Abstract - international conference on topological algebras and their ...

Some generalisations of results about compact or
precompact elements in topological algebras
Mart Abel
University of Tartu
Estonia
In the present talk we generalise several results about compact or precompact
elements of topological algebras (in some cases also of topological rings or topological
groups) for the case where the existence of a norm or a seminorm on a
topological structure is not necessary. We also provide some results concerning
compact of precompact topological algebras.
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Topological algebras in which all one-sided
maximal ideals are closed
Mati Abel
University of Tartu
Estonia
In 2005 Mati Abel and Krzysztof Jarosz described the class of topological unital
algebras in which all maximal two-sided ideals are closed. But how to describe
these topological unital algebras in which all maximal one-sided ideals are closed
was not known until now. A description of such topological unital algebras is given
in this talk.
2

Graded q-Differential Algebra Approach to
Chern-Simons Form
Viktor Abramov
University of Tartu
Estonia
We develop noncommutative approach to a connection which is based on a
notion of graded q-differential algebra, where q is a primitive Nth root of unity. We
define the curvature of connection form and prove Bianchi identity. We construct
a graded q-differential algebra to calculate the curvature of connection. Making
use of Bianchi identity we introduce the Chern character form of connection form
and show that this form is closed. We study the case N = 3 which is the first
non-trivial generalization because in the case N = 2 we have a classical theory.
We calculate the curvature of connection form and show that it can be expressed
in terms of graded q-commutators, where q is a primitive cubic root of unity.
This allows us to prove an infinitesimal homotopy formula, and making use of this
formula we introduce the Chern-Simons form.
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On Banach algebras of continuous bounded
functions with values in a Banach algebra
Hugo Arizmendi, Ángel Carrillo, Alejandra García
National Autonomous University of Mexico
Mexico
Let X be a completely regular Hausdorff space and A be a complex commutative
unital Banach algebra with the norm . We denote by C (X, A) the unital
algebra of all continuous functions on X valued in A and by (Cb (X, A) , ∞ ) and
(Cp (X, A) , ∞ ) the subalgebras of C (X, A) of all bounded continuous functions
and of all functions f ∈ Cb (X, A) such that f (X) is compact in A respectively,
provided with the sup-norm ∞ on X, both are Banach algebras with this norm.
In this talk we will describe the maximal ideal spaces M ((Cb (X, A) , ∞ )) and
M ((Cb (X, A) , ∞ )) of each one of these algebras. We exhibit an example in
which M ((Cb (X, A) , ∞ )) is too large.
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Pseudocompactness and Algebraic Operations on
Spaces
Mitrofan M. Choban
Tiraspol State University
Republic of Moldova
We use the terminology from [3, 5]. Any space is considered to be completely
regular. Classes of spaces related to compact spaces are main objects of distinct
important topological investigations. Pseudocompactness is one of the fragile properties
of spaces related to the class of compact spaces (see [5]). The phenomena
that the property of pseudocompactness is not hereditary with respect to closed
subspaces, create dificult obstacles in the studying of special subspaces of pseudocompact
spaces. There exists a paracompact space Z with one non-isolated point
which is a closed Gδ-subset of a pseudocompact space and Z is not a Čech-complete
space. In this context, there are interesting the next assertions:
A1. If X is a non-empty paracompact Gδ-subspace of a pseudocompact topological
group G, then the group G is compact and X is a Čech-complete space.
A2. Any paracompact Čech-complete space is a closed Gδ-subspace of some
pseudocompact space.
A3. If a paracompact p-space is a Gδ-subspace of some pseudocompact space,
then it is Čech-complete.
For a subspace X of a pseudocompact space Y the conditions under which the
space Y \ X is not pseudocompact are determined. If X is dense in Y , then X
has points of weakly pseudocompactness. A point x ∈ X is a pseudocompctness
(respectively, a weakly pseudocompctness) point of X if there exists a sequence
{Un : n ∈ N = {1, 2, . . . }} of open subsets of X such that: every sequence {Vn : n ∈
N} of non-empty open sets in X, such that Vn ⊆ Un for each n ∈ N, has a point of
accumulation in X; x ∈ Un (respectively, x ∈ clX ∩{Ui : i ≤ n} for each n ∈ N (see
[1 -4]). A topological group with points of weakly pseudocompactness is a space
with points of pseudocompactness. If X is a non-empty paracompact Gδ-subspace
of a topological group G with points of pseudocompactness, then X and G are
paracompact p-spaces.
We also study the problem of continuity of operations in groups with topologies
(see [1 - 3]).
References
[1] A. V. Arhangel’skii and M.M. Choban,Completeness type properties of semitopological
groups, and the theorems of Montgomery and Ellis, Topology
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Proceed. 37 (2011), 33-60.
[2] A. V. Arhangel’skii, M.M. Choban and P. S. Kenderov, Topological games
and continuity of group operations, Topol. Appl. 157 (2010) 2542-2552.
[3] A. V. Arhangelskii and M. G. Tkachenko, Topological groups and related
structures, Atlantis Press. Amsterdam-Paris, 2008.
[4] M.M. Choban, Spaces and mappings with conditions related to paracompactness,
Proceedings ICTA-2011 (Islamabad, Pakistan, July 410, 2011), Cambridge
Scientific Publishers, 2012, 105-132.
[5] R. Engelking, General Topology, PWN. Warszawa, 1977.
6

