African
Journals Online
Quaestiones Mathematicae
VOLUME 24 ISSUE 4 (2001)
Quaestiones Mathematicaes 24 (4) 2001, 425–439
© 2001 NISC Pty Ltd, www.nisc.co.za
Uniform Smoothness Entails Hahn-Banach
Edmond Albius1 and Marianne
Morillon2
1 ERMIT, Département de Mathématiques
et Informatique, Université de La Réunion, 15 avenue René
Cassin - BP 7151 - 97715 Saint-Denis Messag. Cedex 9 France.
2 ERMIT, Département de Mathématiques
et Informatique, Université de La Réunion, 15 avenue René
Cassin - BP 7151 - 97715 Saint-Denis Messag. Cedex 9 France.
e-mail: [email protected]
Abstract: We work in Zermelo-Fraenkel set theory $\bold
ZF$ (without the Axiom of Choice), and we denote by $\bold ZFC$
set theory with the Axiom of Choice. Our paper deals with the
role of the Axiom of Choice in functional analysis, and more
particularly, with the necessity of using the Axiom of Choice
when invoking some consequence of the following Hahn-Banach axiom $\bold
HB$.
Mathematics Subject Classification (2000): Primary
03E25; Secondary 46.
Keywords: axiom of choice, banach space, Hahn-Banach,
uniformly smooth, ZF, functional analysis, Zermelo-Fraenkel,
Mazur property, ZFC, HB, Gâteaux differentiability, Fréchet
differentiability, q-Engel series, John Knopfmacher, partition,
identities, study, Santos polynomials, polynomials
Quaestiones Mathematicaes 24 (4) 2001, 441–452
© 2001 NISC Pty Ltd, www.nisc.co.za
The Completeness Problem in Spaces of Pettis Integrable
Functions
Kazimierz Musial
Instytut Matematyczny, Uniwersytet Wroctawski,
Pl.Grunwaldzki 2/4, 50-384 Wroctaw, Poland.
e-mail: [email protected]
Abstract: Two subspaces of the space of Banach space
valued Pettis integrable functions are considered: the space
$\P(\mu, X, var)$ of Pettis integrable functions with integrals
of finite variation in a Banach space X and $\LLN(\mu, X^{*},
var)$ is always complete and $\P(\mu, X^{*}, var)$ is complete if
Martin's axiom and the perfectness of $\mu$ are assumed.
Moreover, a non-trivial example of a non-conjugate Banach space X
with non-complete $\P(\mu, X, var)$ is presented.
Mathematics Subject Classification (2000): Primary:
46G10; Secondary: 28B05, 28A15.
Keywords: completeness, Pettis integral, lifting,
vector measures, measure preserving transformations,
differentiation theory, differentiation of set functions, vector
valued integration, vector valued measures, Zermelo-Fraenkel,
axiom of choice, functional analysis, Hahn-Banach, Banach, banach
space, Pettis, Pettis integrable, complete
Quaestiones Mathematicaes 24 (4) 2001, 453–480
© 2001 NISC Pty Ltd, www.nisc.co.za
On the Maximal Ideals of Non-Zero-Symmetric Near-Rings and
of Composition Algebras of Polynomial Functions on _-Groups
Erhard Aichinger
Institut für Algebra, Stochastik und wissensbasierte
mathematische Systeme, Johannes Kepler Universität Linz,
Austria.
e-mail: [email protected]
Abstract: Let G be a finite group, and let $\langle
\P(G) ; +, \circ \rangle$ be the near-ring of all unary
polynomial functions on G. We describe the maximal ideals of this
near-ring . Our approach also allows us to determine the maximal
congruences of the composition algebra of polynomial functions on
a finite $\Omega$-group. We apply these results to find the
maximal ideals of a finite near-ring with 1 that is faithful on
its constants. In some occasions, the intersection of the maximal
ideals yields an ideal that has properties that re similar to
nilpotency.
Mathematics Subject Classification (2000): 15Y30
(08A40, 20N99, 16S36).
