## HOUSTON JOURNAL OF MATHEMATICS

### Electronic Edition Vol. 34, No. 1, 2008

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), J. Hausen (Houston), J. A. Johnson (Houston), W. B. Johnson (College Station),  V. I. Paulsen (Houston), M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Contents

Thomas G. Lucas, Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA (tglucas@email.uncc.edu).
The Gaussian property for rings and polynomials, pp. 1-18.
ABSTRACT.
The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. If c(fg)=c(f)c(g) for each polynomial g in R[x], then f is said to be Gaussian. The ring R is Gaussian if each polynomial in R[x] is Gaussian. It is known that f is Gaussian if c(f) is locally principal. The converse is established for polynomials over reduced rings. Also, if the square of the nilradical is zero, then R is Gaussian if and only if the square of each finitely generated ideal is locally principal.

Francisco J. Echarte, Juan Nunez, and Francisco Ramirez, Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad de Sevilla,  Apartado 1160. 41080-Sevilla (Espana) (jnvaldes@us.es).
Description of some families of filiform Lie algebras, pp. 19-32.
ABSTRACT. In this paper we describe some families of filiform Lie algebras by giving a method which allows to obtain them in any arbitrary dimension n starting from the triple (p, q, m), where m = n and p and q are, respectively, invariants z1 and z2 of those algebras. After obtaining the general law of complex filiform Lie algebras corresponding to triples (p, q, m), some concrete examples of this method are shown.

Jackson, Marcel, La Trobe University, Victoria 3086, Australia (m.g.jackson@latrobe.edu.au).
Residual bounds for compact totally disconnected algebras, pp. 33-67.
ABSTRACT. A Boolean topological algebra is a general algebra with a compatible topology that is compact and totally disconnected. It is well known that every Boolean topological semigroup, group or ring is topologically residually finite; that is, every pair of distinct elements can be separated by a continuous homomorphism into a (discretely topologised) finite algebra. We examine the possible residual bounds for Boolean topological algebras in relation to their non-topological residual bound, with particular emphasis given to groups and completely simple semigroups. Amongst the results is the undecidability of the problem of determining if all Boolean topological models of a finite set of identities are profinite.

Erne, Marcel, Faculty for Mathematics and Physics, Leibniz University, Hannover, Germany (erne@math.uni-hannover.de).
Distributors and Wallman locales, pp. 69-98.
ABSTRACT. A distributor in an m-semilattice (a join-semilattice with an isotone multiplication) is a nonempty upper set containing both the join of a and c and the join of b and c iff it contains the join of ab and c. Distributors provide a far-reaching extension of the filter theory for distributive lattices, quantales and similar objects to structures where no distributive laws are assumed a priori. The closure system of all distributors is a universal locale over the given m-semilattice, and the principal distributors form its universal distributive lattice quotient. Moreover, distributors are a helpful tool for the spectral theory of m-semilattices and various related kinds of (ordered) algebras. We present diverse alternative characterizations of Scott-open distributors in complete m-semilattices, for example as kernels of join-preserving homomorphisms onto compact locales, and we establish a one-to-one correspondence between Scott-open distributors and those nuclei whose range is a Wallman locale, the pointfree analogue of a compact T1-topology.

M.  Crampin, Department of Mathematical Physics and Astronomy,  Ghent University, Krijgslaan 281, B-9000 Gent, Belgium and Department of Mathematics, King's College, Strand, London WC2R 2LS, UK (Crampin@btinternet.com).
Kähler and para-Kähler structures associated with Finsler spaces of non-zero constant flag curvature, pp. 99-114.
ABSTRACT.  It was shown by R.,L. Bryant (Houston J. Math. 28 (2002) 221-262) that there is a canonical Kähler structure on the space of geodesics of a Finsler manifold whose flag curvature is constant and positive. A different construction is proposed in the present paper, leading instead to a Kähler structure on the slit tangent bundle of the Finsler space; it is based on the identification of an appropriate complex structure. The construction is easily adapted to apply to a Finsler space with constant negative flag curvature, when it gives a para-Kähler rather than a Kähler structure; the properties of this para-Kähler structure are explored.

Gloria Andablo-Reyes, Facultad de Ciencias de Fisico-Matematicas, UMSNH, F. J. Mujica s/n, Felicitas del Rio, 58060. Morelia, Michoacan, Mexico (gloria@fismat.umich.mx) and Victor Neumann-Lara, Instituto de Matematicas, UNAM, Circuito Exterior, Ciudad Universitaria,04510. Mexico, D. F., Mexico (neumann@math.unam.mx).
Ordered embeddings of symmetric products, pp. 115-122.
ABSTRACT. Let X and Y be metric continua. Let Fn(X) (resp., Fn(Y)) be the hyperspace of nonempty closed subsets of X (resp., Y) which contain at most n elements. We say that the hyperspace Fn(X) can be orderly embedded in Fm(Y) provided that there exists an embedding h from Fn(X) to Fm(Y) such that if A,B are elements of Fn(X) and A is contained in B, then h(A) is contained in h(B). In this paper we prove:
(a) If n is minor or equal than m, m is minor than 2n and Fn(X) can be orderly embedded in Fm(Y), then X can be embedded in Y.
(b) There exist continua X and Y such that, for each n greater than 1, Fn(X) can be orderly embedded in F2n(Y) and X can not be embedded in Y.