On Pythagorean topological algebras
Marina Haralampidou
University of Athens
Greece
In this talk, we introduce the notion of a Pythagorean topological algebra. This
is a locally m-convex algebra (A, (pα)α∈Λ) that satisfies the Pythagorean property.
Namely,
if x, y ∈ A and xy = yx = 0, then pα(x + y) 2 = pα(x) 2 + pα(y) 2 , for all α ∈ Λ.
Our intent is to formulate conditions, under which, that algebra has a pseudo-
H-structure. Moreover, we shall see when topological algebras of this type turn
to be commutative locally m-convex H ∗ -algebras.
7

On Representations of Continuous bundles of
C ∗ -algebras over Stonean Compact
Alexander A. Katz
St. John’s Uiversity
NY, USA
A version of Gelfand-Naimark-Segal theorem is established for representations
of continuous bundles of C*-algebras over Stonean compact.
8

A generalization of a theorem of Kadison for
partially ordered algebras with an order unit
Jukka Kauppi
University of Oulu
Finland
The classical theorem of Stone and Kadison asserts that every partially ordered
real algebra containing an order unit which is a multiplicative identity can be
represented as a dense subalgebra of the algebra of continuous real-valued functions
on a compact Hausdorff space via a norm- and order- preserving mapping that
carries the order unit to the identity function. Motivated by the fact that many
finitely generated ideals of partially ordered algebras contain an order unit, we
generalize this result to the setting of partially ordered algebras with an order
unit but not necessarily with a multiplicative identity. It emerges that the most
natural framework for the representation theory of such algebras is provided by
certain weighted function algebras.
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Function algebras with values in ordered
C ∗ -Segal algebras
Jussi Mattas
University of Oulu
Finland
We study multipliers and order properties of the function algebra C0(X, A)
where X is a locally compact Hausdorff space and A is a C ∗ -Segal algebra, that is, a
Banach algebra which is continuously embedded onto a dense ideal of a C ∗ -algebra.
We generalize to this setting a theorem by Akemann, Pedersen and Tomiyama, who
characterized the multiplier algebra of C0(X, A) when A is a C ∗ -algebra. We also
consider the order unitization of C0(X, A), that is, a homeomorphic embedding
into an order unital C ∗ -Segal algebra.
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Multipliers in locally convex *-algebras
Lourdes Palacios
Universidad Autonoma Metropolitana Iztapalapa
Mexico
Multipliers play an important role in several areas of mathematics where an
algebra structure appears. Due to important applications of non-normed topological
*-algebras in other fields, in this talk we consider a complete locally m-convex
algebra with continuous involution, which is also a “perfect” projective limit, and
describe its multiplier algebra, under a weaker topology, making it a locally C*algebra.
The same is applied in the case of certain locally convex H*-algebras. We
provide some relevant examples.
Joint work with: Marina Haralampidou (University of Athens), Carlos Signoret
(Universidad Autnoma Metropolitana- Iztapalapa).
References
[1] W.M. Ching and J.S.W. Wong, Multipliers and H*-algebras. Pacific J. Math.