Keywords: ideals, near-rings, algebra, composition
algebras, polynomial functions, W-groups, maximal ideals,
Brown-McCoy radical, operations, polynomials, primal algebras,
ordinary polynomial rings, skew polynomial rings, semigroup
rings, Banach, banach space, Pettis, Pettis integrable, complete,
congruences, composition, nilpotency
Quaestiones Mathematicaes 24 (4) 2001, 481–490
© 2001 NISC Pty Ltd, www.nisc.co.za
Brown-McCoy Radicals for General Near-Rings
L Márki1, R Mlitz2
and R Wiegandt3
1 Mathematical Institute, Hungarian
Academy of Sciences, H-1364 Budapest, Pf. 127, Hungary.
e-mail: [email protected]
2 Institut fur Numerische und Angewandte
Mathematik, TU wien, Wiedner Hauptstr. 8-10, A-1040 Wien,
Austria.
e-mail: [email protected]
3 Mathematical Institute,
Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127, Hungary.
e-mail: [email protected]
Abstract: Two possible ways of extending the
Brown-McCoy radical to non-0-symmetric near-rings are discussed.
If a Kurosh-Amitsur semisimple $\mathbb{S}_{0}$ in the class
$\mathbb{N}_{0}$ of 0-symmetric near-rings satisfies a certain
condition on local units, then $\mathbb{S}_{0}$ is a
Kurosh-Amitsur semisimple class also in the class $\mathbb{N}$ of
all near-rings. So the subdirect closure of the class of simple
0-symmetric near-rings with identity is a Kurosh-Amitsur
semisimple class in $\mathbb{N}$ with non-hereditary radical
class. The Hoehnke radical $\varrho_{mathbb{E}}$ determined by
the class of simple near-rings with identity is not idempotent,
though for every $\N \in \mathbb{N}$ the radical
$\varrho_{\mathbb{E}}(N)$ is the unique largest G-regular ideal
of N. In both cases the radical classes are hereditary with
respect to invariant ideals.
Mathematics Subject Classification (2000): Primary:
16Y30; Secondary: 16N99
Keywords: Brown-McCoy, Brown-McCoy radical, radical,
near-rings, non-0-symmetric near-ring, polynomial functions,
maximal ideals, ideals, congruences, composition, algebra,
nilpotency, near rings, semisimple, identities, radical classes,
invariant ideals
Quaestiones Mathematicaes 24 (4) 2001, 491–492
© 2001 NISC Pty Ltd, www.nisc.co.za
Norm One Projections on Banach Spaces
A K Gaur
Department of Mathematics, Duquesne University, Pittsburgh, PA
15282, USA.
e-mail: [email protected]
Abstract: This note deals with a small but an important
observation of hermitian operators on Banach spaces. It is known
that if A is a complex Banach space, B(A) is the set of all
operators on A, and H(B(A)) is the set of all hermitian operators
then B(A) = H(B(A)) + iH(B(A)) implies that A is a Hilbert space.
We give a converse of this using norm one projections on A.
Mathematics Subject Classification (2000): 46H05,
46J10, 4715, 47B48.
Keywords: projections, Banach, banach space, ultra
product of an operator, hermitian operators, direct sum of
spaces, norm one, Banach algebra, continuous functions, function
algebras, normal operators, operators, Brown-McCoy, Brown-McCoy
radical, radical, non-0-symmetric near-ring, near-rings, near
rings, semisimple, identities, radical classes, invariant ideals,
ideals, Banach spaces
Quaestiones Mathematicaes 24 (4) 2001, 493–500
© 2001 NISC Pty Ltd, www.nisc.co.za
Minimum Moduli in Von Neumann Algebras
Perumal Gopalraj1 and Anton Ströh2
1 Department of Mathematics and Applied
Mathematics, University of Pretoria, Pretoria 0002, South Africa.
