HOUSTON JOURNAL OF
MATHEMATICS

 Electronic Edition Vol. 48, No. 1, 2022

Editors:  D. Bao (San Francisco, SFSU), S. Berhanu (Temple), D. Blecher (Houston), B. G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B.  Dacorogna (Lausanne), M. Gehrke (LIAFA, Paris7), R. M. Hardt (Rice), S. Harvey (Rice), A. Haynes (Houston), Y. Hattori (Matsue, Shimane), W. B. Johnson (College Station), M. Marsh (Sacramento), M. Rojas (College Station), Min Ru (Houston), S.W. Semmes (Rice), D. Werner (FU Berlin).
Managing Editors: B. G. Bodmann and A. Haynes (Houston)

 Houston Journal of Mathematics


Contents

Tristan Bice, Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague, Czech Republic (bice@math.cas.cz), and Wiesław Kubiś, Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Prague, Czech Republic (kubis@math.cas.cz).
Wallman duality for semilattice subbases, pp. 1–31.

ABSTRACT. We extend Wallman’s classic duality from lattice bases to semilattice subbases and from compact to locally closed compact spaces. Moreover, we make this duality functorial via appropriate relational morphisms.  

John Harding, New Mexico State University, Department of Mathematical Sciences, 1290 Frenger Mall, Las Cruces, NM 88003 (hardingj@nmsu.edu), and Bert Lindenhovius, Tulane University, Department of Computer Science, 303 Stanley Thomas Hall, New Orleans, LA 70118 (alindenh@tulane.edu).
Orthogeometries and AW*-algebras, pp. 33–58.

ABSTRACT. Based on previous results we give a connection between the category of AW*-algebras and their normal Jordan homomorphisms and a category COG of orthogemetries, which are structures that are somewhat similar to projective geometries, consisting of a set of points and a set of lines, where each line contains exactly 3 points. They are constructed from the commutative AW*-subalgebras of an AW*-algebra that have at most an 8-element Boolean algebra of projections. Morphisms between orthogemetries are partial functions between their sets of points as in projective geometry. The functor we create A∗ : AWJ∗→ COG is injective on non-trivial objects, and full and faithful with respect to morphisms that do not involve type I2 factors.  

ChengChen Zhong, School of Mathematical Sciences, Peking University, Beijing 100871, P.R.China (zhongchengchen@math.pku.edu.cn), LiShuang Pan, Science College, Shijiazhuang University, Hebei 050035, P.R.China (plshuang123@163.com), and An Wang, School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R.China (wangan@cnu.edu.cn).
The representation of holomorphic functions on the quasi-circular domain and the Bergman kernel function on the symmetrized ball, pp. 59–75.

ABSTRACT. In this paper, we construct the relationship between the circular domain and quasi-circular domain by using standard mapping and standard inverse mapping in order to give the quasi-homogeneous representation of holomorphic functions on quasi-circular domain. By using the above result, we obtain the form of orthonormal basis on quasi-circular domain. Especially, we give the Bergman kernel function on symmetrized ball.  

Wenjie Wang, School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046, Henan, P. R. China (wangwj072@163.com).
Almost Kenmotsu (k,μ)′-metrics as η-Ricci solitons, pp. 77–89.

ABSTRACT. In this paper, we prove that if the metric of a non-Kenmotsu almost Kenmotsu (k,μ)′-manifold of dimension 2n + 1, n ≥ 1, represents an η-Ricci soliton, then either the manifold is locally isometric to the product ℍn+1(−4) × ℝn or the potential vector field of the soliton is of strict contact type. However, the later case is impossible if the η-Ricci soliton is changed to a gradient η-Ricci soliton. An concrete example to illustrate the above results is given.  

Hongmei Zhu, College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, P.R. China (zhm403@163.com).
On a class of projectively Ricci-flat Douglas metrics, pp. 91–109.

ABSTRACT. In Finsler geometry, every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, inspired by well-known Berwald square metric, we characterize and classify all of the projectively Ricci-flat Douglas (α,β)-metrics on a manifold of dimension n ≥ 3.  

Willian H. G. Corrêa, Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400 - Centro CEP: 13566-590 - São Carlos - SP, Brazil (willhans@icmc.usp.br).
Complex interpolation of Orlicz sequence spaces and its higher order Rochberg spaces, pp. 111–124.

ABSTRACT. We show that if (ℓϕ0,ℓϕ1) is a couple of suitable Orlicz sequence spaces then the corresponding Rochberg derived spaces of all orders associated to the complex interpolation method are Fenchel-Orlicz spaces. In particular, the induced twisted sums have the (C[0,1], ℂ)-extension property.  

Kui Ji, Department of Mathematics Science, Hebei Normal University, Shijiazhuang, Hebei 050016, China (jikuikui@gmail.com, jikui@hebtu.edu.cn), and Shanshan Ji, Department of Mathematics Science, Hebei Normal University, Shijiazhuang, Hebei 050016, China (jishanshan15@outlook.com).
A note on unitary equivalence of operators acting on reproducing kernel Hilbert spaces, pp. 125–148.