22(1967), 387-396.
[2] M. Haralampidou, On locally H*-algebras, Math. Japon. 38(1993), 451-460.
[3] M. Haralampidou, The Krull nature of locally C*-algebras. Function Spaces
(Edwardsville, IL, 2002), 195-200, Contemp. Math., 328, Amer. Math. Soc.,
Providence, RI, 2003.
[4] A. Inoue, Locally C*-algebras. Mem. Faculty Sci. Kyushu Univ. (Ser. A)
25(1971), 197–235.
[5] M. Joita, On bounded module maps between Hilbert modules over locally
C*-algebras. Acta Math. Univ. Comeneanae vol. LXXIV, 1(2005), 71-78.
[6] T. Husain, Multipliers of topological algebras, Dissertationes Math.
(Rozprawy Mat.) 285 (1989), 40 pp. 263-271.
[7] R. Larsen, The multiplier problem. Lectures Notes in Math. No. 105, Springer-
Verlag, Berlin, 1969.
[8] E.A. Michael, Locally multiplicatively-convex topological algebras, Mem.
Amer. Math. Soc. 11(1952). (Reprinted 1968).
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Smooth manifolds vs differential triads
Maria Papatriantafillou
University of Athens
Greece
We consider differentiable maps in the setting of <strong>Abstract</strong> Differential Geometry
and we study the conditions that assure the uniqueness of differentials in this
setting. In particular, if continuity is considered, we prove that smooth manifolds
form a full subcategory of the category of differential triads, a result with physical
implications.
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The Stone-Čech compactification of a
pseudocompact primitive topological inverse
semigroup
Kateryna Pavlyk
University of Tartu
Estonia
The pseudocompactness is the property of topological space which is not finitely
multiplicative. Therewith Comfort and Ross showed that product of any number
of pseudocompact groups is a pseudocompact topological group [1]. We show
that the Comfort-Ross Theorem can be extended for the class of pseudocompact
primitive topological inverse semigroups and apply this result to show that, as in
the case of topological groups, the Stone-Čech compactification of pseudocompact
primitive topological inverse semigroup is a primitive topological inverse one.
References
[1] W.W. Comfort, K.A. Ross, Pseudocompactness and uniform continuity in
topological groups, Pacific J. Math. 16, 1966, 483–496.
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Some results about the inductive limit of locally
pseudoconvex algebras
Reyna María Pérez Tiscareño
University of Tartu
Estonia
I will talk about inductive limits of locally pseudoconvex algebras, in particular
about LFpg-algebras (LFp-algebras). These are locally pseudoconvex inductive
limits (respectively, locally pseudoconvex inductive limits of increasing sequences)
of locally pseudoconvex F -algebras, which satisfy certain conditions. Properties
and examples of such algebras will be presented.
∗ This research is supported by the European Social Fund (Mobilitas grant No.
MJD247).
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Multipliers in some locally m-convex algebras
Carlos Signoret
Universidad Autonoma Metropolitana Iztapalapa
Mexico
In this talk we present a description of the multiplier algebra of a certain type
of locally m-convex algebras in terms of the multiplier algebras of the factors in
their Arens-Michael decomposition.
15

