Electronic Edition Vol. 43, No. 2, 2017

Editors:  D. Bao (San Francisco, SFSU), D. Blecher (Houston), B. G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke (LIAFA, Paris7), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), W. B. Johnson (College Station),  M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice).
Managing Editors: B. G. Bodmann and K. Kaiser (Houston)

 Houston Journal of Mathematics


Azarang, Alborz, Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz-Iran (a_azarang@scu.ac.ir).
On Fields with only finitely many maximal subrings, pp. 299-324.
ABSTRACT. Fields with only finitely many maximal subrings are completely determined. We show that such fields are certain absolutely algebraic fields and give some characterization of them. In particular, we observed that a field E has only finitely many maximal subrings if and only if every saturated descending chains of subring, beginning from E, is stationary; and have the same length. Moreover, we prove that the last term of such chains is a unique subring of E. We also determine when certain affine rings have only finitely many maximal subrings. In particular, we prove that an affine integral domain R over a field F has only finitely many maximal subrings if and only if F has only finitely many maximal subrings and each generator of R over F is algebraic over F, which is similar to the celebrated Zariski's Lemma.

D.D. Anderson, Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA (dan-anderson@uiowa.edu) and Muhammad Zafrullah, Department of Mathematics, Idaho State University, Pocatello, ID 83209, USA (mzafrullah@usa.net).
Cohen type theorems for a commutative ring, pp. 325-331
ABSTRACT. Let R be a commutative ring with 1 not equal to 0. We show that if every prime ideal containing a proper ideal is principal (resp., invertible, finitely generated locally principal), then I is a finite product of principal (resp., invertible, finitely generated locally principal) prime ideals. Let R be an integral domain and * a finite character star operation on R. We show that if every prime *-ideal containing a proper *-ideal I is *-invertible, then I is a finite *-product of *-invertible prime *-ideals and hence is *-invertible.

Mamoru, Nunokawa, University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan (mamoru_nuno@doctor.nifty.jp), and Janusz, Sokol, Rzeszow University of Technology, Poland (jsokol@prz.edu.pl).
New conditions for starlikeness and strongly starlikeness of order alpha, pp. 333-344
ABSTRACT. Using a modified Nunokawa's lemma new conditions for starlikeness and strongly starlikeness of order alpha are proved in this work.

Xujie, Shi, Nanjing University, Nanjing (yywn2006@sina.com), Liangwen Liao, Nanjing University, Nanjing (maliao@nju.edu.cn), and Jie Zhang, China University of Mining and Technology, Xuzhou (zhangjie1981@cumt.edu.cn).
On a polynomial P such that P(Δηnf) and P(f) sharing a small function, pp. 345-361.
ABSTRACT. In this paper, we investigate the difference counterpart of Brucke's conjecture. We obtain that for entire function f of finite order who has a small Borel exceptional entire function, if p is a polynomial such that P(Δηnf) and P(f) sharing a small function CM, then we obtain the form of f, and we give the necessary and sufficient condition when f equal to Δηnf.

Hamid-Reza Fanaï and Atefeh Hasan-Zadeh,  Department of Mathematical Sciences, Sharif University of Technology, P.O.Box 11155-9415, Tehran, Iran (fanai@sharif.ac.ir), (a-hasanzadeh@mehr.sharif.ac.ir).
A symplectic rigidity problem for 2-step nilmanifolds, pp. 363-374.
ABSTRACT. We study a result of Gordon, Mao and Schueth about compact 2-step nilmanifolds with symplectically conjugate flows, and consider this result as a special case of a problem in Poisson and symplectic structures. In this setting, via Poisson cohomology and other respective notions, we present a proof of their result which extends not only symplectic concepts to Poisson geometry,
but also 2-step nilmanifolds to manifolds with extensible momentum maps.

Adeyemo, H. Praise, University of Ibadan, Ibadan, Oyo State, Nigeria (ph.adeyemo@ui.edu.ng), and Sottile, Frank, Texas A&M University, College Station, Texas, USA, (sottile@math.tamu.edu).
Equivariant cohomology theories and the pattern map, pp. 375-393.
ABSTRACT. Billey and Braden defined a geometric pattern map on flag manifolds which extends the generalized pattern map of Billey and Postnikov on Weyl groups. The interaction of this torus equivariant map with the Bruhat order and its action on line bundles lead to formulas for its pullback on the equivariant cohomology ring and on equivariant K-theory. These formulas are in terms of the Borel presentation, the basis of Schubert classes, and localization at torus fixed points.