Gutiérrez García, J., Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain (javier.gutierrezgarcia@ehu.es),  and  Kubiak, T., Wydzial Matematyki i Informatyki, Uniwersytet im. Adama Mickiewicza, 61-614 Poznań, Poland (tkubiak@amu.edu.pl) and de Prada Vicente, M.A., Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain (mariangeles.deprada@ehu.es).
Generating and inserting continuous functions with values in bounded complete domains and hedgehog-like structures, pp. 123-144.
ABSTRACT.  The paper deals with functions on a topological space having values in a bounded complete domain. Our purpose is two-fold. We first develop a theory of generating such functions from certain scales or prescales of subsets. We then study lower and upper limits of functions having bounded complete domain as a range space. We characterize those limit functions in terms of the (pre)scales generating the original ones. Part of these developments is then used to prove an insertion-type theorem for continuous functions from a topological space to an appropriately based bounded complete domain with its Lawson topology. Examples of those domains include, among others, hedgehogs with countably many spines, their products as well as various ‘mutants’ of the hedgehog.

Banic, Iztok, Department of Mathematics, Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, Maribor 2000, Slovenia (iztok.banic@uni-mb.si).
Continua with kernels, pp. 145-163.
ABSTRACT. In this article we introduce the concept of kernels of continua, obtained by combining inverse limits of inverse sequences of unit intervals and one-valued bonding maps with inverse limits of inverse sequences of unit intervals and upper semicontinuous set-valued bonding functions. We also show some of their properties, with special emphasis on arc-like continua.

Liang-Xue  Peng,  College of Applied  Science,  Beijing University of Technology,  Beijing 100022,  China (pengliangxue@bjut.edu.cn).
On products of certain D-spaces, pp. 165-179.
ABSTRACT. D--spaces were introduced by van Douwen in 1978 and studied by van Douwen and many other topologists. It is not yet clear which topological spaces are D-spaces, and the product theory for D-spaces is not yet complete. In this paper we use certain topological games of Galvin to obtain any countable product of paracompact DC--like spaces is a D-space, and consequently that any countable product of paracompact C-scattered spaces is a D-space. We also show that a special generalized metric space is a D-space, this result extends results of R. Z. Buzyakova. For a fixed integer n, any box product of scattered spaces each with a scattered rank n must be a D-space. The last conclusion extands results of William G. Fleissner and Adrienne M. Stanley.

Joe Corneli, PlanetMath.org, 421 Cedar Ave. S #17 Minneapolis, MN 55454 (jcorneli@planetmath.org ), and Ivan Corwin, 151 2nd Avenue, Apt 4D, NY NY 10003 (ivan.corwin@gmail.com), and Stephanie Hurder, Harvard University,National Bureau of Economic Research, 1050 Massachusetts Avenue Cambridge, MA 02138 ( stephanie.hurder@post.harvard.edu), and Vojislav Sesum, Williams College, Omladinskih brigada 232, Belgrade 11070, Serbia ( sevoja@gmail.com), and Ya Xu, 74 Barnes Ct. #218, Stanford, CA 94305(xulongya@gmail.com), and Elizabeth Adams, 715 Oakland Ave Apt 304, Oakland, CA 94611 ( 06eaa@williams.edu) , and Diana Davis, Williams College, 1637 Baxter Hall, Williamstown, MA 01267 ( 07djd@williams.edu), and Michelle Lee, Williams College, 303 Berkshire Drive, Princeton, NJ 08540 (mishlie@gmail.com), and Regina Pettit, 31001 Floralview Drive South, Apt 203, Farmington Hills, MI 48331(Regina.Pettit@gmail.com) and Neil Hoffman, Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712 (nhoffman@math.utexas.edu)
Double bubbles in Gauss space and spheres, pp. 181-204.
ABSTRACT. We prove that a standard Y is an area-minimizing partition of Gauss space into three given volumes, provided that the standard double bubble is an area-minimizing partition of high-dimensional spheres. We prove that the standard double bubble is the area-minimizing partition of spheres of any dimension where the volumes differ by at most 4%.

Dziubanski, Jacek, Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland (jdziuban@math.uni.wroc.pl).
Hardy spaces associated with semigroups generated by Bessel operators with potentials, pp. 205-234.
ABSTRACT. Let {Tt}t>0 be the semigroup of linear operators generated by the operator -Lu(x)=(1 / 2)u''(x)+(α / 2x )u'(x)-V(x)u(x), x>0, where V is a nonnegative potential satisfying a certain regularity condition, α>1. We say that a function f belongs to the Hardy space H1L associated with the semigroup Tt if the maximal function supt>0|Ttf(x)| belongs to L1( R+, xα dx). We prove atomic decompositions for the elements of the Hardy space H1L.