L-valued bornologies: generalities and examples
related to fuzzy metrics and fuzzy topologies
Alexander ˇ Sostak and Ingrīda Ul¸jane
University of Latvia
Latvia
In order to apply the concept of boundness, so crucial in the theory of metric
spaces, to the case of a general topological space Hu S.T. introduced the notion of
bornology and of a bornological space [5]:
Given a set X a bornology on it is a family B ⊆ 2 X of subsets of X such that (1B)
∀x ∈ X {x} ∈ B;
(2B) if U ⊆ V ⊆ X and V ∈ B then U ∈ B;
(3B) if U, V ⊂ X U, V ∈ B then U ∪ V ∈ B.
The pair (X, B) is called a bornological space and the sets belonging to B are
viewed as bounded in this space.
In the paper [1] the concept of an L-fuzzy bornology, where L is a complete
lattice, was introduced. Actually an L-fuzzy bornology on a set X is a certain
ideal in the family L X of L-fuzzy subsets of the set X. Basics of the theory of
L-fuzzy bornological spaces were worked out there, too. In the present work we
propose an alternative approach to the problem of fuzzification of the concept of
bornology. Namely, here we define an L-valued bornology on a set X as an L-fuzzy
subset B of the powerset 2 X of subsets of X, satisfying certain L-valued analogues
of the axioms of a bornology. Basic properties of the category BOR(L) of Lvalued
bornological spaces and bounded mappings will be discussed. Our special
interest here concerns the L-valued bornologies induced by fuzzy (pseudo-)metrics
[4] and the costruction of an L-valued bornology on an L-fuzzy topological space
(in the sense of C.L. Chang [2] - Goguen [3]). This construction is based on the
concept of the measure of compactness of a set in an L-fuzzy topological space.
We consider this construction as an L-valued counterpart of the bornology in a
topological space defined by the family of relatively compact subsets.
References
[1] M. Abel, A. ˇ Sostak, Towards the theory of L-bornological spaces, Iranian Journal
of Fuzzy Systems, 8 No. 1, (2011) 19–28
[2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182–
190.
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[3] J.A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl., 43 (1973),
734–742.
[4] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets
Syst., 64 (1994) 395–399.
[5] S.-T. Hu, Boundedness in a topological space, J. Math. Pures Appl., 78 (1949),
287–320.
17

Infinite-dimensional Grassmann algebras of
sections of vector bundles
Jaan Vajakas
University of Tartu
Estonia
The notion of infinite-dimensional Grassmann algebra has been introduced by
Berezin to describe generating functionals of quantum field theory in the Fermi
case. We give a general method for constructing infinite-dimensional Grassmann
algebras, satisfying the axioms of infinite-dimensional Grassmann algebra given by
Berezin, using topological tensor products and apply it to construct an infinitedimensional
Grassmann algebra whose elements of degree 1 are antilinear functionals
on the space of sections of a vector bundle. The ghost fields appearing in
the Faddeev-Popov Lagrangian and in the BRST transformations can be identified
with the generators of such infinite-dimensional Grassmann algebras.
18

Functional extenders
Vesko Valov
Nipissing University
Canada
We describe the supports of a class of real-valued maps on C ∗ (X) introduced by
Radul. Using this description, a characterization of compact-valued retracts of a
given space in terms of functional extenders is obtained. Similar characterizations
are obtained for upper (resp., lower) semi-continuous retractions. As an application,
we provide a characterization of absolute extensors for zero-dimensional
spaces, as well as absolute extensors for one-dimensional spaces, involving nonlinear
functional extenders.
19

Extensions of topological algebras
Wies̷law ˙ Zelazko
Polish Academy of Sciences
Poland
A topological algebra is a topological vector space equipped with a jointly
continuous associative multiplication. Thus a completion of a topological algebra
is again such an algebra. Unless otherwise stated all algebras considered in my
talk are commutative complex complete unital topological algebras. Let K be a
class of topological algebras and A ∈ K. An algebra B ∈ K is said a K-extension
of A if A is topologically isomorphic to a subalgebra of B containing its unity.
More precisely, an extension is an algebra B together with an imbedding of A into
B (different imbeddings give different extensions). An element x ∈ A ∈ K is said
K-singular if it is non-invertible in every K-extension of A. An ideal I ⊂ A ∈ K
is said K-non removable if it is contained in a proper ideal of B for every Kextension
B of A. An element x ∈ A is in the K-radical of A if it belongs to
the radical rad(B) of every K-extension B of A. The set of all such elements is
denoted by radK(A) and it is an ideal contained in rad(A). The following classes
will be considered:the class B of Banach algebras,the class LB of locally bounded
algebras,the class MLC of (complete) locally m-convex algebras, the class MPC
of locally m-pseudoconvex algebras, the class LC of locally convex algebras, the
class LPC of locally pseudoconvex algebras, the class F of completely metrizable
topological algebras, the class T of topological algebras, and the the class ST
of semi-topological algebras. The T -singular elements will be called permanently
singular, and similarly we shall be talking about permanently non-removable ideals
and about the permanent radical.
The talk will be a survey on characterizations of above concepts and a presentation
of related still open problems. The presented results belong to M. Abel,
R.F. Arens, B. Bollobas, A. Fernandez, M. Florencio, V.Müller, G.E. Shoilov, Z.
S̷lodkowski and the author.
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