Ghawadrah, Ghadeer, Université Paris VI, Boîte 186, 4 Place Jussieu, 75252 paris cedex 05, France (ghawadrah@math.jussieu.fr), (g.ghawadrah@najah.edu).
The descriptive complexity of the family of Banach spaces with the bounded approximation property, pp. 395-401.
ABSTRACT. We show that the set of all separable Banach spaces that have the bounded approximation property (BAP) is a Borel subset of the set of all closed subspaces of C(Δ), where Δ is the Cantor set, equipped with the standard Effros-Borel structure.

Søren Eilers, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark (eilers@math.ku.dk)Gunnar Restorff, Department of Science and Technology, University of the Faroe Islands, Nóatún 3, FO-100 Tórshavn, Faroe Islands (gunnarr@setur.fo), and Efren Ruiz,  Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hawaii, 96720-4091, USA 9 (ruize@hawaii.edu)
Ideal related K-theory with coefficients, 403-458.
ABSTRACT. In this paper, we define an invariant, which we believe should be the substitute for total K-theory in the case when there is one distinguished ideal. Moreover, some diagrams relating the new groups to the ordinary K-groups with coefficients are constructed. These diagrams will in most cases help to determine the new groups, and will in a companion paper be used to prove a universal multi-coefficient theorem for the one distinguished ideal case for a large class of algebras.

Hui Li, Research Center for Operator Algebras, Department of Mathematics, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China (hli@math.ecnu.edu.cn).
Twisted Topological Graph Algebras, pp. 459-494.
ABSTRACT. We define the notion of a twisted topological graph algebra associated to a topological graph and a 1-cocycle on its edge set. We prove a stronger version of a Vasselli's result. We expand Katsura's results to study twisted topological graph algebras. We prove a version of the Cuntz-Krieger uniqueness theorem, describe the gauge-invariant ideal structure. We find that a twisted topological graph algebra is simple if and only if the corresponding untwisted one is simple.

Pons, Matthew A., North Central College, Naperville, IL 60540, USA (mapons@noctrl.edu ).
The Adjoint of a linear fractional composition operator on the Dirichlet space, pp. 495-508.
ABSTRACT. Here we revisit the investigation of the adjoint of a composition operator with linear fractional symbol acting on the Dirichlet space of the unit disk. Earlier work shows that the adjoint can be expressed as another composition operator plus a rank two operator. We work on the Dirichlet space equipped with an equivalent norm, and hence a different inner product, to determine how the adjoint representation differs in this setting.

Bartosz K. Kwasniewski, Department of Mathematics and Computer Science, The University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark (bartoszk@math.uwb.edu.pl).
Exel's crossed product and crossed products by completely positive maps, pp. 509-567
ABSTRACT. We introduce crossed products of a C*-algebra A by a completely positive map relative to an ideal in A. When the map is multiplicative, they generalize various crossed products by endomorphisms. When A is commutative they include C*-algebras associated to Markov operators by Ionescu, Muhly, Vega, and to topological relations by Brenken, but in general they are not modeled by topological quivers popularized by Muhly and Tomforde.
We show that Exel's crossed product, generalized to the case where A is not necessarily unital, is a relative crossed product of A by the transfer operator L. We give natural conditions under which this crossed product depends only on L. Moreover, the C*-algebra associated to an Exel system by Exel and Royer always coincides with our unrelative crossed product by L.
As another non-trivial application of our construction we extend a result of Brownlowe, Raeburn and Vittadello, by showing that the C*-algebra C*(E) of an arbitrary infinite graph E can be realized as a crossed product of the diagonal algebra D by a `Perron-Frobenious' operator L. The important difference to the previous result is that in general there is no endomorphism α of D making (D, α, L) an Exel system. 

Li, Yongjin, Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China (stslyj@mail.sysu.edu.cn), and Bu, Qingying, Department of Mathematics, University of Mississippi, Oxford, MS 38677, USA (qbu@olemiss.edu).
New examples of  non-reflexive Grothendieck spaces, pp. 569-575.
ABSTRACT. In this short paper, we characterize the Fremlin projective tensor product X⊗Y that is a Grothendieck space, where X is a Banach sequence lattice and Y is a Banach lattice. Then by using this characterization we provide new examples of non-reflexive Grothendieck spaces.