2 Department of Mathematics and Applied
Mathematics, University of Pretoria, Pretoria 0002, South Africa.
e-mail: [email protected]
Abstract: In this paper we answer a question raised in
[12] in the affirmative, namely that the essential minimum
modulus of an element in a von Neumann algebra, relative to any
norm closed two-sided ideal, is equal to the minimum modulus of
the element perturbed by an element from the ideal. As a
corollary of this result, we extend some basic perturbation
results on semi-Fredholm elements to a von Neumann algebra
setting. We can characterize the semi-Fredholm elements in terms
of the points of continuity of the essential minimum modulus
function.
Mathematics Subject Classification (2000): 46L
Keywords: algebra, selfadjoint operator algebras,
Fredholm theory, von Neumann algebra, minimum modulus,
semi-Fredholm
Quaestiones Mathematicaes 24 (4) 2001, 501–517
© 2001 NISC Pty Ltd, www.nisc.co.za
A Category of L-Fuzzy Convergence Spaces
Gunther Jäger
Department of Mathematics (Pure and Applied), Rhodes
University, 6140 Grahamstown, South Africa.
e-mail: [email protected]
Abstract: In this paper we take convergence of
stratified L-filters as a primitive notion and construct in this
way a Cartesian closed category, which contains the category of
stratified L-topological spaces as reflexive subcategory. The
class of spaces with non-idempotent stratified fuzzy interior
operator is characterized as subclass of the class of our
stratified L-fuzzy convergence spaces and a first
characterization, which fuzzy convergences stem from stratified
L-topologies is established.
Mathematics Subject Classification (2000): 54A40
Keywords: category, L-filter, fuzzy convergence,
L-Topological space, function space, fuzzy topology, L-topology,
convergence, closed categories, categories, interior operator,
convergence spaces
Quaestiones Mathematicaes 24 (4) 2001, 519–533
© 2001 NISC Pty Ltd, www.nisc.co.za
Observations About the Projective Tensor Product of Banach
Spaces, $\ell^{p}\hat{otimes}\X, 1 < p < \infty$
Qingying Bu1 and Joe Diestel2
1 Department of
Mathematics, Kent State University, Kent, Ohio 44242, USA.
e-mail: [email protected]
2 Banach Center, Department of
Mathematical Sciences, Kent State University, Kent, Ohio 44242,
USA.
e-mail: [email protected]
Abstract: A sequential description of the projective
tensor product $\ell^{p}\hat{otimes}\X$ is given. This
description allows us to show the inclusion of
$\ell^{p}\hat{otimes}\X$ into $\ell_{X}^{p}$ is a semi-embedding.
As a consequence, if X has the Radon-Nikodym property so does
$\ell^{p}\hat{otimes}\X$. A discussion of difficulties with
$\L^{p}(0, 1)\hat{otimes}\X$ follows.
Mathematics Subject Classification (2000): 46A, 46B,
46E, 47B.
Keywords: tensor, Banach, banach space, tensor product,
projective norm, greatest crossnorm, semi-embedding,
Radon-Nikodym property, absolutely p-summable sequence, strongly
p-summable sequence, topological linear spaces, normed linear
spaces, Banach lattices, linear function spaces, duals, function
algebras, linear operators, product, Banach spaces
Quaestiones Mathematicaes 24 (4) 2001, 535–548
© 2001 NISC Pty Ltd, www.nisc.co.za
Strongly Sequentially Continuous Functions
A Borichev1 R Deville2
and E Matheron3
1 Laboratoire de Mathématiques,
Université Bordeaux 1, 351, cours de la Libération, 33400
Talence, France.
e-mail: [email protected]
2 Laboratoire de Mathématiques,
Université Bordeaux 1, 351, cours de la Libération, 33400
Talence, France.
e-mail: [email protected]
3 Laboratoire de Mathématiques,
Université Bordeaux 1, 351, cours de la Libération, 33400
Talence, France.
e-mail: [email protected]
Abstract: Given a topological abelian group G, we study
the class of strongly sequentially continuous functions on G.
