to reveal some quantitative intrinsic relations between L(G), VN(G) and the underlying group G. A Kac algebraic Kakutani-Kodaira theorem on the dual pair L(G) and VN(G) is obtained.

Pawel Kolwicz, Institute of Mathematics, University of Technology, ul. Piotrowo 3a, 60-965 Poznañ, Poland ( kolwicz@math.put.poznan.pl ) .
Rotundity properties in Calderón-Lozanovskií spaces, pp. 883-912.
ABSTRACT. We find criteria for strict and uniform convexity of Calderón-Lozanovskií spaces solving problem XII from [3] and generalizing several theorems, which give only the sufficient (or necessity) conditions (see [2], [4]). In particular we obtain the respective criteria for Orlicz-Lorentz spaces which has been proved directly in [1], [5], [6]. We give also applications to Orlicz spaces generated by the composing of Orlicz functions.

References: [1] J. Cerdá, H. Hudzik, A. Kamiñska and M. Mastylo, Geometric properties of symmetric spaces with applications to Orlicz-Lorentz spaces , Positivity 2 (1998), 311-337. [2] J. Cerdá, H. Hudzik and M. Mastylo, On the geometry of some Calderón-Lozanovskií interpolation spaces, Indag. Math. N.S. 6(1), (1995), 35-49. [3] S. Chen, Y. Cui, H. Hudzik and T. Wang, On some solved and unsolved problems in geometry of certain classes of Banach function spaces, Unsolved Problems on Mathematics for the 21st Century, J. M. Abe and Tanaka (Eds.) IOS Press, 2001. [4] H. Hudzik, A. Kamiñska and M. Mastylo, Geometric properties of some Calderón-Lozanovskií spaces and Orlicz-Lorentz spaces, Houston J. Math. 22(3), (1996), 639-663. [5] A. Kamiñska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147, (1990), 29-38. [6] A. Kamiñska, Uniform convexity of generalized Lorentz spaces, Arch. Math. 56 (1991), 181-188.

Guyan Robertson, School of Mathematics and Statistics, University of Newcastle, NE1 7RU, U.K.} (a.g.robertson@newcastle.ac.uk).
Boundary operator algebras for free uniform tree lattices, pp. 913-935.
ABSTRACT. Let X be a finite connected graph, each of whose vertices has degree at least three. The fundamental group of X is a free group which acts on the boundary of the universal covering tree, endowed with a natural topology and Borel measure. The corresponding crossed product C*-algebra depends only on the rank of the free group and is a Cuntz-Krieger algebra whose structure is explicitly determined. The crossed product von Neumann algebra does not possess this rigidity. If the tree is homogeneous then the von Neumann algebra is a purely infinite hyperfinite factor whose exact type depends on whether X is bipartite or not.

Zhong, Hualiang, Robarts Research Institute, London, Ontario, Canada   N6A 5K8 (hzhong@imaging.robarts.ca) and Boivin, André, University of Western Ontario, London, Ontario, Canada   N6A 5B7 (boivin@uwo.ca).
On a class of non-harmonic Fourier series, pp. 937-956.
ABSTRACT. In the theory of non-harmonic Fourier series, one-quarter theorems deal with basis, frames and/or series expansion properties under some extreme conditions. In this paper we show the existence of a series representation analogous to the Fourier series for square integrable functions in terms of system of complex exponentials when the sequence of exponents is close to the "extreme case" sequence {n+sign(n)¼}.

S. Eidelman, Department of Mathematics, International Solomon University, Sheludenko 1b, Kiev, Ukraine, and Y. Eidelman, School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel (eideyu@post.tau.ac.il).
On regularity of the extremal solution of the Dirichlet problem for some semilinear elliptic equations of the second order, pp. 957-960.
ABSTRACT. We consider the problem on the regularity of the extremal positive solution of the Dirichlet problem for superlinear elliptic equations with a positive parameter. We prove the smoothness of the extremal solution for a certain class of nonlinearities. In this class we distinguish a subclass of functions for which the extremal solution is classical in the space of any dimension.

Zhaoyang Yin, Department of Mathematics, Zhongshan University, 510275 Guangzhou, China (mcsyzy@zsu.edu.cn).
Well-posedness and blowup phenomena for a class of nonlinear third-order partial differential equations, pp. 961-972.
ABSTRACT. We establish the local well-posedness for a class of nonlinear third-order partial differential equations. We also present a blow up scenario and prove that the equation has strong solutions that blow up in finite time.