Benjamin Espinoza, Department of Mathematics, University of Pittsburgh at Greensburg, 236 Frank A. Cassell Hall, 150 Finoli Drive, Greensburg, PA 15601, USA (bee1@pitt.edu), Paul Gartside, Department of Mathematics, University of Pittsburgh, 406 Thackeray Hall, Pittsburgh, PA 15260, USA (gartside@math.pitt.edu), Merve Kovan-Bakan, Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA (mervekovan@gmail.com), and Ana Mamatelashvili, Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA (anm137@math.pitt.edu).
Strong arcwise connectedness, pp. 577-610.
ABSTRACT. A space is n-strong arc connected (n-sac) if for any n points in the space there is an arc in the space visiting them in order. A space is ω-strong arc connected (ω-sac) if it is n-sac for all n. We study these properties in finite graphs, regular continua, and rational continua. There are no 4-sac graphs, but there are 3-sac graphs and graphs which are 2-sac but not 3-sac. For every n there is an n-sac regular continuum, but no regular continuum is ω-sac. There is an ω-sac rational continuum. For graphs we give a simple characterization of those graphs which are 3-sac. It is shown, using ideas from descriptive set theory, that there is no simple characterization of n-sac, or ω-sac, rational continua.

Li-Hong Xie, School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, P.R. China (xielihong2011@aliyun.com) and Peng-Fei Yan, School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, P.R. China (ypengfei@sina.com).
Expansions of set-valued mappings on stratifiable spaces, pp. 611-624.
ABSTRACT. In this paper, we give some characterizations of stratifiable and semistratifiable spaces by expansions of set-valued mappings.

Alejandro Illanes, Instituto de Matemáticas, Universidad Nacional Autónoma de México, 04510, México, D.F. (illanes@matem.unam.mx) and Jorge M. Martínez-Montejano, Facultad de Ciencías, Universidad Nacional Autónoma de México, 04510, México, D.F. (jorge@matematicas.unam.mx).
Z-sets in symmetric products, pp. 625-647.
ABSTRACT. A closed subset A of a continuum X (with metric D) is a Z-set in X, provided that for each ε>0 there exists a map fε:X→X\A such that d(x,fε(x))<ε for each x∈X. Given a continuum X and a positive integer n, we consider the hyperspace Fn(X) of the nonempty closed subsets of X with at most n points. We prove that F1([0,1]) ) is a Z-set in Fn([0,1])) if and only if n is even. Also, we consider the problem of determining the continua X for which F1(X) is a Z-set in F2(X). We solve this problem for finite graphs and compactifications of the ray with locally connected remainder different of a simple closed curve. Lastly, we study several significant examples.

Liang-Xue Peng (Corresponding author), Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn) and Hui Li,  Beijing University of Technology, Beijing 100124, China (lihui86@emails.bjut.edu.cn).
On monotone n-star covering properties, pp. 649-667.
In this note, we discuss some properties of monotonically n-star Ρ spaces. We give an example to show that there is a space X which has a monotonically star closed-and-discrete dense subspace but X is not monotonically star closed-and-discrete. This gives a partial answer to a question posed by S.G. Popvassilev and J.E. Porter ([Question 35(b), in: Topology Appl. 169 (2014) 87-98]). We point out that there is a monotonically 2-star finite space which is monotonically star closed-and-discrete but it is not star finite. We show that if X is a monotonically star closed-and-discrete GO-space and g∈X, then (←,g) is monotonically 2-star closed-and-discrete. Every open paracompact subspace of a regular monotonically n-star closed-and-discrete space is monotonically n-star closed-and-discrete, where n∈N. Every monotonically D-space is monotonically star closed-and-discrete.

Er-Guang Yang, School of Mathematics & Physics, Anhui University of Technology, Maanshan 243002, P.R. China (egyang@126.com).  
On MCP-spaces and mcb-spaces, 669-677.
ABSTRACT. In [Properties defined with semi-continuous functions and some related spaces, Houston J. Math., 41 (2015), 1097-1106], the relationships between properties defined with real-valued functions and some covering properties were studied. In this paper, we shall continue with this study. Some other properties, such as (UL)mwl and (UC)m are introduced and we show that spaces with these properties coincide with MCP-spaces and mcb-spaces respectively. As an application, an insertion theorem of MCP-spaces is obtained which corrects a mistake in [Monotone countable paracompactness, Top. Appl., 101 (2000), 281-298]. 

Er-Guang Yang, School of Mathematics & Physics, Anhui University of Technology, Maanshan 243002, P.R. China (egyang@126.com).
Characterizations of some spaces with maps to ordered topological vector spaces, pp. 679-689.
ABSTRACT. In this paper, we generalize real-valued functions in some earlier results to maps with values into ordered topological vector spaces. Some characterizations of countably paracompact spaces and cb-spaces in terms of maps to ordered topological vector spaces are obtained which extend some known results in the literature.