Strong sequential continuity is a property intermediate between
sequential continuity and uniform sequential continuity, which
appeared naturally in the study of smooth functions on Banach
spaces. In this paper, we shall mainly concentrate on the
"gap" between strong sequential continuity and uniform
sequential continuity. It turns out that if G has some
"completeness" property - for example, if it is
completely metrizable - then all strongly sequentially continuous
functions on G are uniformly sequentially continuous. On the
other hand, we exhibit a large and natural class of groups for
which the two notion differ. This class is defined by a property
reminiscent of the classical Dirichlet theorem; it includes all
dense subgroups of $\mathbb{R}$ generated by an increasing
sequence of Dirichlet sets, and groups of the form (X, $\omega$),
where X is a separable Banach space failing the Schur property.
Finally, we show that the family of bounded, real-valued strongly
sequentially continuous functions on G is a closed subalgebra of
$\l_{\infty}(G)$.
Mathematics Subject Classification (2000): 26A99,
45E99, 46B99
Keywords: continuous functions, topological groups,
completeness, Schur property, Dirichlet's Theorem, strongly
sequential continuity, study, Banach, Banach spaces, banach
space, dense, sets
Quaestiones Mathematicaes 24 (4) 2001, 549–554
© 2001 NISC Pty Ltd, www.nisc.co.za
Sobriety and Semi-Sobriety of L-Topological Spaces
Wesley Kotzé
Department of Mathematics, Rhodes University, 6140
Grahamstown, South Africa.
e-mail: [email protected]
Abstract: This is a sequel to a paper "Fuzzy
sobriety and fuzzy Hausdorff" which appeared in this journal
vol. 20, 1997. The relationships between sobriety and fuzzy
Hausdorff concepts are more subtle than was envisaged there and a
concept of semi-sobriety is appropriate.
Mathematics Subject Classification (2000): 54A40, 54D15
Keywords: L-Topological space, separation axioms,
sobriety, fuzzy topology, higher separation axioms,
semi-sobriety, Hausdorff properties
Quaestiones Mathematicaes 24 (4) 2001, 555–564
© 2001 NISC Pty Ltd, www.nisc.co.za
Graded Matlis Duality and Applications to Covers
Edgar E Enochs1 and J A
López-Ramos2
1 Department of Mathematics, University
of Kentucky, Lexington, KY 40506 USA.
e-mail: [email protected]
2 Departamento de. Álgebra y Análisis
Matemático, Universídad de Almería, 04120 Almería, Spain.
e-mail: [email protected]
Abstract: We study homological properties of graded
Matlis duality and apply them to get covers by Gorenstein
gr-projective modules. We show that these covers are minimal
graded maximal Cohen-Macaulay approximations in some cases.
Mathematics Subject Classification (2000): Primary:
16D90; Secondary: 18G25.
Keywords: graded Matlis duality, covers, Gorenstein
gr-projective module, module categories, relative homological
algebra, projective classes, study, duality, cover, modules,
module, approximation
Quaestiones Mathematicaes 24 (4) 2001, 565–573
© 2001 NISC Pty Ltd, www.nisc.co.za
On Relatively Prime Decompositions and Related Results
Temba Shonhiwa
Department of Mathematics, University of Zimbabwe, PO Box
MP 167, Mt. Pleasant, Harare, Zimbabwe.
e-mail: [email protected]
Abstract: In this paper we prove the inversion theorem,
f(n,k) = $\sum_{j=k}^{n}g(n,j)\P_{r}(j,k)$ if and only if g(n,k)
= $\sum_{j=k}^{n}f(n,j)\Q_{r}(j,k)$
for any ordered sequence pair $\langle\f(n,k),g(n,k)\rangle$,
where $\P_{r }(n,k)$ is the number of partitions of n into k
relatively prime parts and $\Q_{r}(n,k)$ is its inverse.
Mathematics Subject Classification (2000): 11A25, 11P81
Keywords: partition, composition, relatively prime,
arithmetic functions, related numbers, inversion formulas,
elementary theory of partitions, inverse
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