ABSTRACT. A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this theorem, we obtain a different but equivalent criterion for the unitary equivalence of operators acting on reproducing kernel Hilbert spaces. As an application, we describe the structure of intertwining operator and prove that the decomposition of Cowen-Douglas operators is unique up to unitary equivalence.  

Hang Zhang, School of Mathematics, Southwest Jiaotong University, Chengdu 610756, China (zhanghangzh@sina.com; hzhangzh@gmail.com), and Shuguo Zhang, College of Mathematics, Sichuan University, Chengdu 610064, China (zhangsg@scu.edu.cn).
Capturing topological spaces by countable elementary submodels, pp. 149–170.

ABSTRACT. Given a Tychonoff space (X,τ) and an elementary submodel M of a sufficiently large fragment of set theory such that (X,τ) ∈ M, one can define a quotient-like topological space X∕M. The space X∕M is always separable metrizable if M is countable. We investigate conditions under which X∕M is homeomorphic to some special subspaces of the real line. To this end, we establish several preservation theorems of the form “if (X,τ) has property P, then X∕M has property P for any countable M such that (X,τ) ∈ M”. We also pose questions.  

Vladimir V. Tkachuk, Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa, 09340, Mexico City, Mexico (vova@xanum.uam.mx).
Lindelöf scattered subspaces of nice σ-products are σ-compact, pp. 171–181.

ABSTRACT. We show that there exists an Eberlein compact space K such that some Lindelöf subspace of K fails to be a Lindelöf Σ-space. We also prove that any scattered Lindelöf subspace of a σ-product of first countable spaces is σ-compact. It is established that if X is the Gδ-modification of a scattered compact space, then ext(Cp(X)) = ø. Our results solve several published open questions.  

Alexander J. Izzo, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403 (aizzo@bgsu.edu).
The set of bounded continuous nowhere locally uniformly continuous functions is not Borel, pp. 183–187.

ABSTRACT. It is known that for X a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on X contains a dense Gδ set in the space Cb(X) of all bounded continuous real-valued functions on X in the supremum norm. Furthermore, when X is separable, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on X is itself a Gδ set. We show that in contrast, when X is nonseparable, this set of functions is not even a Borel set.  

Vlasta Matijević, Department of Mathematics, Faculty of Science, University of Split, 21 000 Split, Croatia (vlasta@pmfst.hr), and Leonard R. Rubin, Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA (lrubin@ou.edu).
Čech systems and spaces that are not strongly 0-dimensional, pp. 189–204.

ABSTRACT. We shall prove that if a nonempty Hausdorff paracompactum X is not strongly 0-dimensional and U is a Čech system for X, then U is not an approximate (inverse) system in the sense of Mardešić and Watanabe. This result generalizes those of its predecessors where much more restrictive properties on the classes of spaces in consideration had been required. On the other side there are two distinct classes of Hausdorff paracompacta that are strongly 0-dimensional, those that are discrete and those that are not. Čech systems for spaces in the former class cannot be approximate systems because their indexing sets are not unbounded. We proved in earlier research that those in the latter class always have a Čech system that is an approximate system in the Mardešić-Watanabe sense. Consequently it will be shown that if X is a nonempty nondiscrete Hausdorff paracompactum, then it has a Čech system that is an approximate system (in the preceding sense) if and only if it is strongly 0-dimensional.  

Germán Montero-Rodríguez, Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla. Avenida San Claudio y 18 Sur, Colonia San Manuel. Edificio FM1-101B, Ciudad Universitaria, C.P. 72570, Puebla, Mexico (gmontero.fcfm.buap@gmail.com), David Herrera-Carrasco, Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla. Avenida San Claudio y 18 Sur, Colonia San Manuel. Edificio FM1-101B, Ciudad Universitaria, C.P. 72570, Puebla, Mexico (dherrera@fcfm.buap.mx), María de J. López, Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla. Avenida San Claudio y 18 Sur, Colonia San Manuel. Edificio FM1-101B, Ciudad Universitaria, C.P. 72570, Puebla, Mexico (mjlopez@fcfm.buap.mx), and Fernando Macías-Romero, Facultad de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla. Avenida San Claudio y 18 Sur, Colonia San Manuel. Edificio FM1-101B, Ciudad Universitaria, C.P. 72570, Puebla, Mexico (fmacias@fcfm.buap.mx).
Finite graphs have unique n-fold symmetric product suspension, pp. 205–225.

ABSTRACT. Let Z be a metric continuum and n be a positive integer. We consider the hyperspace Fn(Z) of all nonempty closed subsets of Z with at most n points. Given n > 1, the n-fold symmetric product suspension of Z is the quotient space Fn(Z)∕F1(Z), denoted by SFn(Z). In this paper we prove that if n ≥ 4, X is a finite graph, and Y is a continuum such that SFn(X) is homeomorphic to SFn(Y ), then X is homeomorphic to Y . This result answers a question posed by Alejandro Illanes.