Some generalisations of results about compact or precompact elements in topological algebras

Transcription

1 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. 1

2 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

3 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. 3

4 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 (C b (X, A), ) and (C p (X, A), ) the subalgebras of C (X, A) of all bounded continuous functions and of all functions f C b (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 ((C b (X, A), )) and M ((C b (X, A), )) of each one of these algebras. We exhibit an example in which M ((C b (X, A), )) is too large. 4

5 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 G δ -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 G δ -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 G δ-subspace of some pseudocompact space. A3. If a paracompact p-space is a G δ -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 {U n : n N = {1, 2,... }} of open subsets of X such that: every sequence {V n : n N} of non-empty open sets in X, such that V n U n for each n N, has a point of accumulation in X; x U n (respectively, x cl X {U i : 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 G δ -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 5

6 Proceed. 37 (2011), [2] A. V. Arhangel skii, M.M. Choban and P. S. Kenderov, Topological games and continuity of group operations, Topol. Appl. 157 (2010) [3] A. V. Arhangelskii and M. G. Tkachenko, Topological groups and related structures, Atlantis Press. Amsterdam-Paris, [4] M.M. Choban, Spaces and mappings with conditions related to paracompactness, Proceedings ICTA-2011 (Islamabad, Pakistan, July 410, 2011), Cambridge Scientific Publishers, 2012, [5] R. Engelking, General Topology, PWN. Warszawa,

7 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, (p α ) α Λ ) that satisfies the Pythagorean property. Namely, if x, y A and xy = yx = 0, then p α (x + y) 2 = p α (x) 2 + p α (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

8 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

9 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. 9

10 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 C 0 (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 C 0 (X, A) when A is a C -algebra. We also consider the order unitization of C 0 (X, A), that is, a homeomorphic embedding into an order unital C -Segal algebra. 10

11 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), [2] M. Haralampidou, On locally H*-algebras, Math. Japon. 38(1993), [3] M. Haralampidou, The Krull nature of locally C*-algebras. Function Spaces (Edwardsville, IL, 2002), , Contemp. Math., 328, Amer. Math. Soc., Providence, RI, [4] A. Inoue, Locally C*-algebras. Mem. Faculty Sci. Kyushu Univ. (Ser. A) 25(1971), [5] M. Joita, On bounded module maps between Hilbert modules over locally C*-algebras. Acta Math. Univ. Comeneanae vol. LXXIV, 1(2005), [6] T. Husain, Multipliers of topological algebras, Dissertationes Math. (Rozprawy Mat.) 285 (1989), 40 pp [7] R. Larsen, The multiplier problem. Lectures Notes in Math. No. 105, Springer- Verlag, Berlin, [8] E.A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11(1952). (Reprinted 1968). 11

12 Smooth manifolds vs differential triads Maria Papatriantafillou University of Athens Greece We consider differentiable maps in the setting of Abstract 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. 12

13 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,

14 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). 14

15 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

16 L-valued bornologies: generalities and examples related to fuzzy metrics and fuzzy topologies Alexander Šostak and Ingrīda Uļjane 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 L- valued 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. Šostak, Towards the theory of L-bornological spaces, Iranian Journal of Fuzzy Systems, 8 No. 1, (2011) [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968),

17 [3] J.A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl., 43 (1973), [4] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994) [5] S.-T. Hu, Boundedness in a topological space, J. Math. Pures Appl., 78 (1949),

18 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

19 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

20 Extensions of topological algebras Wies law Żelazko 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 K- extension 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 rad K (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. S lodkowski and the author. 20