Janez Bernik,  Roman Drnovsek,  Damjana Kokol Bukovsek, Tomaz  Kosir,  and  Matjaz Omladic, Department of Mathematics, University of Ljubljana, 1000 Ljubljana, Slovenia, (janez.bernik@fmf.uni-lj.si).
Reducibility and triangularizability of semitransitive spaces of operators, pp. 235-247.
ABSTRACT. A linear space L of operators on a vector space X is called semitransitive if, given two nonzero vectors x, y in X, there exists an element A in L such that either y=Ax or x=Ay. In this paper we consider semitransitive spaces of operators on a finite dimensional vector space X over an algebraically closed field. In particular, we are interested in the existence of nontrivial invariant subspaces of X for a semitransitive space L. We are able to relate the existence of an invariant subspace for L to the properties of some rank varieties that we associate to L. Using this relation we show that, if the dimension of L is the same as the dimension of X, which is minimal possible, then L is triangularizable. By contrast we show that, from n=3 onwards, there exists a minimal semitransitive space L of dimension n+1 of operators on an n-dimensional vector space X which is also irreducible. We also give a new characterization of semitransitive spaces of operators on finite dimensional vector spaces.

Fangyan Lu,  Department of  Mathematics, Suzhou University, Suzhou 215006,  P. R. China (fylu@suda.edu.cn)(fylu@pub.sz.jsinfo.net).
Jordan triple isomorphisms of nest algebras and applications, pp. 249-267.
ABSTRACT. Let  Ai be a subalgebra of the nest algebra  T( Ni)  which  contains all finite rank operators in T( Ni) , i=1, 2. Let φ  be a linear bijection from A1 onto A2 which satisfies  φ (ABA)= φ (A) φ (B) φ (A) for all A,  B  in   A1 .  Then φ is proved to be  an isomorphism,  or a negative of an isomorphism, or an anti-isomorphism, or a  negative of an anti-isomorphism. As applications, Jordan  isomorphisms and  isometries are characterized.

Skalski, A.  and Zacharias, J.  University of Nottingham, University Park, Nottingham NG7 2RD, (adam.skalski@maths.nottingham.ac.uk), (joachim.zacharias@nottingham.ac.uk)..
Entropy of shifts on higher-rank graph C*-algebras, pp. 269-282.
ABSTRACT. Let OΛ be a higher-rank graph C*-algebra. For every p in  Z+r there is a canonical completely positive map Φp on OΛ and a subshift Tp on the path space X=Λ. We show that ht(Φp)=h(Tp), where ht is Voiculescu's approximation entropy and h the classical topological entropy. For a higher rank Cuntz-Krieger algebra OM we obtain ht(Φp)= log rad(M1 p1... Mr pr), rad being the spectral radius. This generalizes Boca and Goldstein's result for Cuntz-Krieger algebras.

Kenneth R. Davidson  and Dilian Yang, University of Waterloo, Waterloo, ON N2L 3G1, CANADA,  (krdavids@uwaterloo.ca), (dyang@uwaterloo.ca).
A note on absolute continuity in free semigroup algebras , pp. 283-288.
ABSTRACT.
A free semigroup algebra is the weak operator topology closed (nonself-adjoint, unital) algebra generated by n isometries with pairwise orthogonal ranges. The prototype is the algebra generated by the left regular representation of the free semigroup on n letters. A free semigroup algebra which is isomorphic to the left regular algebra is called type L. If the infinite ampliation of the isometries generates a type L algebra, it is called weak-* type L. A free semigroup algebra is absolutely continuous if the vector functionals on it are equivalent to (some) vector functionals on the left regular representation.
The purpose of this note is to show that absolutely continuous free semigroup algebras are weak-* type L.

Sakai, Katsuro, Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan (sakaiktr@sakura.cc.tsukuba.ac.jp). Let X be an infinite-dimensional Banach space with density τ and let CC(X) and BC(X) be the spaces of all non-empty compact convex sets in X and of all non-empty bounded closed convex sets admitting the Hausdorff metric, respectively. In this note, it is proved that (i) the space CC(X) is homeomorphic to the Hilbert space with density τ; (ii) the space BC(X) is homeomorphic to the Hilbert space with density τ if it has the density τ; (iii) the space BC(X) is homeomorphic to the Hilbert space with density 2τ if X is separable (i.e., τ is countable) or reflective.

Guangcun Lu, Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China (gclu@bnu.edu.cn).
Symplectic fixed points and Lagrangian intersections on weighted projective spaces, pp. 301-316.
ABSTRACT.
In this note we show that Arnold conjecture for the Hamiltonian maps still holds on weighted complex projective spaces. For the real part of the weighted complex projective space, a Lagrange orbifold we also prove that Arnold conjecture for the Lagrange intersections for it is also true if each weight of this weighted complex projective space is odd.

Editorial Addendum concerning the paper "The BMO-1 space and its application to Schechter's Inequality", by Sadek Gala, Houston Journal of Mathematics, Vol. 33(4), pp. 1059-1066, p.317.
This addendum replicates in print the Editorial Statement made on Gala's paper in HJM Vol. 33(4).